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dk儿童数学思维手册
《DK儿童数学思维手册》是英国DK公司的青少年数学知识科普图书,它从我们的生活说起,展示了数学在日常生活中的应用,让小读者感到数学并不遥远,更不可怕,同时还能更好的给孩子们学习数学,有需要的就快来吧
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适读人群
7-10岁
《DK儿童数学思维手册》提倡多元化培养孩子的思维能力——
观察力:通过丰富生动的图片,训练孩子的观察能力,进而提升其认知水平,对各种新知识产生兴趣。
数字游戏:激发孩子对数量、图像、逻辑推理的兴趣,提升孩子探索能力、分析、解决问题的能力,为思维进阶打下坚实的基础。
动手操作:可以多方位开发孩子的左右脑潜能,提升孩子的学习能力、解决问题能力和创造力,帮助幼儿学会思考,主动探索。
几何与空间:通过对图形的认识,对积木的拼堆,培养孩子的空间想象能力。为今后立体几何奠定扎实的基础。
逻辑推理:通过一些数量代换、文字推理。培养孩子理解抽象问题的能力。
感知探索:通过对一些实物和一些科学书籍的阅读,来培养孩子对世界的多角度的认识,并且积极去探索其根源。
书中包含知识讲解图、场景大图、具体切面图、手绘图等多类图片,配合文字解说,吸引儿童进行快乐的自主学习。
数学不仅仅是背公式、做习题,只要你认真耐心地阅读本书,就会看到那些有趣的数学实验、数学知识背后的小故事。
《DK儿童数学思维手册》用有趣、发散性思维来培养想象力和求知欲
目录
数学大脑 6
创造数字 22
神奇的数字 46
形状和空间 62
数学世界 86
内容简介
用一把小小的尺子测算出地球的周长,你行吗?
你知道密码是怎样编写出来的吗?
你见过只有一个面的纸条吗?
你想设计一个自己的迷宫吗?
如果你对上面这些内容感兴趣,就快点翻开这本《DK儿童数学思维手册》吧!
《DK儿童数学思维手册》是英国DK公司出版的一本训练青少年数学思维、逻辑思维的精装全彩图书,书中仔细研究了历史上一些最强的数学大脑
——从古希腊数学家阿基米德到第二次世界大战期间破译密码的专家阿兰•图灵;还包含了一系列的数字游戏、逻辑问答、图形谜题和其他各种能激发大脑细胞的好玩活动,让我们一起来看看,你的数学思维能力到底有多强!
激发内在潜能,感受数学的神奇。准备好迎接一场头脑风暴吧!
作者简介
迈克·戈德史密斯是一位著名的英国物理学家、科普作家,英国国家物理实验室声学部前负责人,他的作品非常受读者欢迎,已被译成多国文字在全球发行。
dk儿童数学思维手册截图
MA TH
TRAIN your BRAIN to be a
GENIUS LONDON, NEW YORK,MELBOURNE, MUNICH, AND DELHI
Senior editor Francesca Baines
Project editors Clare Hibbert, James Mitchem
Designer Hoa Luc
Senior art editors Jim Green, Stefan Podhorodecki
Additional designers Dave Ball, Jeongeun Yule Park
Managing editor Linda Esposito
Managing art editor Diane Peyton Jones
Category publisher Laura Buller
Production editor Victoria Khroundina
Senior production controller Louise Minihane
Jacket editor Manisha Majithia
Jacket designer Laura Brim
Picture researcher Nic Dean
DK picture librarian Romaine Werblow
Publishing director Jonathan Metcalf
Associate publishing director Liz Wheeler
Art director Phil Ormerod
First American edition, 2012
Published in the United States by
DK Publishing
375 Hudson Street
New York, New York 10014
Copyright ? 2012 Dorling Kindersley Limited
12 13 14 15 16 10 9 8 7 6 5 4 3 2 1
001—182438 —0912
All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the prior written permission of the copyright owner.
Published in Great Britain by Dorling Kindersley Limited.
A catalog record for this book is available from the Library of Congress.
ISBN: 978-0-7566-9796-9
DK books are available at special discounts when purchased in bulk for sales promotions,premiums, fund-raising, or educational use. For details, contact: DK Publishing Special Markets,375 Hudson Street, New York, New York 10014 or SpecialSales@dk.com.
Printed and bound in China by Hung Hing
Discover more at
www.dk.com
This book is full of puzzles and
activities to boost your brain
power. The activities are a lot of
fun, but you should always check
with an adult before you do any
of them so that they know what
you’re doing and are sure
that you’re safe. Written by
Consultant
Illustrated by
Dr. Mike Goldsmith
Branka Surla
Seb Burnett
MA TH
TRAIN your BRAIN to be a
GENIUS 4
CONTENTS 6 A world of math
MATH BRAIN
10 Meet your brain
12 Math skills
14 Learning math
16 Brain vs. machine
18 Problems with numbers
20 Women and math
22 Seeing the solution
INVENTING NUMBERS
26 Learning to count
28 Number systems
30 Big zero
32 Pythagoras
34 Thinking outside the box
36 Number patterns
38 Calculation tips
40 Archimedes
42 Math that measures
44 How big? How far?
46 The size of the problem
MAGIC NUMBERS
50 Seeing sequences
52 Pascal’s triangle
54 Magic squares
56 Missing numbers
58 Karl Gauss
60 In?nity
62 Numbers with meaning
64 Number tricks
66 Puzzling primes5
SHAPES AND SPACE
70 Triangles
72 Shaping up
74 Shape shifting
76 Round and round
78 The third dimension
80 3-D shape puzzles
82 3-D fun
84 Leonhard Euler
86 Amazing mazes
88 Optical illusions
90 Impossible shapes
A WORLD OF MATH
94 Interesting times
96 Mapping
98 Isaac Newton
100 Probability
102 Displaying data
104 Logic puzzles and paradoxes
106 Breaking codes
108 Codes and ciphers
110 Alan Turing
112 Algebra
114 Brainteasers
116 Secrets of the Universe
118 The big quiz
120 Glossary
122 Answers
126 Index
128 Credits
The book is full of
problems and puzzles for
you to solve. To check
the answers, turn to
pages 122–125. 6
It is impossible to imagine our world without
math. We use it, often without realizing, for a
whole range of activities —when we tell time,go shopping, catch a ball, or play a game. This
book is all about how to get your math brain
buzzing, with lots of things to do, many of the
big ideas explained, and stories about how the
great math brains have changed our world.
MATH A WORLD OF
Calculation
You need math to make
just about everything, from
a cake to a car. Quantities,costs, and timings must
all be worked out using
calculation and estimation.
I wonder what would
happen if the ride spun
even faster?
People are hungry
tonight. At this rate, I’ll
run out of hot dogs in
half an hour.
Science
Math is essential for
scientists—it helps them
test theories and make
them exact. Some theories
are then put to practical use,to build bridges, machines,and even carnival rides!
I′ll be in this line for
10 minutes, so I should still
be in time to catch the next
bus home.
Panel puzzle
These shapes form a square panel, used
in one of the carnival stalls. However, an
extra shape has somehow been mixed
up with them. Can you ?gure out which
piece does not belong?
There’s a height restriction
on this ride, sonny. Try
coming back next year.
D
E
F
C
B
A7
One in four people are
hitting a coconut. Grr! I’m
making a loss.
Shapes
Understanding shapes and
space helps us make
sense of the world around
us. You need to know about
this area of math to create
and design anything—
including tricky games.
Patterns
Many areas of math
involve looking for patterns,such as how numbers
repeat or how shapes behave.
Often these patterns can be
used to help us and inspire
new ways of thinking.
Profit margin
It costs 144 a day to run the
bumper cars, accounting for
wages, electricity, transportation,and so on. There are 12 bumper
cars, and, on average, 60 percent
of them are occupied each session.
The ride is open for eight hours a
day, with four sessions an hour,and each driver pays 2 per
session. How much pro?t is
the owner making?
A game of chance
Everyone loves to try to knock down
a coconut—but what are your chances
of success? The stall owner needs to
know so he can make sure he’s got
enough coconuts, and to work out how
much to charge. He’s discovered that, on
average, he has 90 customers a day, each
throwing three balls, and the total number
of coconuts won is 30. So what is the
likelihood of you winning a coconut?
Gulp! The slide looks
even steeper from the top.
I wonder what speed I’ll
be going when I get to
the bottom?
Look at me! I’m
loating in the air and
I’ve got two tongues!
I think I’ve got the
angle just right... one
more go and I’ll win
a prize.Math
brain10
Cerebrum Where thinking
occurs and memories
are stored
Hypothalamus Controls
sleep, hunger, and body
temperature
Thalamus Receives
sensory nerve signals
and sends them on
to the cerebrum
Meninges
Protective layers
that cushion the
brain against shock
Skull Forms a
tough casing
around the brain
Cerebellum Helps
control balance
and movement
Medulla Controls
breathing, heartbeat,blood pressure,and vomiting
Your brain is the most complex organ
in your body—a spongy, pink mass made
up of billions of microscopic nerve cells. Its
largest part is the cauli?ower-like cerebrum,made up of two hemispheres, or halves,linked by a network of nerves. The cerebrum
is the part of the brain where math is
understood and calculations are made.
BRAIN
MEET YOUR
Looking inside
This cross-section of the skull
reveals the thinking part of the
brain, or cerebrum. Beneath its
outer layers is the “white matter,”
which transfers signals between
different parts of the brain.
LEFT-BRAIN SKILLS
The left side of your cerebrum is
responsible for the logical, rational
aspects of your thinking, as well as for
grammar and vocabulary. It’s here that
you work out the answers to calculations.
A BRAIN OF TWO HALVES
The cerebrum has two hemispheres. Each deals
mainly with the opposite side of the body—data
from the right eye, for example, is handled in
the brain’s left side. For some functions,including math, both halves work
together. For others, one half is
more active than the other.
Writing
skills
Like spoken language, writing
involves both hemispheres. The right
organizes ideas, while the left ?nds
the words to express them.
Scientific
thinking
Logical thinking is the job
of the brain’s left side, but
most science also involves
the creative right side.
Mathematical
skills
The left brain oversees
numbers and calculations,while the right processes
shapes and patterns.
Rational thought
Thinking and reacting in a
rational way appears to be
mainly a left-brain activity.
It allows you to analyze a
problem and ?nd an answer.
Language
The left side handles the meanings
of words, but it is the right half that
puts them together into sentences
and stories.
Left visual cortex Processes
signals from the right eye
Corpus callosum Links the
two sides of the brain
Pituitary gland Controls
the release of hormones11
Frontal lobe Vital to
thought, personality,speech, and emotion
Temporal lobe Where
sounds are recognized,and where long-term
memories are stored
RIGHT-BRAIN SKILLS
The right side of your cerebrum is where
creativity and intuition take place, and is
also used to understand shapes and motion.
You carry out rough calculations here, too.
The outer surface
Thinking is carried out on the surface
of the cerebrum, and the folds and
wrinkles are there to make this surface
as large as possible. In preserved
brains, the outer layer is gray, so it
is known as “gray matter.”
Right eye Collects data on
light-sensitive cells that is
processed in the opposite
side of the brain—the left
visual cortex in the
occipital lobe
Right optic nerve
Carries information from
the right eye to the left
visual cortex
Spatial skills
Understanding the shapes of
objects and their positions in
space is a mainly right-brain
activity. It provides you your
ability to visualize.
Imagination
The right side of the brain
directs your imagination.
Putting your thoughts into
words, however, is the job
of the left side of the brain.
Music
The brain’s right side is
where you appreciate music.
Together with the left side,it works to make sense of
the patterns that make the
music sound good.
Insight
Moments of insight occur
in the right side of the brain.
Insight is another word for
those “eureka!” moments
when you see the connections
between very different ideas.
Art
The right side of the brain
looks after spatial skills.
It is more active during
activities such as drawing,painting, or looking at art.
Parietal lobe Gathers
together information
from senses such as
touch and taste
Occipital lobe Processes
information from the
eyes to create images
Spinal cord Joins the
brain to the system
of nerves that runs
throughout the body
Neurons and numbers
Neurons are brain cells that link up to
pass electric signals to each other.
Every thought, idea, or feeling that
you have is the result of neurons
triggering a reaction in your brain.
Scientists have found that when you
think of a particular number, certain
neurons ?re strongly.
Doing the math
This brain scan was carried out on a
person who was working out a series
of subtraction problems. The yellow
and orange areas show the parts of
the brain that were producing the
most electrical nerve signals. What’s
interesting is that areas all over
the brain are active—not just one.
Cerebellum Tucked
beneath the cerebrum’s
two halves, this
structure coordinates
the body’s muscles12
Many parts of your brain are involved in math, with big
differences between the way it works with numbers (arithmetic),and the way it grasps shapes and patterns (geometry). People
who struggle in one area can often be strong in another. And
sometimes there are several ways to tackle the same problem,using different math skills.
SKILLS
MATH
BRAIN GAMES
A quick glance
Our brains have evolved to grasp key
facts quickly—from just a glance at
something—and also to think things
over while examining them.
