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

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    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 ......