How do you count?
When you count in your head, do
you imagine the sounds of the
numbers, or the way they look?
Try these two experiments and
see which you ?nd easiest.
Step 1
Try counting backward in 3s from
100 in a noisy place with your eyes
shut. First, try “hearing” the
numbers, then visualizing them.
Step 1
Look at the sequences below—
just glance at them brie?y without
counting—and write down the
number of marks in each group.
Step 2
Now count the marks in each group
and then check your answers.
Which ones did you get right?
Step 2
Next, try both methods again
while watching TV with the sound
off. Which of the four exercises
do you ?nd easier?
About 10 percent of people think of
numbers as having colors. With
some friends, try scribbling the
irst number between 0 and 9 that
pops into your head when you
think of red, then black,then blue. Do any of you
get the same
answers?
The part of the brain that can “see” numbers
at a glance only works up to three or four, so
you probably got groups less than ?ve right.
You only roughly estimate higher numbers,so are more likely to get these wrong.
97...94...
88...85...
There are four main styles
of thinking, any of which can
be used for learning math: seeing
the words written, thinking in
pictures, listening to the sounds
of words, and hands-on activities.13
Spot the shape
In each of these sequences,can you ?nd the shape on the
far left hidden in one of the
·ve shapes to the right?
You will need:
· Pack of at least 40 small
pieces of candy
· Three bowls
· Stopwatch
· A friend
Eye test
This activity tests your ability
to judge quantities by eye. You
should not count the objects—
the idea is to judge equal
quantities by sight alone.
Step 1
Set out the three bowls in front
of you and ask your friend to
time you for ?ve seconds. When
he says “go,” try to divide the
candy evenly between them.
Step 2
Now count up the number of
candy pieces you have in each
bowl. How equal were the
quantities in all three?
Number cruncher
Your short-term memory can store a certain
amount of information for a limited time.
This exercise reveals your brain’s ability to
remember numbers. Starting at the top,read out loud a line of numbers one at a
time. Then cover up the line and try to
repeat it. Work your way down the list
until you can’t remember all the numbers.
438
7209
18546
907513
2146307
50918243
480759162
1728406395
Most people can hold about
seven numbers at a time in their
short-term memory. However, we
usually memorize things by saying
them in our heads. Some digits take
longer to say than others and this
affects the number we can remember.
So in Chinese, where the sounds of the
words for numbers are very short, it
is easier to memorize more numbers.
We have a natural sense of
pattern and shape. The Ancient
Greek philosopher Plato discovered
this a long time ago, when he
showed his slaves some shape
puzzles. The slaves got the answers
right, even though they’d had
no schooling.
You’ll probably be surprised how
accurately you have split up the
candy. Your brain has a strong sense
of quantity, even though it is not
thinking about it in terms
of numbers.
1
2
3
4
A B C D E
A B C D E
A B C D E
A B C D E14
For many, the thought of learning
math is daunting. But have you
ever wondered where math came
from? Did people make it up as they
went along? The answer is yes and
no. Humans—and some animals—
are born with the basic rules of
math, but most of it was invented.
Brain size and evolution
Compared with the size of the body, the human
brain is much bigger than those of other animals.
We also have larger brains than our apelike
ancestors. A bigger brain indicates a greater
capacity for learning and problem solving. Frog Bird Human
Baby at six months
In one study, a baby was shown
two toys, then a screen was put
up and one toy was taken away.
The activity of the baby’s brain
revealed that it knew something
was wrong, and understood the
difference between one and two.
ACTIVITY
Can your pet count?
All dogs can “count” up to about three. To test your dog,or the dog of a friend, let the dog see you throw one, two,or three treats somewhere out of sight. Once the dog
has found the number of treats you threw, it will usually
stop looking. But throw ?ve or six treats and the dog will
“lose count” and not know when to stop. It will keep on
looking even after ?nding all the treats. Use dry treats
with no smell and make sure they fall out of sight.
Baby at 48 hours
Newborn babies have some sense
of numbers. They can recognize
that seeing 12 ducks is different
from 4 ducks.
A sense of numbers
Over the last few years, scientists have tested
babies and young children to investigate their
math skills. Their ?ndings show that we humans
are all born with some knowledge of numbers.
Animal antics
Many animals have a sense of
numbers. A crow called Jakob
could identify one among many
identical boxes if it had ?ve dots
on it. And ants seem to know
exactly how many steps there
are between them and their nest.
MATH
LEARNING15
How memory works
Memory is essential to math. It allows us to keep
track of numbers while we work on them, and to
learn tables and equations. We have different
kinds of memory. As we do a math problem, for
example, we remember the last few numbers
only brie?y (short-term memory), but we will
remember how to count from 1 to 10 and so on
for the rest of our lives (long-term memory).
From five to nine
When a ?ve-year-old is asked to
put numbered blocks in order,he or she will tend to space
the lower numbers farther
apart than the higher ones.
By the age of about nine,children recognize that the
difference between numbers
is the same—one—and space
the blocks equally.
Clever Hans
Just over a century ago, there was a mathematical horse
named Hans. He seemed to add, subtract, multiply, and
divide, then tap out his answer with his hoof. However,Hans wasn’t good at math. Unbeknownst to his owner, the
horse was actually excellent at “reading” body language.
He would watch his owner’s face change when he had
made the right number of taps, and then stop.
Child at age four
The average four-year-old
can count to 10, though the
numbers may not always
be in the right order. He
or she can also estimate
larger quantities, such
as hundreds. Most
importantly, at four
a child becomes
interested in making
marks on paper,showing numbers
in a visual way.
Sensory memory
We keep a memory
of almost everything
we sense, but only for
half a second or so.
Sensory memory can
store about a dozen
things at once.
Short-term memory
We can retain a handful
of things (such as a
few digits or words)
in our memory for about
a minute. After that,unless we learn them,they are forgotten.
Long-term memory
With effort, we can
memorize and learn an
impressive number of
facts and skills. These
long-term memories
can stay with us for
our whole lives.
It can help you memorize
your tables if you speak or sing
them. Or try writing them down,looking out for any patterns. And,of course, practice them again
and again.
I’m going
to draw hundreds and
hundreds of dots!16
In a battle of the superpowers—brain versus
machine—the human brain would be the winner!
Although able to perform calculations at lightning
speeds, the supercomputer, as yet, is unable to
think creatively or match the mind of a genius.
So, for now, we humans remain one step ahead.
BRAIN Prodigies
A prodigy is someone who has an incredible
skill from an early age—for example, great
ability in math, music, or art. India’s
Srinivasa Ramanujan (1887–1920) had hardly
any schooling, yet became an exceptional
mathematician. Prodigies have active memories
that can hold masses of data at once.
Savants
Someone who is incredibly skilled in a
specialized ?eld is known as a savant.
Born in 1979, Daniel Tammet is a British savant
who can perform mind-boggling feats of
calculation and memory, such as memorizing
22,514 decimal places of pi (3.141...), see pages
76–77. Tammet has synesthesia, which means
he sees numbers with colors and shapes.
Your brain:
· Has about 100 billion neurons
· Each neuron, or brain cell, can
send about 100 signals per second
· Signals travel at speeds of about
33 ft (10 m) per second
· Continues working and transmitting
signals even while you sleep
Hard work
More often than not, dedication and
hard work are the key to exceptional
success. In 1637, a mathematician
named Pierre de Fermat proposed
a theorem but did not prove it. For
more than three centuries, many
great mathematicians tried and
failed to solve the problem. Britain’s
Andrew Wiles became fascinated
by Fermat’s Last Theorem when he
was 10. He ?nally solved it more
than 30 years later in 1995.
What about your brain?
If someone gives you some numbers to add
up in your head, you keep them all “in mind”
while you do the math. They are held in your
short-term memory (see page 15). If you can
hold more than eight numbers in your head,you've got a great math brain.
VS.17
MACHINE
Your computer:
· Has about 10 billion transistors
· Each transistor can send about
one billion signals per second
· Signals travel at speeds of about
120 million miles (200 million km)
per second
· Stops working when it is
turned off
Artificial
intelligence
An arti?cially intelligent computer
is one that seems to think like a
person. Even the most powerful
computer has nothing like the
all-round intelligence of a human
being, but some can carry out
certain tasks in a humanlike
way. The computer system
Watson, for example, learns
from its mistakes, makes choices,and narrows down options. In
2011, it beat human contestants
to win the quiz show Jeopardy.
Missing ingredient
Computers are far better than humans
at calculations, but they lack many of
our mental skills and cannot come up with
original ideas. They also ?nd it almost
impossible to disentangle the visual world—
even the most advanced computer would
be at a loss to identify the contents
of a messy bedroom!
Computers
When they were ?rst invented, computers were
called electronic brains. It is true that, like the
human brain, a computer’s job is to process
data and send out control signals. But, while
computers can do some of the same things
as brains, there are more differences than
similarities between the two. Machines are
not ready to take over the world just yet.18
NUMBERS
PROBLEMS WITH
Numerophobia
A phobia is a fear of something that there is no reason to
be scared of, such as numbers. The most feared numbers
are 4, especially in Japan and China, and 13. Fear of the
number 13 even has its own name—triskaidekaphobia.
Although no one is scared of all numbers, a lot of people
are scared of using them!
Dyscalculia
Which of these two numbers is higher? 76 46
If you can’t tell within a second, you might have dyscalculia,where the area of your brain that compares numbers does
not work properly. People with dyscalculia can also have
dif?culty telling time. But remember, dyscalculia is very
rare, so it is not a good excuse for missing the bus.
A life without math
Although babies are born with a sense of
numbers, more complicated ideas need to
be taught. Most societies use and teach
these mathematical ideas—but not all of them.
Until recently, the Hadza people of Tanzania,for example, did not use counting, so their
language had no numbers beyond 3 or 4.
Too late to learn?
Math is much easier to learn when
young than as an adult. The great
19th-century British scientist
Michael Faraday was never taught
math as a child. As a result, he
was unable to complete or prove
his more advanced work. He just
didn’t have a thorough enough
grasp of mathematics.19
A lot of people think math is tricky, and many try
to avoid the subject. It is true that some people have
learning dif?culties with math, but they are very
rare. With a little time and practice, you can soon get
to grips with the basic rules of math, and once you’ve
mastered those, then the skills are yours for life!
1 x 7 = 7 2 x 7 = 14
3 x 7 = 21 4 x 7 = 28
5 x 7 = 35 6 x 7 = 42
7 x 7 = 49 8 x 7 = 56
Visualizing math
Sometimes math questions sound complicated or use
unfamiliar words or symbols. Drawing or visualizing
(picturing in your head) can help with understanding and
solving math problems. Questions about dividing shapes
equally, for example, are simple enough to draw, and a
rough sketch is all you need to get an idea of the answer.
Practice makes perfect
For those of us who struggle with calculations, the contestants
who take part in TV math contests can seem like geniuses.
In fact, anyone can be a math whizz if they follow the three
secrets to success: practice, learning some basic calculations
by heart (such as multiplication tables), and using tips
and shortcuts.
Misleading numbers
Numbers can in?uence how and what you think.
You need to be sure what numbers mean so they
cannot be used to mislead you. Look at these two
stories. You should be suspicious of the numbers
in both of them—can you ?gure out why?
A useful survey?
Following a survey carried out by the
Association for More Skyscrapers (AMS),it is suggested that most of the 30 parks
in the city should close. The survey found
that, of the three parks surveyed, two had
fewer than 25 visitors all day. Can you
identify four points that should make you
think again about AMS’s survey?
The bigger picture
In World War I, soldiers wore cloth hats, which
contributed to a high number of head injuries.
Better protection was required, so cloth hats
were replaced by tin helmets. However, this
led to a dramatic rise in head injuries. Why
do you think this happened?
HEAD INJURIES
ON THE RISE!
PARKS TO CLOSE!
The 13th-century thinker
Roger Bacon said, “He who
is ignorant of [math] cannot
know the other sciences, nor
the affairs of this world.”
ACTIVITY20
Historically, women have always had
a tough time breaking into the ?elds of
math and science. This was mainly
because, until a century or so ago, they
received little or no education in these
subjects. However, the most determined
women did their homework and went on
to make signi?cant discoveries in some
highly sophisticated areas of math.
WOMEN AND MATH
Sofia Kovalevskaya
Born in Russia in 1850, Kovalevskaya’s fascination with
math began when her father used old math notes as
temporary wallpaper for her room! At the time, women
could not attend college but Kovalevskaya managed
to ?nd math tutors, learned rapidly, and soon made
her own discoveries. She developed the math of
spinning objects, and ?gured out how Saturn’s rings
move. By the time she died, in 1891, she was
a university professor.
Amalie Noether
German mathematician Amalie “Emmy” Noether
received her doctorate in 1907, but at ?rst no university
would offer her—or any woman—a job in math.
Eventually her supporters (including Einstein) found
her work at the University of Gottingen, although at ?rst
her only pay was from students. In 1933, she was forced
to leave Germany and went to the United States, where
she was made a professor. Noether discovered how to
use scienti?c equations to work out new facts, which
could then be related to entirely different ?elds of study.
Noether showed how
the many symmetries
that apply to all kinds
of objects, including
atoms, can reveal basic
laws of physics.
Kovalevskaya took
discoveries in physics
and turned them into
math, so that tops
and other spinning
objects could be
understood exactly.Hypatia
Daughter of a mathematician and philospher,Hypatia was born around 355 CE in Alexandria,which was then part of the Roman Empire. Hypatia
became the head of an important “school,” where
great thinkers tried to ?gure out the nature of the
world. It is believed she was murdered in 415 CE by
a Christian mob who found her ideas threatening.
Augusta Ada King
Born in 1815, King was the only child of the poet
Lord Byron, but it was her mother who encouraged
her study of math. She later met Charles Babbage
and worked with him on his computer machines.
Although Babbage never completed a working
computer, King had written what we would now
call its program—the ?rst in the world. There is
a computer language called Ada, named after her.
Grace Hopper
A rear admiral in the U.S. Navy, Hopper
developed the world’s ?rst compiler—
a program that converts ordinary language
into computer code. Hopper also developed the
·rst language that could be used by more than
one computer. She died in 1992, and the
destroyer USS Hopper was named after her.
Florence Nightingale
This English nurse made many improvements
in hospital care during the 19th century.
She used statistics to convince of?cials that
infections were more dangerous to soldiers
than wounds. She even invented her own
mathematical charts, similar to pie charts,to give the numbers greater impact.
Although Babbage’s
computer was not
built during his
lifetime, it was
eventually made
according to his
plans, nearly two
centuries later. If
he had built it, it
would have been
steam-powered!
Hypatia studied the
way a cone can be cut
to produce different
types of curves.
Nightingale’s chart
compared deaths from
different causes in the
Crimean War between 1854
and 1855. Each segment
stands for one month.
Blue represents
deaths from
preventable
diseases
Pink represents
deaths from
wounds
Hopper popularized
the term computer
“bug” to mean a coding
error, after a moth
became trapped in
part of a computer.
Black represents
deaths from all
other causes22
SOLUTION SEEING THE
BRAIN GAMES
What do you see?
The ?rst step to sharpening the
visual areas of your brain is to practice
recognizing visual information. Each
of these pictures is made up of the
outlines of three different objects.
Can you ?gure out what they are?
Thinking in 2-D
Lay out 16 matches to make ?ve squares
as shown here. By moving only two
matches, can you turn the ?ve squares
into four? No matches can be removed.
Visual sequencing
To do this puzzle, you need to visualize objects and
imagine moving them around. If you placed these three
tiles on top of each other, starting with the largest at
the bottom, which of the four images at the bottom
would you see?
1
2
3
4
1 2 3 423
Math doesn't have to be just strings of
numbers. Sometimes, it's easier to solve
a math problem when you can see it
as a picture—a technique known as
visualization. This is because visualizing
math uses different parts of the brain,which can make it easier to ?nd logical
solutions. Can you see the answers
to these six problems?
Recent studies show that
playing video games
develops visual
awareness and increases
short-term memory and
attention span.
3-D vision
Test your skills at mentally rotating a
3-D shape. If you folded up this shape
to make a cube, which of the four
options below would you see?
Illusion confusion
Optical illusions, such as this elephant,put your brain to work as it tries to
make sense of an image that is in fact
nonsense. Illusions also stimulate
the creative side of your brain and
force you to see things differently.
Can you ?gure out how many legs
this elephant has?
Seeing is understanding
A truly enormous snake has been spotted climbing
up a tree. One half of the snake is yet to arrive at the
tree. Two-thirds of the other half is wrapped around
the tree trunk and 5 ft (1.5 m) of snake is hanging
down from the branch. How long is the snake?
Forty percent of your
brain is dedicated to
seeing and processing
visual material.
1 2 3 4Inventingnumbers26
We are born with some understanding of
numbers, but almost everything else about
math needs to be learned. The rules and skills
we are taught at school had to be worked out
over many centuries. Even rules that seem
simple, such as which number follows 9, how
to divide a cake in three, or how to draw a
square, all had to be invented, long ago.
COUNT
LEARNING TO
1. Fingers and tallies
People have been counting on their ?ngers for more than
100,000 years, keeping track of their herds, or marking the days.
Since we humans have 10 ?ngers, we use 10 digits to count—
the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In fact, the word
digit means “?nger.” When early peoples ran out of ?ngers, they
made scratches called tallies instead. The earliest-known tally
marks, on a baboon’s leg bone, are 37,000 years old.
4. Egyptian math
Fractions tell us how to divide things— for example, how
to share a loaf between four people. Today, we would say
each person should get one quarter, or ?. The Egyptians,working out fractions 4,500 years ago, used the eye of
a god called Horus. Different parts of the eye stood for
fractions, but only those produced by halving a number
one or more times.
5. Greek math
Around 600 BCE, the Greeks started to develop the type
of math we use today. A big breakthrough was that they
didn’t just have ideas about numbers and shapes—they
also proved those ideas were true. Many of the laws that
the Greeks proved have stood the test of time—we still rely
on Euclid’s ideas on shapes (geometry) and Pythagoras’s
work on triangles, for example.
·
1 8
1 64
1 16
1 32
·27
2. From counters to numbers
The ?rst written numbers were used in the Near East
about 10,000 years ago. People there used clay counters to
stand for different things: For instance, eight oval-shaped
counters meant eight jars of oil. At ?rst, the counters were
wrapped with a picture, until people realized that the
pictures could be used without the counters. So the picture
that meant eight jars became the number 8.
3. Babylonian number rules
The place-value system (see page 31) was invented in
Babylon about 5,000 years ago. This rule allowed the position
of a numeral to affect its value—that’s why 2,200 and 2,020
mean different things. We count in base-10, using single
digits up to 9 and then double digits (10, 11, 12, and so on),but the Babylonians used base-60. They wrote their numbers
as wedge-shaped marks.
6. New math
Gradually, the ideas of the Greeks spread far and wide,leading to new mathematical developments in the Middle
East and India. In 1202, Leonardo of Pisa (an Italian
mathematician also known as Fibonacci) introduced
the eastern numbers and discoveries to Europe in his
Book of Calculation. This is why our numbering system
is based on an ancient Indian one.
Fizz-Buzz!
Try counting with a difference.
The more people there are, the
more fun it is. The idea is that you
all take turns counting, except
that when someone gets to a
multiple of three they shout
“Fizz,” and when they get to
a multiple of ?ve they
shout “Buzz.” If a number
is a multiple of both
three and ?ve, shout
“Fizz-Buzz.”If you get it
wrong, you’re out. The
last remaining player is
the winner.
ACTIVITY
The Egyptians used symbols of
walking feet to represent addition
and subtraction. They understood
calculation by imagining a person
walking right (addition) or left
(subtraction) a number line.
Fizz-Buzz!
Fizz-Buzz!28
The numbers we know and love today
developed over many centuries from
ancient systems. The earliest system
of numbers that we know today is the
Babylonian one, invented in Ancient
Iraq at least 5,000 years ago.
NUMBER
Counting in tens
Most of us learn to count
using our hands. We have
10 ?ngers and thumbs
(digits), so we have 10
numerals (also called
digits). This way of counting
is known as the base-10 or
decimal system, after
decem, Latin for “ten.”
Base-60
The Babylonians counted in base-60.
They gave their year 360 days (6 x 60).
We don’t know for sure how they used
their hands to count. One
theory is that they used
a thumb to count in units
up to 12 on one hand,and the ?ngers and thumb
of the other hand to count
in 12s up to a total of 60.
Table of
numbers
Ancient number
systems were nearly
all based on the
same idea: a symbol
for 1 was invented
and repeated to
represent small
or low numbers.
For larger numbers,usually starting at
10, a new symbol
was invented. This,too, could be written
down several times.
12
48
60
36 24
Their other hand kept
track of the 12s—one
12 per ?nger or thumb.
Intelligent eight-tentacled
creatures would almost
certainly count in base-8.
1
2
3
4
5
6
7
8
9 10
11
12
SYSTEMS
The Babylonians
counted in 12s on
one hand, using
·nger segments.
Babylonian
Mayan
Ancient
Egyptian
· ? ? ? ? ? ? ? ? Ancient
Greek
ú ? ü y t? ā ā ? ? Roman
Chinese
1 2 3 4 5 6 7 8 9 10
FACTS AND FIGURES29
Tech talk
Computers have their own
two-digit system, called
binary. This is because
computer systems are
made of switches that
have only two positions:
on (1) or off (0).
Building by numbers
The Ancient Egyptians used their mathematical
knowledge for building. For instance, they knew how
to work out the volume of a pyramid of any height or
width. The stones used to build the Pyramids at Giza
were measured so precisely that you cannot ?t a credit
card between them.
No dates, and no birthdays
No money, no buying
or selling
Sports would be either
chaotic or very boring
without any scores
No way of measuring
distance—just keep
walking until you
get there!
No measurements of
heights or angles, so your
house would be unstable
No science, so no amazing
inventions or technology,and no phone numbers
No numbers
Imagine a world with no
numbers. There would be…
Roman numerals
In the Roman number system, if a
numeral is placed before a larger one,it means it should be subtracted from
it. So IV is four (“I” less than “V”). This
can get tricky, though. The Roman way
of writing 199, for example, is CXCIX.
Going Greek
Oddly enough, the Ancient Greeks used the
same symbols for numbers as for letters.
So β was 2—when it wasn’t being b!
alpha and 1
beta and 2
gamma and 3
digamma and 6
zeta and 7
eta and 8
theta and 9
iota and 10
delta and 4
epsilon and 5
· ? ?? ? ? ? ?
·? ?? ? ?- --? -?? -??? ?
20 30 40 50 60 70 80 90 10030
Although it may seem like nothing, zero
is probably the most important number
of all. It was the last digit to be discovered
and it’s easy to see why—just try counting
to zero on your ?ngers! Even after its
introduction, this mysterious number wasn’t
properly understood. At ?rst it was used as a
placeholder but later became a full number.
ZERO
BIG
Brahmagupta
Indian mathematicians were the ?rst
people to use zero as a true number,not just a placeholder. Around 650 CE,an Indian mathematician named
Brahmagupta worked out how
zero behaved in calculations. Even
though some of Brahmagupta’s answers
were wrong, this was a big step forward.
Filling the gap
An early version of zero was
invented in Babylon more
than 5,000 years ago. It
looked like this pictogram
(right) and it played one of the
roles that zero does for us—it
spaced out other numbers. Without it, the numbers
12, 102, and 120 would all be written in the same
way: 12. But this Babylonian symbol did not have all
the other useful characteristics zero has today.
What is zero?
Zero can mean nothing, but not always! It can also
be valuable. Zero plays an important role in calculations
and in everyday life. Temperature, time, and football
scores can all have a value of zero—without it, everything
would be very confusing!
Yes, but it’s neither
odd nor even.
Zero isn’t
positive or
negative.
Is zero a
number?
A number
minus itself
is zero.
And you can’t
divide numbers
by zero.
Any number times
zero is zero.31
Place value
In our decimal system, the value of a digit depends
on its place in the number. Each place has a value of
10 times the place to the right. This place-value system
only works when you have zero to “hold” the place for
a value when no other digit goes in that position. So
on this abacus, the 2 represents the thousands in the
number, the 4 represents the hundreds, the 0 holds
the place for tens, and the 6 represents the ones,making the number 2,406.
Absolute zero
We usually measure temperatures
in degrees Celsius or Fahrenheit,but scientists often use the Kelvin
scale. The lowest number on this
scale, 0K, is known as absolute
zero. In theory, this is the lowest
possible temperature in the
Universe, but in reality the closest
scientists have achieved are
temperatures a few millionths of a
Kelvin warmer than absolute zero.
Roman homework
The Romans had no zero and used
letters to represent numbers: I was
1, V was 5, X was 10, C was 100, and
D was 500 (see pages 28–29). But
numbers weren’t always what they
seemed. For example, IX means
“one less than 10,” or 9. Without
zero, calculations were dif?cult.
Try adding 309 and 805 in Roman
numerals (right) and you’ll
understand why they didn’t catch on.
In a countdown,a rocket launches
at “zero!”
At zero hundred
hours—00:00—
it’s midnight.
Zero height is sea
level and zero gravity
exists in space.
ZERO
ACTIVITY
2 4 0 6
212°F
(100°C)
373K
Water boils
273K
Water freezes
195K
C02
freezes
(dry ice)
32°F
(0°C)
-108°F
(-78°C)
-459°F
(-273°C)
0K
Absolute zero
Without zero, we wouldn’t
be able to tell the difference
between numbers such
as 11 and 101…… and there’d be the same
distance between –1 and 1
as between 1 and 2.32
Pythagoras
Pythagoras thought
of odd numbers as
male, and even
numbers as female.
Early travels
Born around 570 BCE on the Greek island
of Samos, it is thought that Pythagoras
traveled to Egypt, Babylon (modern-day
Iraq), and perhaps even India in search of
knowledge. When he was in his forties, he
·nally settled in Croton, a town in Italy that
was under Greek control.
Strange society
In Croton, Pythagoras formed a school where
mainly math but also religion and mysticism were
studied. Its members, now called Pythagoreans,had many curious rules, from “let no swallows nest
in your eaves” to “do not sit on a quart pot” and
“eat no beans.” They became involved in local
politics and grew unpopular with the leaders of
Croton. After of?cials burned down their meeting
places, many of them ?ed, including Pythagoras.
The school of Pythagoras was made up of an inner circle of
mathematicians, and a larger group who came to listen to them
speak. According to some accounts, Pythagoras did his work in
the peace and quiet of a cave.
Pythagoras is perhaps the most famous mathematician
of the ancient world, and is best known for his theorem
on right-angled triangles. Ever curious about the world
around him, Pythagoras learned much on his travels.
He studied music in Egypt and may have been the ?rst
to invent a musical scale.
Pythagorean theorem
Pythagoras’s name lives on today in
his famous theorem. It says that, in a
right-angled triangle, the square of the
hypotenuse (the longest side, opposite
the right angle) is equal to the sum of
the squares of the other two sides.
The theorem can be written
mathematically as a2 + b2 = c2.
a
a
The square of the
long side (c), the
hypotenuse, can
be made by adding
the squares of the
other two sides
(a and b).
For Pythagoras,the most perfect
shape-making
number was 10,its dots forming
a triangle known
as the tetractys.
c
b
a
b
b
b
c
c
a
The triangle’s
right angle is
opposite the
longest side,the hypotenuse.
c33
Pythagoras believed that the
Earth was at the center of a set of
spheres that made a harmonious
sound as they turned.
Dangerous numbers
Pythagoras believed that all
numbers were rational—that
they could be written as a
fraction. For example, 5 can
be written as 5?
1, and 1.5 as 3?
2.
But one of his cleverest
students, Hippasus, is said
to have proved that the
square root of 2 could not
be shown as a fraction and
was therefore irrational.
Pythagoras could not accept
this, and by some accounts
was so upset he committed
suicide. Rumor also has it
that Hippasus was drowned
for proving the existence
of irrational numbers.
Pythagoreans realized that sets of pots
of water sounded harmonious if they
were ?lled according to simple ratios.
Math and music
Pythagoras showed that musical
notes that sound harmonious
(pleasant to the ear) obey simple
mathematical rules. For example,a harmonious note can be made
by plucking two strings where one
is twice the length of the other—
in other words, where the strings
are in a ratio of 2:1.
The number legacy
Pythagoreans believed that the world
contained only ?ve regular polyhedra (solid
objects with identical ?at faces), each with
a particular number of sides, as shown here.
For them, this was proof of their idea that
numbers explained everything. This theory
lives on, as today’s scientists all explain the
world in terms of mathematics.
Cube
6 square faces
Octahedron
8 triangular faces
Icosahedron
20 triangular faces
Dodecahedron
12 pentagonal faces
Pythagoras was one of the ?rst to propose
the idea that the Earth may be a sphere.
Tetrahedron
4 triangular faces34
BRAIN GAMES
Some problems can’t be
solved by working through
them step-by-step, and need
to be looked at in a different
way—sometimes we can
simply “see” the answer. This
intuitive way of ?guring things
out is one of the most dif?cult
parts of the brain’s workings
to explain. Sometimes, seeing
an answer is easier if you try
to approach the problem in an
unusual way—this is called
lateral thinking.
THINKING OUTSIDE
THE BOX
5. In the money
You have two identical money
bags. One is ?lled with small
coins. The other is ?lled with
coins that are twice the size
and value of the others. Which
of the bags is worth more?
6. How many?
If 10 children can eat 10
bananas in 10 minutes,how many children would
be needed to eat 100
bananas in 100 minutes?
1. Changing places
You are running in a race and
overtake the person in second place.
What position are you in now?
8. The lonely man
There was a man who never left his
house. The only visitor he had was
someone delivering supplies every two
weeks. One dark and stormy night, he
lost control of his senses, turned off
all the lights, and went to sleep. The
next morning it was discovered that
his actions had resulted in the deaths
of several people. Why?
7. Left or right?
A left-handed glove can be
changed into a right-handed
one by looking at it in a mirror.
Can you think of another way?
4. Sister act
A mother and father have
two daughters who were
born on the same day of
the same month of the
same year, but are not
twins. How are they
related to each other?
2. Pop!
How can you stick
10 pins into a balloon
without popping it?
3. What are the odds?
You meet a mother with two children. She
tells you that one of them is a boy. What is
the probability that the other is also a boy?35
15. Leave it to them
Some children are raking leaves in their
street. They gather seven piles at one house,four piles at another, and ?ve piles at
another. When the children put all the piles
together, how many will they have?
12. Whodunnit?
Acting on an anonymous phone call, the police raid
a house to arrest a suspected murderer. They don’t
know what he looks like but they know his name is
John and that he is inside the house. Inside they ?nd
a carpenter, a truck driver, a mechanic, and a ?reman
playing poker. Without hesitation or communication of
any kind, they immediately arrest the ?reman. How do
they know they have their man?
11. At a loss
A man buys sacks of rice
for 1 a pound from
American farmers and then
sells them for 0.05 a pound
in India. As a result, he
becomes a millionaire. How?
14. Crash!
A plane takes off from London headed
for Japan. After a few hours there is an
engine malfunction and the plane
crashes on the ItalianSwiss border.
Where do they bury the survivors?
10. Half full
Three of the glasses below are ?lled with orange
juice and the other three are empty. By touching
just one glass, can you arrange it so that the full
and empty glasses alternate?
13. Frozen!
You are trapped in a cabin on a cold snowy
mountain with the temperature falling and night
coming on. You have a matchbox containing just
a single match. You ?nd the following things in
the cabin. What do you light ?rst?
· A candle
· A gas lamp
· A windproof lantern
· A wood ?re with ?re starters
· A signal ?are to attract rescuers
9. A cut above
A New York City hairdresser recently
said that he would rather cut the hair
of three Canadians than one New
Yorker. Why would he say this?
16. Home
A man built a rectangular
house with all four sides
facing south. One morning
he looked out of the window
and spotted a bear. What
color was it?
FRAGILE36
Thousands of years ago, some Ancient
Greeks thought of numbers as having shapes,perhaps because different shapes can be made
by arranging particular numbers of objects.
Sequences of numbers can make patterns, too.
PATTERNS
NUMBER
Square numbers
If a particular number of objects can be
arranged to make a square with no gaps,that number is called a square number.
You can also make a square number by
“squaring” a number—which means
multiplying a number by itself: 1 x 1 = 1,2 x 2 = 4, 3 x 3 = 9, and so on.
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
111112 = 123454321
1111112 = 12345654321
16 objects can
be arranged to
make a 4 x 4
square.
Something odd
The ?rst ?ve square numbers
are 1, 4, 9, 16, and 25. Work
out the difference between
each pair in the sequence
(the difference between 1
and 4 is 3, for example). Write
your answers out in order.
Can you spot an odd pattern?
The magic ones
By squaring numbers made
of nothing but ones, you
can make all the other digits
appear—eventually! Stranger
still, those digits appear in
numbers that read the same
whether you look at them
forward or backward.
1 3 4
1 2
1
7 8
4 5
9
6
2 3 6 7
1 2 3
8
4
5
13 14
9 10
15
11
16
12
1 2 3
4
5
6 7 8 9 10
12 13 14 15 11
16 17 18 19 20
21 22 23 24 25
1 4 9 16 25
3 5 7 937
Prison break
It’s lights-out time at the prison, where
50 prisoners are locked in 50 cells. Not
realizing the cells’ doors are locked, a
guard comes along and turns the key to
each cell once, unlocking them all. Ten
minutes later, a second guard comes and
turns the keys of cells 2, 4, 6, and so on.
A third guard does the same, stopping
at cells 3, 6, 9, and so on. This carries on
until 50 guards have passed the cells.
How many prisoners escape? Look
out for a pattern that will give you
a shortcut to solving the problem.
Shaking hands
A group of three friends meet and
everyone shakes hands with everyone
else once. How many handshakes are
there in total? Try drawing this out, with
a dot for each person and lines between
them for handshakes. Now work out the
handshakes for groups of four, ?ve, or six
people. Can you spot a pattern?
Triangular numbers
If you can make an equilateral triangle (a triangle
with sides of equal length) from a particular
number of objects, that number is known as
triangular. You can make triangular numbers by
adding numbers that are consecutive (next to each
other): 0 + 1 = 1, 0 + 1 + 2 = 3, 0 + 1 + 2 + 3 = 6, and
so on. Many Ancient Greek mathematicians were
fascinated by triangular numbers, but we don’t
use them much today, except to admire the pattern!
Cubic numbers
If a number of objects, such as building blocks,can be assembled to make a cube shape, then that
number is called a cubic number. Cubic numbers
can also be made by multiplying a number by itself
twice. For example, 2 x 2 x 2 = 8.
1
1
3 1
2 3
6
1
2 3
4 5 6
10
1
2 3
4 5 6
7 8 9 10
15
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
A perfect solution?
The numbers 1, 2, 3, and 6 all divide
into the number 6, so we call them its
factors. A perfect number is one that’s
the sum of its factors (other than itself).
So, 1 + 2 + 3 = 6, making 6 a perfect
number. Can you ?gure out the next
perfect number?
ACTIVITY
1
8
2738
Mathematicians use all kinds of tricks
and shortcuts to reach their answers
quickly. Most can be learned easily
and are worth learning to save time
and impress your friends and teachers.
CALCULATION TIPS
To work out 9 x 9,bend down your
ninth ?nger.
Multiply by 9 with your hands
Here’s a trick that will make multiplying by 9 a breeze.
Step 1
Hold your hands face up in front of you. Find out
what number you need to multiply by 9 and bend the
corresponding ?nger. So to work out 9 x 9, turn down
your ninth ?nger.
Step 2
Take the number of ?ngers on the left side of the
bent ?nger, and combine (not add) it with the one on
the right. For example, if you bent your ninth ?nger,you’d combine the number of ?ngers on the left, 8,with the number of ?ngers on the right, 1. So you’d
have 81 (9 x 9 is 81).
1
2 3
4
5 6
7 8
9
10
Alex Lemaire
With plenty of practice, people can solve
amazing math problems without a
calculator. In 2007, French mathematician
Alex Lemaire worked out the number that,if multiplied by itself 13 times, gives a
particular 200 digit number. He gave the
correct answer in 70 seconds!
BRAIN GAMES
Multiplication tips
Mastering your times tables is an essential math
skill, but these tips will also help you out in a pinch:
· To quickly multiply by 4, simply double
the number, and then double it again.
· If you have to multiply a number by 5,?nd the answer by halving the number and then
multiplying it by 10. So 24 x 5 would be 24 ÷ 2 = 12,then 12 x 10 = 120.
· An easy way to multiply a number by 11 is
to take the number, multiply it by 10, and then
add the original number once more.
· To multiply large numbers when one is even, halve
the even number and double the other one. Repeat if
the halved number is still even. So, 32 x 125 is the
same as 16 x 250, which is the same as 8 x 500, which
is the same as 4 x 1,000. They all equal 4,000. 39
Division tips
There are lots of tips that can help
speed up your division:
· To ?nd out if a number is divisible by 3,add up the digits. If they add up to a multiple
of 3, the number will be divisible by 3.
For instance, 5,394 must be divisible
by 3 because 5 + 3 + 9 + 4 = 21,and 21 is divisible by 3.
· A number is divisible by 6
if it’s divisible by 3 and the
last digit is even.
· A number is divisible by 9 if all the digits add up to
a multiple of 9. For instance, 201,915 must be divisible
by 9 because 2 + 0 + 1 + 9 + 1 + 5 = 18, and 18 is
divisible by 9.
· To ?nd out if a number is divisible by 11, start with the
digit on the left, subtract the next digit from it, then add the
next, subtract the next, and so on. If the answer is 0 or 11,then the original number is divisible by 11. For example,35,706 is divisible by 11 because 3 – 5 + 7 – 0 + 6 = 11.
Beat the clock
Test your powers of mental arithmetic in this game against
the clock. It’s more fun if you play with a group of friends.
Step 1
First, one of you must choose two of the following numbers: 25, 50, 75, 100.
Next, someone else selects four numbers between 1 and 10. Now get
a friend to pick a number between
100 and 999. Write this down next
to the six smaller numbers.
Step 2
You all now have two minutes to
add, subtract, multiply, or divide
your chosen numbers—which you
can use only once—to get as close
as possible to the big number. The
winner is the person with the
exact or closest number.
Calculating a tip
If you need to leave a 15 percent tip after
a meal at a restaurant, here’s an easy
shortcut: Just work out 10 percent (divide
the number by 10), then add that number to
half its value, and you have your answer.
Fast squaring
If you need to square a two-digit
number that ends in 5, just multiply
the ?rst digit by itself plus 1, then
put 25 on the end. So to square
15, do: 1 x (1 + 1) = 2 , then attach
25 to give 225. This is how you can
work out the square of 25:
2 x (2+1) = 6
6 and 25 = 625
In Asia, children use an abacus
(a frame of bars of beads) to
add and subtract faster than
an electronic calculator.
10% of 35 = 3.50
3.50 ÷ 2 = 1.75
3.50 + 1.75 = 5.2540
Ingenious inventions
Archimedes is credited with building the
world’s ?rst planetarium—a machine that
shows the motions of the Sun, Moon, and
planets. One thing he didn’t invent, despite it
bearing his name, is the Archimedean screw.
It is more likely that he introduced this design
for a water pump, having seen it in Egypt.
Eureka!
Archimedes’ most famous discovery came
about when the king asked him to check if
his crown was pure gold. To answer this, he
had to measure the crown’s volume, but
how? Stepping into a full bath, Archimedes
realized that the water that spilled from the
tub could be measured to ?nd out the volume
of his body—or a crown.
Early life
Archimedes was born in Syracuse, Sicily,in 287 BCE. As a young man he traveled to
Egypt and worked with mathematicians there.
According to one story, when Archimedes
returned home to Syracuse, he heard that the
Egyptian mathematicians were claiming some
of his discoveries as their own. To catch them,he sent them some calculations with errors in
them. The Egyptians claimed these new
discoveries too, but were caught when people
discovered that the calculations were wrong.
An Archimedean screw is
a cylinder with a screw
inside. The screw raises
water as it turns.
Archimedes
Archimedes once said, “Give me a lever long
enough... and I shall move the world.”
Archimedes was probably the greatest
mathematician of the ancient world. Unlike
most of the others, he was a highly practical
person too, using his math skills to build
all kinds of contraptions, including some
extraordinary war machines.
On discovering how to measure volume, Arch ......
TRAIN your BRAIN to be a
GENIUS LONDON, NEW YORK,MELBOURNE, MUNICH, AND DELHI
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Publishing director Jonathan Metcalf
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First American edition, 2012
Published in the United States by
DK Publishing
375 Hudson Street
New York, New York 10014
Copyright ? 2012 Dorling Kindersley Limited
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001—182438 —0912
All rights reserved. No part of this publication may be reproduced, stored
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Published in Great Britain by Dorling Kindersley Limited.
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ISBN: 978-0-7566-9796-9
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Printed and bound in China by Hung Hing
Discover more at
www.dk.com
This book is full of puzzles and
activities to boost your brain
power. The activities are a lot of
fun, but you should always check
with an adult before you do any
of them so that they know what
you’re doing and are sure
that you’re safe. Written by
Consultant
Illustrated by
Dr. Mike Goldsmith
Branka Surla
Seb Burnett
MA TH
TRAIN your BRAIN to be a
GENIUS 4
CONTENTS 6 A world of math
MATH BRAIN
10 Meet your brain
12 Math skills
14 Learning math
16 Brain vs. machine
18 Problems with numbers
20 Women and math
22 Seeing the solution
INVENTING NUMBERS
26 Learning to count
28 Number systems
30 Big zero
32 Pythagoras
34 Thinking outside the box
36 Number patterns
38 Calculation tips
40 Archimedes
42 Math that measures
44 How big? How far?
46 The size of the problem
MAGIC NUMBERS
50 Seeing sequences
52 Pascal’s triangle
54 Magic squares
56 Missing numbers
58 Karl Gauss
60 In?nity
62 Numbers with meaning
64 Number tricks
66 Puzzling primes5
SHAPES AND SPACE
70 Triangles
72 Shaping up
74 Shape shifting
76 Round and round
78 The third dimension
80 3-D shape puzzles
82 3-D fun
84 Leonhard Euler
86 Amazing mazes
88 Optical illusions
90 Impossible shapes
A WORLD OF MATH
94 Interesting times
96 Mapping
98 Isaac Newton
100 Probability
102 Displaying data
104 Logic puzzles and paradoxes
106 Breaking codes
108 Codes and ciphers
110 Alan Turing
112 Algebra
114 Brainteasers
116 Secrets of the Universe
118 The big quiz
120 Glossary
122 Answers
126 Index
128 Credits
The book is full of
problems and puzzles for
you to solve. To check
the answers, turn to
pages 122–125. 6
It is impossible to imagine our world without
math. We use it, often without realizing, for a
whole range of activities —when we tell time,go shopping, catch a ball, or play a game. This
book is all about how to get your math brain
buzzing, with lots of things to do, many of the
big ideas explained, and stories about how the
great math brains have changed our world.
MATH A WORLD OF
Calculation
You need math to make
just about everything, from
a cake to a car. Quantities,costs, and timings must
all be worked out using
calculation and estimation.
I wonder what would
happen if the ride spun
even faster?
People are hungry
tonight. At this rate, I’ll
run out of hot dogs in
half an hour.
Science
Math is essential for
scientists—it helps them
test theories and make
them exact. Some theories
are then put to practical use,to build bridges, machines,and even carnival rides!
I′ll be in this line for
10 minutes, so I should still
be in time to catch the next
bus home.
Panel puzzle
These shapes form a square panel, used
in one of the carnival stalls. However, an
extra shape has somehow been mixed
up with them. Can you ?gure out which
piece does not belong?
There’s a height restriction
on this ride, sonny. Try
coming back next year.
D
E
F
C
B
A7
One in four people are
hitting a coconut. Grr! I’m
making a loss.
Shapes
Understanding shapes and
space helps us make
sense of the world around
us. You need to know about
this area of math to create
and design anything—
including tricky games.
Patterns
Many areas of math
involve looking for patterns,such as how numbers
repeat or how shapes behave.
Often these patterns can be
used to help us and inspire
new ways of thinking.
Profit margin
It costs 144 a day to run the
bumper cars, accounting for
wages, electricity, transportation,and so on. There are 12 bumper
cars, and, on average, 60 percent
of them are occupied each session.
The ride is open for eight hours a
day, with four sessions an hour,and each driver pays 2 per
session. How much pro?t is
the owner making?
A game of chance
Everyone loves to try to knock down
a coconut—but what are your chances
of success? The stall owner needs to
know so he can make sure he’s got
enough coconuts, and to work out how
much to charge. He’s discovered that, on
average, he has 90 customers a day, each
throwing three balls, and the total number
of coconuts won is 30. So what is the
likelihood of you winning a coconut?
Gulp! The slide looks
even steeper from the top.
I wonder what speed I’ll
be going when I get to
the bottom?
Look at me! I’m
loating in the air and
I’ve got two tongues!
I think I’ve got the
angle just right... one
more go and I’ll win
a prize.Math
brain10
Cerebrum Where thinking
occurs and memories
are stored
Hypothalamus Controls
sleep, hunger, and body
temperature
Thalamus Receives
sensory nerve signals
and sends them on
to the cerebrum
Meninges
Protective layers
that cushion the
brain against shock
Skull Forms a
tough casing
around the brain
Cerebellum Helps
control balance
and movement
Medulla Controls
breathing, heartbeat,blood pressure,and vomiting
Your brain is the most complex organ
in your body—a spongy, pink mass made
up of billions of microscopic nerve cells. Its
largest part is the cauli?ower-like cerebrum,made up of two hemispheres, or halves,linked by a network of nerves. The cerebrum
is the part of the brain where math is
understood and calculations are made.
BRAIN
MEET YOUR
Looking inside
This cross-section of the skull
reveals the thinking part of the
brain, or cerebrum. Beneath its
outer layers is the “white matter,”
which transfers signals between
different parts of the brain.
LEFT-BRAIN SKILLS
The left side of your cerebrum is
responsible for the logical, rational
aspects of your thinking, as well as for
grammar and vocabulary. It’s here that
you work out the answers to calculations.
A BRAIN OF TWO HALVES
The cerebrum has two hemispheres. Each deals
mainly with the opposite side of the body—data
from the right eye, for example, is handled in
the brain’s left side. For some functions,including math, both halves work
together. For others, one half is
more active than the other.
Writing
skills
Like spoken language, writing
involves both hemispheres. The right
organizes ideas, while the left ?nds
the words to express them.
Scientific
thinking
Logical thinking is the job
of the brain’s left side, but
most science also involves
the creative right side.
Mathematical
skills
The left brain oversees
numbers and calculations,while the right processes
shapes and patterns.
Rational thought
Thinking and reacting in a
rational way appears to be
mainly a left-brain activity.
It allows you to analyze a
problem and ?nd an answer.
Language
The left side handles the meanings
of words, but it is the right half that
puts them together into sentences
and stories.
Left visual cortex Processes
signals from the right eye
Corpus callosum Links the
two sides of the brain
Pituitary gland Controls
the release of hormones11
Frontal lobe Vital to
thought, personality,speech, and emotion
Temporal lobe Where
sounds are recognized,and where long-term
memories are stored
RIGHT-BRAIN SKILLS
The right side of your cerebrum is where
creativity and intuition take place, and is
also used to understand shapes and motion.
You carry out rough calculations here, too.
The outer surface
Thinking is carried out on the surface
of the cerebrum, and the folds and
wrinkles are there to make this surface
as large as possible. In preserved
brains, the outer layer is gray, so it
is known as “gray matter.”
Right eye Collects data on
light-sensitive cells that is
processed in the opposite
side of the brain—the left
visual cortex in the
occipital lobe
Right optic nerve
Carries information from
the right eye to the left
visual cortex
Spatial skills
Understanding the shapes of
objects and their positions in
space is a mainly right-brain
activity. It provides you your
ability to visualize.
Imagination
The right side of the brain
directs your imagination.
Putting your thoughts into
words, however, is the job
of the left side of the brain.
Music
The brain’s right side is
where you appreciate music.
Together with the left side,it works to make sense of
the patterns that make the
music sound good.
Insight
Moments of insight occur
in the right side of the brain.
Insight is another word for
those “eureka!” moments
when you see the connections
between very different ideas.
Art
The right side of the brain
looks after spatial skills.
It is more active during
activities such as drawing,painting, or looking at art.
Parietal lobe Gathers
together information
from senses such as
touch and taste
Occipital lobe Processes
information from the
eyes to create images
Spinal cord Joins the
brain to the system
of nerves that runs
throughout the body
Neurons and numbers
Neurons are brain cells that link up to
pass electric signals to each other.
Every thought, idea, or feeling that
you have is the result of neurons
triggering a reaction in your brain.
Scientists have found that when you
think of a particular number, certain
neurons ?re strongly.
Doing the math
This brain scan was carried out on a
person who was working out a series
of subtraction problems. The yellow
and orange areas show the parts of
the brain that were producing the
most electrical nerve signals. What’s
interesting is that areas all over
the brain are active—not just one.
Cerebellum Tucked
beneath the cerebrum’s
two halves, this
structure coordinates
the body’s muscles12
Many parts of your brain are involved in math, with big
differences between the way it works with numbers (arithmetic),and the way it grasps shapes and patterns (geometry). People
who struggle in one area can often be strong in another. And
sometimes there are several ways to tackle the same problem,using different math skills.
SKILLS
MATH
BRAIN GAMES
A quick glance
Our brains have evolved to grasp key
facts quickly—from just a glance at
something—and also to think things
over while examining them.
How do you count?
When you count in your head, do
you imagine the sounds of the
numbers, or the way they look?
Try these two experiments and
see which you ?nd easiest.
Step 1
Try counting backward in 3s from
100 in a noisy place with your eyes
shut. First, try “hearing” the
numbers, then visualizing them.
Step 1
Look at the sequences below—
just glance at them brie?y without
counting—and write down the
number of marks in each group.
Step 2
Now count the marks in each group
and then check your answers.
Which ones did you get right?
Step 2
Next, try both methods again
while watching TV with the sound
off. Which of the four exercises
do you ?nd easier?
About 10 percent of people think of
numbers as having colors. With
some friends, try scribbling the
irst number between 0 and 9 that
pops into your head when you
think of red, then black,then blue. Do any of you
get the same
answers?
The part of the brain that can “see” numbers
at a glance only works up to three or four, so
you probably got groups less than ?ve right.
You only roughly estimate higher numbers,so are more likely to get these wrong.
97...94...
88...85...
There are four main styles
of thinking, any of which can
be used for learning math: seeing
the words written, thinking in
pictures, listening to the sounds
of words, and hands-on activities.13
Spot the shape
In each of these sequences,can you ?nd the shape on the
far left hidden in one of the
·ve shapes to the right?
You will need:
· Pack of at least 40 small
pieces of candy
· Three bowls
· Stopwatch
· A friend
Eye test
This activity tests your ability
to judge quantities by eye. You
should not count the objects—
the idea is to judge equal
quantities by sight alone.
Step 1
Set out the three bowls in front
of you and ask your friend to
time you for ?ve seconds. When
he says “go,” try to divide the
candy evenly between them.
Step 2
Now count up the number of
candy pieces you have in each
bowl. How equal were the
quantities in all three?
Number cruncher
Your short-term memory can store a certain
amount of information for a limited time.
This exercise reveals your brain’s ability to
remember numbers. Starting at the top,read out loud a line of numbers one at a
time. Then cover up the line and try to
repeat it. Work your way down the list
until you can’t remember all the numbers.
438
7209
18546
907513
2146307
50918243
480759162
1728406395
Most people can hold about
seven numbers at a time in their
short-term memory. However, we
usually memorize things by saying
them in our heads. Some digits take
longer to say than others and this
affects the number we can remember.
So in Chinese, where the sounds of the
words for numbers are very short, it
is easier to memorize more numbers.
We have a natural sense of
pattern and shape. The Ancient
Greek philosopher Plato discovered
this a long time ago, when he
showed his slaves some shape
puzzles. The slaves got the answers
right, even though they’d had
no schooling.
You’ll probably be surprised how
accurately you have split up the
candy. Your brain has a strong sense
of quantity, even though it is not
thinking about it in terms
of numbers.
1
2
3
4
A B C D E
A B C D E
A B C D E
A B C D E14
For many, the thought of learning
math is daunting. But have you
ever wondered where math came
from? Did people make it up as they
went along? The answer is yes and
no. Humans—and some animals—
are born with the basic rules of
math, but most of it was invented.
Brain size and evolution
Compared with the size of the body, the human
brain is much bigger than those of other animals.
We also have larger brains than our apelike
ancestors. A bigger brain indicates a greater
capacity for learning and problem solving. Frog Bird Human
Baby at six months
In one study, a baby was shown
two toys, then a screen was put
up and one toy was taken away.
The activity of the baby’s brain
revealed that it knew something
was wrong, and understood the
difference between one and two.
ACTIVITY
Can your pet count?
All dogs can “count” up to about three. To test your dog,or the dog of a friend, let the dog see you throw one, two,or three treats somewhere out of sight. Once the dog
has found the number of treats you threw, it will usually
stop looking. But throw ?ve or six treats and the dog will
“lose count” and not know when to stop. It will keep on
looking even after ?nding all the treats. Use dry treats
with no smell and make sure they fall out of sight.
Baby at 48 hours
Newborn babies have some sense
of numbers. They can recognize
that seeing 12 ducks is different
from 4 ducks.
A sense of numbers
Over the last few years, scientists have tested
babies and young children to investigate their
math skills. Their ?ndings show that we humans
are all born with some knowledge of numbers.
Animal antics
Many animals have a sense of
numbers. A crow called Jakob
could identify one among many
identical boxes if it had ?ve dots
on it. And ants seem to know
exactly how many steps there
are between them and their nest.
MATH
LEARNING15
How memory works
Memory is essential to math. It allows us to keep
track of numbers while we work on them, and to
learn tables and equations. We have different
kinds of memory. As we do a math problem, for
example, we remember the last few numbers
only brie?y (short-term memory), but we will
remember how to count from 1 to 10 and so on
for the rest of our lives (long-term memory).
From five to nine
When a ?ve-year-old is asked to
put numbered blocks in order,he or she will tend to space
the lower numbers farther
apart than the higher ones.
By the age of about nine,children recognize that the
difference between numbers
is the same—one—and space
the blocks equally.
Clever Hans
Just over a century ago, there was a mathematical horse
named Hans. He seemed to add, subtract, multiply, and
divide, then tap out his answer with his hoof. However,Hans wasn’t good at math. Unbeknownst to his owner, the
horse was actually excellent at “reading” body language.
He would watch his owner’s face change when he had
made the right number of taps, and then stop.
Child at age four
The average four-year-old
can count to 10, though the
numbers may not always
be in the right order. He
or she can also estimate
larger quantities, such
as hundreds. Most
importantly, at four
a child becomes
interested in making
marks on paper,showing numbers
in a visual way.
Sensory memory
We keep a memory
of almost everything
we sense, but only for
half a second or so.
Sensory memory can
store about a dozen
things at once.
Short-term memory
We can retain a handful
of things (such as a
few digits or words)
in our memory for about
a minute. After that,unless we learn them,they are forgotten.
Long-term memory
With effort, we can
memorize and learn an
impressive number of
facts and skills. These
long-term memories
can stay with us for
our whole lives.
It can help you memorize
your tables if you speak or sing
them. Or try writing them down,looking out for any patterns. And,of course, practice them again
and again.
I’m going
to draw hundreds and
hundreds of dots!16
In a battle of the superpowers—brain versus
machine—the human brain would be the winner!
Although able to perform calculations at lightning
speeds, the supercomputer, as yet, is unable to
think creatively or match the mind of a genius.
So, for now, we humans remain one step ahead.
BRAIN Prodigies
A prodigy is someone who has an incredible
skill from an early age—for example, great
ability in math, music, or art. India’s
Srinivasa Ramanujan (1887–1920) had hardly
any schooling, yet became an exceptional
mathematician. Prodigies have active memories
that can hold masses of data at once.
Savants
Someone who is incredibly skilled in a
specialized ?eld is known as a savant.
Born in 1979, Daniel Tammet is a British savant
who can perform mind-boggling feats of
calculation and memory, such as memorizing
22,514 decimal places of pi (3.141...), see pages
76–77. Tammet has synesthesia, which means
he sees numbers with colors and shapes.
Your brain:
· Has about 100 billion neurons
· Each neuron, or brain cell, can
send about 100 signals per second
· Signals travel at speeds of about
33 ft (10 m) per second
· Continues working and transmitting
signals even while you sleep
Hard work
More often than not, dedication and
hard work are the key to exceptional
success. In 1637, a mathematician
named Pierre de Fermat proposed
a theorem but did not prove it. For
more than three centuries, many
great mathematicians tried and
failed to solve the problem. Britain’s
Andrew Wiles became fascinated
by Fermat’s Last Theorem when he
was 10. He ?nally solved it more
than 30 years later in 1995.
What about your brain?
If someone gives you some numbers to add
up in your head, you keep them all “in mind”
while you do the math. They are held in your
short-term memory (see page 15). If you can
hold more than eight numbers in your head,you've got a great math brain.
VS.17
MACHINE
Your computer:
· Has about 10 billion transistors
· Each transistor can send about
one billion signals per second
· Signals travel at speeds of about
120 million miles (200 million km)
per second
· Stops working when it is
turned off
Artificial
intelligence
An arti?cially intelligent computer
is one that seems to think like a
person. Even the most powerful
computer has nothing like the
all-round intelligence of a human
being, but some can carry out
certain tasks in a humanlike
way. The computer system
Watson, for example, learns
from its mistakes, makes choices,and narrows down options. In
2011, it beat human contestants
to win the quiz show Jeopardy.
Missing ingredient
Computers are far better than humans
at calculations, but they lack many of
our mental skills and cannot come up with
original ideas. They also ?nd it almost
impossible to disentangle the visual world—
even the most advanced computer would
be at a loss to identify the contents
of a messy bedroom!
Computers
When they were ?rst invented, computers were
called electronic brains. It is true that, like the
human brain, a computer’s job is to process
data and send out control signals. But, while
computers can do some of the same things
as brains, there are more differences than
similarities between the two. Machines are
not ready to take over the world just yet.18
NUMBERS
PROBLEMS WITH
Numerophobia
A phobia is a fear of something that there is no reason to
be scared of, such as numbers. The most feared numbers
are 4, especially in Japan and China, and 13. Fear of the
number 13 even has its own name—triskaidekaphobia.
Although no one is scared of all numbers, a lot of people
are scared of using them!
Dyscalculia
Which of these two numbers is higher? 76 46
If you can’t tell within a second, you might have dyscalculia,where the area of your brain that compares numbers does
not work properly. People with dyscalculia can also have
dif?culty telling time. But remember, dyscalculia is very
rare, so it is not a good excuse for missing the bus.
A life without math
Although babies are born with a sense of
numbers, more complicated ideas need to
be taught. Most societies use and teach
these mathematical ideas—but not all of them.
Until recently, the Hadza people of Tanzania,for example, did not use counting, so their
language had no numbers beyond 3 or 4.
Too late to learn?
Math is much easier to learn when
young than as an adult. The great
19th-century British scientist
Michael Faraday was never taught
math as a child. As a result, he
was unable to complete or prove
his more advanced work. He just
didn’t have a thorough enough
grasp of mathematics.19
A lot of people think math is tricky, and many try
to avoid the subject. It is true that some people have
learning dif?culties with math, but they are very
rare. With a little time and practice, you can soon get
to grips with the basic rules of math, and once you’ve
mastered those, then the skills are yours for life!
1 x 7 = 7 2 x 7 = 14
3 x 7 = 21 4 x 7 = 28
5 x 7 = 35 6 x 7 = 42
7 x 7 = 49 8 x 7 = 56
Visualizing math
Sometimes math questions sound complicated or use
unfamiliar words or symbols. Drawing or visualizing
(picturing in your head) can help with understanding and
solving math problems. Questions about dividing shapes
equally, for example, are simple enough to draw, and a
rough sketch is all you need to get an idea of the answer.
Practice makes perfect
For those of us who struggle with calculations, the contestants
who take part in TV math contests can seem like geniuses.
In fact, anyone can be a math whizz if they follow the three
secrets to success: practice, learning some basic calculations
by heart (such as multiplication tables), and using tips
and shortcuts.
Misleading numbers
Numbers can in?uence how and what you think.
You need to be sure what numbers mean so they
cannot be used to mislead you. Look at these two
stories. You should be suspicious of the numbers
in both of them—can you ?gure out why?
A useful survey?
Following a survey carried out by the
Association for More Skyscrapers (AMS),it is suggested that most of the 30 parks
in the city should close. The survey found
that, of the three parks surveyed, two had
fewer than 25 visitors all day. Can you
identify four points that should make you
think again about AMS’s survey?
The bigger picture
In World War I, soldiers wore cloth hats, which
contributed to a high number of head injuries.
Better protection was required, so cloth hats
were replaced by tin helmets. However, this
led to a dramatic rise in head injuries. Why
do you think this happened?
HEAD INJURIES
ON THE RISE!
PARKS TO CLOSE!
The 13th-century thinker
Roger Bacon said, “He who
is ignorant of [math] cannot
know the other sciences, nor
the affairs of this world.”
ACTIVITY20
Historically, women have always had
a tough time breaking into the ?elds of
math and science. This was mainly
because, until a century or so ago, they
received little or no education in these
subjects. However, the most determined
women did their homework and went on
to make signi?cant discoveries in some
highly sophisticated areas of math.
WOMEN AND MATH
Sofia Kovalevskaya
Born in Russia in 1850, Kovalevskaya’s fascination with
math began when her father used old math notes as
temporary wallpaper for her room! At the time, women
could not attend college but Kovalevskaya managed
to ?nd math tutors, learned rapidly, and soon made
her own discoveries. She developed the math of
spinning objects, and ?gured out how Saturn’s rings
move. By the time she died, in 1891, she was
a university professor.
Amalie Noether
German mathematician Amalie “Emmy” Noether
received her doctorate in 1907, but at ?rst no university
would offer her—or any woman—a job in math.
Eventually her supporters (including Einstein) found
her work at the University of Gottingen, although at ?rst
her only pay was from students. In 1933, she was forced
to leave Germany and went to the United States, where
she was made a professor. Noether discovered how to
use scienti?c equations to work out new facts, which
could then be related to entirely different ?elds of study.
Noether showed how
the many symmetries
that apply to all kinds
of objects, including
atoms, can reveal basic
laws of physics.
Kovalevskaya took
discoveries in physics
and turned them into
math, so that tops
and other spinning
objects could be
understood exactly.Hypatia
Daughter of a mathematician and philospher,Hypatia was born around 355 CE in Alexandria,which was then part of the Roman Empire. Hypatia
became the head of an important “school,” where
great thinkers tried to ?gure out the nature of the
world. It is believed she was murdered in 415 CE by
a Christian mob who found her ideas threatening.
Augusta Ada King
Born in 1815, King was the only child of the poet
Lord Byron, but it was her mother who encouraged
her study of math. She later met Charles Babbage
and worked with him on his computer machines.
Although Babbage never completed a working
computer, King had written what we would now
call its program—the ?rst in the world. There is
a computer language called Ada, named after her.
Grace Hopper
A rear admiral in the U.S. Navy, Hopper
developed the world’s ?rst compiler—
a program that converts ordinary language
into computer code. Hopper also developed the
·rst language that could be used by more than
one computer. She died in 1992, and the
destroyer USS Hopper was named after her.
Florence Nightingale
This English nurse made many improvements
in hospital care during the 19th century.
She used statistics to convince of?cials that
infections were more dangerous to soldiers
than wounds. She even invented her own
mathematical charts, similar to pie charts,to give the numbers greater impact.
Although Babbage’s
computer was not
built during his
lifetime, it was
eventually made
according to his
plans, nearly two
centuries later. If
he had built it, it
would have been
steam-powered!
Hypatia studied the
way a cone can be cut
to produce different
types of curves.
Nightingale’s chart
compared deaths from
different causes in the
Crimean War between 1854
and 1855. Each segment
stands for one month.
Blue represents
deaths from
preventable
diseases
Pink represents
deaths from
wounds
Hopper popularized
the term computer
“bug” to mean a coding
error, after a moth
became trapped in
part of a computer.
Black represents
deaths from all
other causes22
SOLUTION SEEING THE
BRAIN GAMES
What do you see?
The ?rst step to sharpening the
visual areas of your brain is to practice
recognizing visual information. Each
of these pictures is made up of the
outlines of three different objects.
Can you ?gure out what they are?
Thinking in 2-D
Lay out 16 matches to make ?ve squares
as shown here. By moving only two
matches, can you turn the ?ve squares
into four? No matches can be removed.
Visual sequencing
To do this puzzle, you need to visualize objects and
imagine moving them around. If you placed these three
tiles on top of each other, starting with the largest at
the bottom, which of the four images at the bottom
would you see?
1
2
3
4
1 2 3 423
Math doesn't have to be just strings of
numbers. Sometimes, it's easier to solve
a math problem when you can see it
as a picture—a technique known as
visualization. This is because visualizing
math uses different parts of the brain,which can make it easier to ?nd logical
solutions. Can you see the answers
to these six problems?
Recent studies show that
playing video games
develops visual
awareness and increases
short-term memory and
attention span.
3-D vision
Test your skills at mentally rotating a
3-D shape. If you folded up this shape
to make a cube, which of the four
options below would you see?
Illusion confusion
Optical illusions, such as this elephant,put your brain to work as it tries to
make sense of an image that is in fact
nonsense. Illusions also stimulate
the creative side of your brain and
force you to see things differently.
Can you ?gure out how many legs
this elephant has?
Seeing is understanding
A truly enormous snake has been spotted climbing
up a tree. One half of the snake is yet to arrive at the
tree. Two-thirds of the other half is wrapped around
the tree trunk and 5 ft (1.5 m) of snake is hanging
down from the branch. How long is the snake?
Forty percent of your
brain is dedicated to
seeing and processing
visual material.
1 2 3 4Inventingnumbers26
We are born with some understanding of
numbers, but almost everything else about
math needs to be learned. The rules and skills
we are taught at school had to be worked out
over many centuries. Even rules that seem
simple, such as which number follows 9, how
to divide a cake in three, or how to draw a
square, all had to be invented, long ago.
COUNT
LEARNING TO
1. Fingers and tallies
People have been counting on their ?ngers for more than
100,000 years, keeping track of their herds, or marking the days.
Since we humans have 10 ?ngers, we use 10 digits to count—
the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In fact, the word
digit means “?nger.” When early peoples ran out of ?ngers, they
made scratches called tallies instead. The earliest-known tally
marks, on a baboon’s leg bone, are 37,000 years old.
4. Egyptian math
Fractions tell us how to divide things— for example, how
to share a loaf between four people. Today, we would say
each person should get one quarter, or ?. The Egyptians,working out fractions 4,500 years ago, used the eye of
a god called Horus. Different parts of the eye stood for
fractions, but only those produced by halving a number
one or more times.
5. Greek math
Around 600 BCE, the Greeks started to develop the type
of math we use today. A big breakthrough was that they
didn’t just have ideas about numbers and shapes—they
also proved those ideas were true. Many of the laws that
the Greeks proved have stood the test of time—we still rely
on Euclid’s ideas on shapes (geometry) and Pythagoras’s
work on triangles, for example.
·
1 8
1 64
1 16
1 32
·27
2. From counters to numbers
The ?rst written numbers were used in the Near East
about 10,000 years ago. People there used clay counters to
stand for different things: For instance, eight oval-shaped
counters meant eight jars of oil. At ?rst, the counters were
wrapped with a picture, until people realized that the
pictures could be used without the counters. So the picture
that meant eight jars became the number 8.
3. Babylonian number rules
The place-value system (see page 31) was invented in
Babylon about 5,000 years ago. This rule allowed the position
of a numeral to affect its value—that’s why 2,200 and 2,020
mean different things. We count in base-10, using single
digits up to 9 and then double digits (10, 11, 12, and so on),but the Babylonians used base-60. They wrote their numbers
as wedge-shaped marks.
6. New math
Gradually, the ideas of the Greeks spread far and wide,leading to new mathematical developments in the Middle
East and India. In 1202, Leonardo of Pisa (an Italian
mathematician also known as Fibonacci) introduced
the eastern numbers and discoveries to Europe in his
Book of Calculation. This is why our numbering system
is based on an ancient Indian one.
Fizz-Buzz!
Try counting with a difference.
The more people there are, the
more fun it is. The idea is that you
all take turns counting, except
that when someone gets to a
multiple of three they shout
“Fizz,” and when they get to
a multiple of ?ve they
shout “Buzz.” If a number
is a multiple of both
three and ?ve, shout
“Fizz-Buzz.”If you get it
wrong, you’re out. The
last remaining player is
the winner.
ACTIVITY
The Egyptians used symbols of
walking feet to represent addition
and subtraction. They understood
calculation by imagining a person
walking right (addition) or left
(subtraction) a number line.
Fizz-Buzz!
Fizz-Buzz!28
The numbers we know and love today
developed over many centuries from
ancient systems. The earliest system
of numbers that we know today is the
Babylonian one, invented in Ancient
Iraq at least 5,000 years ago.
NUMBER
Counting in tens
Most of us learn to count
using our hands. We have
10 ?ngers and thumbs
(digits), so we have 10
numerals (also called
digits). This way of counting
is known as the base-10 or
decimal system, after
decem, Latin for “ten.”
Base-60
The Babylonians counted in base-60.
They gave their year 360 days (6 x 60).
We don’t know for sure how they used
their hands to count. One
theory is that they used
a thumb to count in units
up to 12 on one hand,and the ?ngers and thumb
of the other hand to count
in 12s up to a total of 60.
Table of
numbers
Ancient number
systems were nearly
all based on the
same idea: a symbol
for 1 was invented
and repeated to
represent small
or low numbers.
For larger numbers,usually starting at
10, a new symbol
was invented. This,too, could be written
down several times.
12
48
60
36 24
Their other hand kept
track of the 12s—one
12 per ?nger or thumb.
Intelligent eight-tentacled
creatures would almost
certainly count in base-8.
1
2
3
4
5
6
7
8
9 10
11
12
SYSTEMS
The Babylonians
counted in 12s on
one hand, using
·nger segments.
Babylonian
Mayan
Ancient
Egyptian
· ? ? ? ? ? ? ? ? Ancient
Greek
ú ? ü y t? ā ā ? ? Roman
Chinese
1 2 3 4 5 6 7 8 9 10
FACTS AND FIGURES29
Tech talk
Computers have their own
two-digit system, called
binary. This is because
computer systems are
made of switches that
have only two positions:
on (1) or off (0).
Building by numbers
The Ancient Egyptians used their mathematical
knowledge for building. For instance, they knew how
to work out the volume of a pyramid of any height or
width. The stones used to build the Pyramids at Giza
were measured so precisely that you cannot ?t a credit
card between them.
No dates, and no birthdays
No money, no buying
or selling
Sports would be either
chaotic or very boring
without any scores
No way of measuring
distance—just keep
walking until you
get there!
No measurements of
heights or angles, so your
house would be unstable
No science, so no amazing
inventions or technology,and no phone numbers
No numbers
Imagine a world with no
numbers. There would be…
Roman numerals
In the Roman number system, if a
numeral is placed before a larger one,it means it should be subtracted from
it. So IV is four (“I” less than “V”). This
can get tricky, though. The Roman way
of writing 199, for example, is CXCIX.
Going Greek
Oddly enough, the Ancient Greeks used the
same symbols for numbers as for letters.
So β was 2—when it wasn’t being b!
alpha and 1
beta and 2
gamma and 3
digamma and 6
zeta and 7
eta and 8
theta and 9
iota and 10
delta and 4
epsilon and 5
· ? ?? ? ? ? ?
·? ?? ? ?- --? -?? -??? ?
20 30 40 50 60 70 80 90 10030
Although it may seem like nothing, zero
is probably the most important number
of all. It was the last digit to be discovered
and it’s easy to see why—just try counting
to zero on your ?ngers! Even after its
introduction, this mysterious number wasn’t
properly understood. At ?rst it was used as a
placeholder but later became a full number.
ZERO
BIG
Brahmagupta
Indian mathematicians were the ?rst
people to use zero as a true number,not just a placeholder. Around 650 CE,an Indian mathematician named
Brahmagupta worked out how
zero behaved in calculations. Even
though some of Brahmagupta’s answers
were wrong, this was a big step forward.
Filling the gap
An early version of zero was
invented in Babylon more
than 5,000 years ago. It
looked like this pictogram
(right) and it played one of the
roles that zero does for us—it
spaced out other numbers. Without it, the numbers
12, 102, and 120 would all be written in the same
way: 12. But this Babylonian symbol did not have all
the other useful characteristics zero has today.
What is zero?
Zero can mean nothing, but not always! It can also
be valuable. Zero plays an important role in calculations
and in everyday life. Temperature, time, and football
scores can all have a value of zero—without it, everything
would be very confusing!
Yes, but it’s neither
odd nor even.
Zero isn’t
positive or
negative.
Is zero a
number?
A number
minus itself
is zero.
And you can’t
divide numbers
by zero.
Any number times
zero is zero.31
Place value
In our decimal system, the value of a digit depends
on its place in the number. Each place has a value of
10 times the place to the right. This place-value system
only works when you have zero to “hold” the place for
a value when no other digit goes in that position. So
on this abacus, the 2 represents the thousands in the
number, the 4 represents the hundreds, the 0 holds
the place for tens, and the 6 represents the ones,making the number 2,406.
Absolute zero
We usually measure temperatures
in degrees Celsius or Fahrenheit,but scientists often use the Kelvin
scale. The lowest number on this
scale, 0K, is known as absolute
zero. In theory, this is the lowest
possible temperature in the
Universe, but in reality the closest
scientists have achieved are
temperatures a few millionths of a
Kelvin warmer than absolute zero.
Roman homework
The Romans had no zero and used
letters to represent numbers: I was
1, V was 5, X was 10, C was 100, and
D was 500 (see pages 28–29). But
numbers weren’t always what they
seemed. For example, IX means
“one less than 10,” or 9. Without
zero, calculations were dif?cult.
Try adding 309 and 805 in Roman
numerals (right) and you’ll
understand why they didn’t catch on.
In a countdown,a rocket launches
at “zero!”
At zero hundred
hours—00:00—
it’s midnight.
Zero height is sea
level and zero gravity
exists in space.
ZERO
ACTIVITY
2 4 0 6
212°F
(100°C)
373K
Water boils
273K
Water freezes
195K
C02
freezes
(dry ice)
32°F
(0°C)
-108°F
(-78°C)
-459°F
(-273°C)
0K
Absolute zero
Without zero, we wouldn’t
be able to tell the difference
between numbers such
as 11 and 101…… and there’d be the same
distance between –1 and 1
as between 1 and 2.32
Pythagoras
Pythagoras thought
of odd numbers as
male, and even
numbers as female.
Early travels
Born around 570 BCE on the Greek island
of Samos, it is thought that Pythagoras
traveled to Egypt, Babylon (modern-day
Iraq), and perhaps even India in search of
knowledge. When he was in his forties, he
·nally settled in Croton, a town in Italy that
was under Greek control.
Strange society
In Croton, Pythagoras formed a school where
mainly math but also religion and mysticism were
studied. Its members, now called Pythagoreans,had many curious rules, from “let no swallows nest
in your eaves” to “do not sit on a quart pot” and
“eat no beans.” They became involved in local
politics and grew unpopular with the leaders of
Croton. After of?cials burned down their meeting
places, many of them ?ed, including Pythagoras.
The school of Pythagoras was made up of an inner circle of
mathematicians, and a larger group who came to listen to them
speak. According to some accounts, Pythagoras did his work in
the peace and quiet of a cave.
Pythagoras is perhaps the most famous mathematician
of the ancient world, and is best known for his theorem
on right-angled triangles. Ever curious about the world
around him, Pythagoras learned much on his travels.
He studied music in Egypt and may have been the ?rst
to invent a musical scale.
Pythagorean theorem
Pythagoras’s name lives on today in
his famous theorem. It says that, in a
right-angled triangle, the square of the
hypotenuse (the longest side, opposite
the right angle) is equal to the sum of
the squares of the other two sides.
The theorem can be written
mathematically as a2 + b2 = c2.
a
a
The square of the
long side (c), the
hypotenuse, can
be made by adding
the squares of the
other two sides
(a and b).
For Pythagoras,the most perfect
shape-making
number was 10,its dots forming
a triangle known
as the tetractys.
c
b
a
b
b
b
c
c
a
The triangle’s
right angle is
opposite the
longest side,the hypotenuse.
c33
Pythagoras believed that the
Earth was at the center of a set of
spheres that made a harmonious
sound as they turned.
Dangerous numbers
Pythagoras believed that all
numbers were rational—that
they could be written as a
fraction. For example, 5 can
be written as 5?
1, and 1.5 as 3?
2.
But one of his cleverest
students, Hippasus, is said
to have proved that the
square root of 2 could not
be shown as a fraction and
was therefore irrational.
Pythagoras could not accept
this, and by some accounts
was so upset he committed
suicide. Rumor also has it
that Hippasus was drowned
for proving the existence
of irrational numbers.
Pythagoreans realized that sets of pots
of water sounded harmonious if they
were ?lled according to simple ratios.
Math and music
Pythagoras showed that musical
notes that sound harmonious
(pleasant to the ear) obey simple
mathematical rules. For example,a harmonious note can be made
by plucking two strings where one
is twice the length of the other—
in other words, where the strings
are in a ratio of 2:1.
The number legacy
Pythagoreans believed that the world
contained only ?ve regular polyhedra (solid
objects with identical ?at faces), each with
a particular number of sides, as shown here.
For them, this was proof of their idea that
numbers explained everything. This theory
lives on, as today’s scientists all explain the
world in terms of mathematics.
Cube
6 square faces
Octahedron
8 triangular faces
Icosahedron
20 triangular faces
Dodecahedron
12 pentagonal faces
Pythagoras was one of the ?rst to propose
the idea that the Earth may be a sphere.
Tetrahedron
4 triangular faces34
BRAIN GAMES
Some problems can’t be
solved by working through
them step-by-step, and need
to be looked at in a different
way—sometimes we can
simply “see” the answer. This
intuitive way of ?guring things
out is one of the most dif?cult
parts of the brain’s workings
to explain. Sometimes, seeing
an answer is easier if you try
to approach the problem in an
unusual way—this is called
lateral thinking.
THINKING OUTSIDE
THE BOX
5. In the money
You have two identical money
bags. One is ?lled with small
coins. The other is ?lled with
coins that are twice the size
and value of the others. Which
of the bags is worth more?
6. How many?
If 10 children can eat 10
bananas in 10 minutes,how many children would
be needed to eat 100
bananas in 100 minutes?
1. Changing places
You are running in a race and
overtake the person in second place.
What position are you in now?
8. The lonely man
There was a man who never left his
house. The only visitor he had was
someone delivering supplies every two
weeks. One dark and stormy night, he
lost control of his senses, turned off
all the lights, and went to sleep. The
next morning it was discovered that
his actions had resulted in the deaths
of several people. Why?
7. Left or right?
A left-handed glove can be
changed into a right-handed
one by looking at it in a mirror.
Can you think of another way?
4. Sister act
A mother and father have
two daughters who were
born on the same day of
the same month of the
same year, but are not
twins. How are they
related to each other?
2. Pop!
How can you stick
10 pins into a balloon
without popping it?
3. What are the odds?
You meet a mother with two children. She
tells you that one of them is a boy. What is
the probability that the other is also a boy?35
15. Leave it to them
Some children are raking leaves in their
street. They gather seven piles at one house,four piles at another, and ?ve piles at
another. When the children put all the piles
together, how many will they have?
12. Whodunnit?
Acting on an anonymous phone call, the police raid
a house to arrest a suspected murderer. They don’t
know what he looks like but they know his name is
John and that he is inside the house. Inside they ?nd
a carpenter, a truck driver, a mechanic, and a ?reman
playing poker. Without hesitation or communication of
any kind, they immediately arrest the ?reman. How do
they know they have their man?
11. At a loss
A man buys sacks of rice
for 1 a pound from
American farmers and then
sells them for 0.05 a pound
in India. As a result, he
becomes a millionaire. How?
14. Crash!
A plane takes off from London headed
for Japan. After a few hours there is an
engine malfunction and the plane
crashes on the ItalianSwiss border.
Where do they bury the survivors?
10. Half full
Three of the glasses below are ?lled with orange
juice and the other three are empty. By touching
just one glass, can you arrange it so that the full
and empty glasses alternate?
13. Frozen!
You are trapped in a cabin on a cold snowy
mountain with the temperature falling and night
coming on. You have a matchbox containing just
a single match. You ?nd the following things in
the cabin. What do you light ?rst?
· A candle
· A gas lamp
· A windproof lantern
· A wood ?re with ?re starters
· A signal ?are to attract rescuers
9. A cut above
A New York City hairdresser recently
said that he would rather cut the hair
of three Canadians than one New
Yorker. Why would he say this?
16. Home
A man built a rectangular
house with all four sides
facing south. One morning
he looked out of the window
and spotted a bear. What
color was it?
FRAGILE36
Thousands of years ago, some Ancient
Greeks thought of numbers as having shapes,perhaps because different shapes can be made
by arranging particular numbers of objects.
Sequences of numbers can make patterns, too.
PATTERNS
NUMBER
Square numbers
If a particular number of objects can be
arranged to make a square with no gaps,that number is called a square number.
You can also make a square number by
“squaring” a number—which means
multiplying a number by itself: 1 x 1 = 1,2 x 2 = 4, 3 x 3 = 9, and so on.
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
111112 = 123454321
1111112 = 12345654321
16 objects can
be arranged to
make a 4 x 4
square.
Something odd
The ?rst ?ve square numbers
are 1, 4, 9, 16, and 25. Work
out the difference between
each pair in the sequence
(the difference between 1
and 4 is 3, for example). Write
your answers out in order.
Can you spot an odd pattern?
The magic ones
By squaring numbers made
of nothing but ones, you
can make all the other digits
appear—eventually! Stranger
still, those digits appear in
numbers that read the same
whether you look at them
forward or backward.
1 3 4
1 2
1
7 8
4 5
9
6
2 3 6 7
1 2 3
8
4
5
13 14
9 10
15
11
16
12
1 2 3
4
5
6 7 8 9 10
12 13 14 15 11
16 17 18 19 20
21 22 23 24 25
1 4 9 16 25
3 5 7 937
Prison break
It’s lights-out time at the prison, where
50 prisoners are locked in 50 cells. Not
realizing the cells’ doors are locked, a
guard comes along and turns the key to
each cell once, unlocking them all. Ten
minutes later, a second guard comes and
turns the keys of cells 2, 4, 6, and so on.
A third guard does the same, stopping
at cells 3, 6, 9, and so on. This carries on
until 50 guards have passed the cells.
How many prisoners escape? Look
out for a pattern that will give you
a shortcut to solving the problem.
Shaking hands
A group of three friends meet and
everyone shakes hands with everyone
else once. How many handshakes are
there in total? Try drawing this out, with
a dot for each person and lines between
them for handshakes. Now work out the
handshakes for groups of four, ?ve, or six
people. Can you spot a pattern?
Triangular numbers
If you can make an equilateral triangle (a triangle
with sides of equal length) from a particular
number of objects, that number is known as
triangular. You can make triangular numbers by
adding numbers that are consecutive (next to each
other): 0 + 1 = 1, 0 + 1 + 2 = 3, 0 + 1 + 2 + 3 = 6, and
so on. Many Ancient Greek mathematicians were
fascinated by triangular numbers, but we don’t
use them much today, except to admire the pattern!
Cubic numbers
If a number of objects, such as building blocks,can be assembled to make a cube shape, then that
number is called a cubic number. Cubic numbers
can also be made by multiplying a number by itself
twice. For example, 2 x 2 x 2 = 8.
1
1
3 1
2 3
6
1
2 3
4 5 6
10
1
2 3
4 5 6
7 8 9 10
15
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
A perfect solution?
The numbers 1, 2, 3, and 6 all divide
into the number 6, so we call them its
factors. A perfect number is one that’s
the sum of its factors (other than itself).
So, 1 + 2 + 3 = 6, making 6 a perfect
number. Can you ?gure out the next
perfect number?
ACTIVITY
1
8
2738
Mathematicians use all kinds of tricks
and shortcuts to reach their answers
quickly. Most can be learned easily
and are worth learning to save time
and impress your friends and teachers.
CALCULATION TIPS
To work out 9 x 9,bend down your
ninth ?nger.
Multiply by 9 with your hands
Here’s a trick that will make multiplying by 9 a breeze.
Step 1
Hold your hands face up in front of you. Find out
what number you need to multiply by 9 and bend the
corresponding ?nger. So to work out 9 x 9, turn down
your ninth ?nger.
Step 2
Take the number of ?ngers on the left side of the
bent ?nger, and combine (not add) it with the one on
the right. For example, if you bent your ninth ?nger,you’d combine the number of ?ngers on the left, 8,with the number of ?ngers on the right, 1. So you’d
have 81 (9 x 9 is 81).
1
2 3
4
5 6
7 8
9
10
Alex Lemaire
With plenty of practice, people can solve
amazing math problems without a
calculator. In 2007, French mathematician
Alex Lemaire worked out the number that,if multiplied by itself 13 times, gives a
particular 200 digit number. He gave the
correct answer in 70 seconds!
BRAIN GAMES
Multiplication tips
Mastering your times tables is an essential math
skill, but these tips will also help you out in a pinch:
· To quickly multiply by 4, simply double
the number, and then double it again.
· If you have to multiply a number by 5,?nd the answer by halving the number and then
multiplying it by 10. So 24 x 5 would be 24 ÷ 2 = 12,then 12 x 10 = 120.
· An easy way to multiply a number by 11 is
to take the number, multiply it by 10, and then
add the original number once more.
· To multiply large numbers when one is even, halve
the even number and double the other one. Repeat if
the halved number is still even. So, 32 x 125 is the
same as 16 x 250, which is the same as 8 x 500, which
is the same as 4 x 1,000. They all equal 4,000. 39
Division tips
There are lots of tips that can help
speed up your division:
· To ?nd out if a number is divisible by 3,add up the digits. If they add up to a multiple
of 3, the number will be divisible by 3.
For instance, 5,394 must be divisible
by 3 because 5 + 3 + 9 + 4 = 21,and 21 is divisible by 3.
· A number is divisible by 6
if it’s divisible by 3 and the
last digit is even.
· A number is divisible by 9 if all the digits add up to
a multiple of 9. For instance, 201,915 must be divisible
by 9 because 2 + 0 + 1 + 9 + 1 + 5 = 18, and 18 is
divisible by 9.
· To ?nd out if a number is divisible by 11, start with the
digit on the left, subtract the next digit from it, then add the
next, subtract the next, and so on. If the answer is 0 or 11,then the original number is divisible by 11. For example,35,706 is divisible by 11 because 3 – 5 + 7 – 0 + 6 = 11.
Beat the clock
Test your powers of mental arithmetic in this game against
the clock. It’s more fun if you play with a group of friends.
Step 1
First, one of you must choose two of the following numbers: 25, 50, 75, 100.
Next, someone else selects four numbers between 1 and 10. Now get
a friend to pick a number between
100 and 999. Write this down next
to the six smaller numbers.
Step 2
You all now have two minutes to
add, subtract, multiply, or divide
your chosen numbers—which you
can use only once—to get as close
as possible to the big number. The
winner is the person with the
exact or closest number.
Calculating a tip
If you need to leave a 15 percent tip after
a meal at a restaurant, here’s an easy
shortcut: Just work out 10 percent (divide
the number by 10), then add that number to
half its value, and you have your answer.
Fast squaring
If you need to square a two-digit
number that ends in 5, just multiply
the ?rst digit by itself plus 1, then
put 25 on the end. So to square
15, do: 1 x (1 + 1) = 2 , then attach
25 to give 225. This is how you can
work out the square of 25:
2 x (2+1) = 6
6 and 25 = 625
In Asia, children use an abacus
(a frame of bars of beads) to
add and subtract faster than
an electronic calculator.
10% of 35 = 3.50
3.50 ÷ 2 = 1.75
3.50 + 1.75 = 5.2540
Ingenious inventions
Archimedes is credited with building the
world’s ?rst planetarium—a machine that
shows the motions of the Sun, Moon, and
planets. One thing he didn’t invent, despite it
bearing his name, is the Archimedean screw.
It is more likely that he introduced this design
for a water pump, having seen it in Egypt.
Eureka!
Archimedes’ most famous discovery came
about when the king asked him to check if
his crown was pure gold. To answer this, he
had to measure the crown’s volume, but
how? Stepping into a full bath, Archimedes
realized that the water that spilled from the
tub could be measured to ?nd out the volume
of his body—or a crown.
Early life
Archimedes was born in Syracuse, Sicily,in 287 BCE. As a young man he traveled to
Egypt and worked with mathematicians there.
According to one story, when Archimedes
returned home to Syracuse, he heard that the
Egyptian mathematicians were claiming some
of his discoveries as their own. To catch them,he sent them some calculations with errors in
them. The Egyptians claimed these new
discoveries too, but were caught when people
discovered that the calculations were wrong.
An Archimedean screw is
a cylinder with a screw
inside. The screw raises
water as it turns.
Archimedes
Archimedes once said, “Give me a lever long
enough... and I shall move the world.”
Archimedes was probably the greatest
mathematician of the ancient world. Unlike
most of the others, he was a highly practical
person too, using his math skills to build
all kinds of contraptions, including some
extraordinary war machines.
On discovering how to measure volume, Arch ......
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