Numbers

Further than you think

How Far Away Is the Moon?

The Moon is 384,000 km away — far enough to fit every other planet in between.

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Bigger than your brain wants

How Big Is the Sun?

The Sun is so big that a million Earths would fit inside it.

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One kind of atom, all the way through

Why Is There a Table of Everything?

An element is stuff made of just one kind of atom, and the whole world is built from about ninety of them.

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Putting together

What Does Adding Do?

Adding puts two groups together into one bigger total.

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Levelling everyone off

What Does an Average Actually Tell You?

An average is what everyone would get if the total were shared equally, which can hide a lot.

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A thousand times, not a bit

How Much Bigger Is a Billion Than a Million?

A billion is a thousand times a million, and turning both into seconds shows how vast that gap is.

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On and off is enough

How Do Computers Count With Only Two Digits?

Computers count in binary because a switch has just two states, and place value does the rest.

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The small things add up first

Why Does the Money Run Out Before the Month?

Budgets go wrong because small, frequent spending adds up faster than the large purchases you notice.

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More orders than atoms nearby

Has Anyone Ever Had This Hand Before?

A pack of cards has so many possible orders that a well-shuffled deck has probably never existed before.

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Objects have no memory

Why Does a Coin Not Owe You Heads?

Chance has no memory: past results cannot influence a fair coin or die on its next throw.

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Counting up, not taking away

How Do You Work Out Your Change?

Change is easiest worked out by counting up from the price to the note you handed over.

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Interest on the interest

Why Does Saved Money Grow Faster Each Year?

Compound interest pays you on last year's interest as well, so growth speeds up every year.

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Two numbers, one place

How Do You Describe Exactly One Square?

Coordinates pin down a place with two numbers: how far across, then how far up.

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One thing, one word

How Does Counting Work?

Counting works by pairing each object with one number word, in a fixed order.

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Count a little, multiply a lot

How Do You Count a Huge Crowd?

Crowds are counted by counting one small area and multiplying by the number of areas.

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More ways to get there

Why Is Seven the Commonest Roll?

Seven comes up most on two dice because more combinations add up to it than to any other total.

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Taking a slice off the top

How Do Shops Work Out a Sale Price?

A discount takes a percentage off the old price, so the saving depends on what the price was.

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Fair shares

How Does Division Work?

Division shares a total into equal groups and tells you how much each group gets.

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Two of the same

What Does Doubling Mean?

Doubling means adding a number to itself, and it grows shockingly fast.

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Balance, not 'the answer'

What Does the Equals Sign Really Mean?

The equals sign means both sides balance, which is why 7 = 4 + 3 is perfectly correct.

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Close enough on purpose

What Is Estimating?

Estimating gives you a fast, roughly right answer instead of a slow exact one.

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Bigger bottom, smaller piece

Why Is a Third Bigger Than a Quarter?

A third beats a quarter because the bottom number counts cuts, and more cuts make each piece smaller.

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Seeing beats reading

Why Draw Numbers as a Picture?

Graphs work because eyes compare lengths and heights far faster than they read columns of numbers.

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Comparing

Which Number Is Bigger?

A bigger number is simply one you reach later when counting.

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Small steps that stop being small

Why Does Doubling Get Out of Hand?

Repeated doubling grows slowly at first and then explodes, because each step is bigger than everything before it.

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Two equal shares

What Does Half Mean?

A half is one of two pieces that are exactly the same size.

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Never running out

What Is Infinity?

Infinity means there is no end — you can always add one more.

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Everything divided by the same number

How Does a Map Fit a Country on a Page?

A map shrinks every distance by the same factor, which is what keeps the shapes right.

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Everyone agreeing

Why Do We Measure With Units?

Measuring means counting how many agreed units something is.

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One giant number spoils it

Why Is the Average Sometimes a Lie?

An average is dragged by extreme values, while the middle value ignores them.

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Making any amount

Why Do Coins Have Different Values?

Coin values are chosen so you can make any amount with only a few coins.

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Many equal groups

What Does Multiplying Do?

Multiplying is repeated adding of equal groups, done in a single step.

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The line carries on

How Can a Number Be Less Than Nothing?

Negative numbers exist because the number line does not stop at zero; it carries straight on.

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Numbers as places

What Is a Number Line?

A number line turns numbers into positions, so arithmetic becomes movement.

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Can everyone find a partner?

Odd and Even Numbers

Even numbers split into pairs with none left over; odd numbers always leave one.

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Something that repeats

What Is a Pattern?

A pattern is a repeating rule, and finding the rule lets you predict what comes next.

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Out of a hundred, always

What Does Per Cent Actually Mean?

Per cent means out of a hundred, which lets you compare quantities of different sizes.

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Every circle's hidden number

What Is Pi and Why Does It Never End?

Pi is the number of diameters that fit around any circle, and it is the same for all of them.

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Position carries meaning

Why Does 25 Mean Twenty-Five?

In our number system, where a digit sits decides how much it is worth.

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No rectangle but a line

Which Numbers Refuse to Split?

Primes are numbers that cannot be split into equal groups, so they only ever form one long row.

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One of four

What Is a Quarter?

A quarter is one of four exactly equal shares of a whole.

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You are usually in the slow one

Why Is the Other Queue Always Faster?

With several queues, the odds are against yours being the quickest — and one shared queue fixes it.

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Two out of a million is still nearly none

Why Does Buying Two Tickets Barely Help?

Doubling your tickets doubles a chance so small that twice it is still almost nothing.

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Position is not distance

Why Do Rankings Change When Nothing Does?

A ranking keeps only the order, discarding how far apart the things really are.

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The bit left over

What Is a Remainder?

A remainder is what is left after making as many equal groups as possible.

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Balance beat arithmetic

Why Do Clocks Sometimes Say IIII?

Clocks use IIII instead of IV because the heavier shape balances VIII across the dial.

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Close enough, on purpose

Why Do We Round Numbers?

Rounding deliberately gives up small accuracy to get a number that is easier to think with.

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Length halves, weight drops by eight

Why Is a Model Never Quite Right?

Scaling changes length, area and volume by different factors, so models never behave like the real thing.

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Faster than one at a time

Counting in Jumps

Skip counting means counting in equal jumps instead of one at a time.

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A third direction

What Makes a Shape 3D?

A solid shape adds a third direction to a flat one, which is what gives it faces, edges and corners.

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Order makes finding easy

Why Do We Sort Things?

Sorting things into order makes finding any one of them far faster.

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Sharing, with an awkward remainder

How Do You Split a Bill Fairly?

Splitting a bill is division until there is a remainder, and then it is a decision.

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What is left

What Does Taking Away Do?

Subtraction finds what is left, or how far apart two numbers are.

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Counting without losing your place

Why Count in Groups of Five?

Tally marks bundle counts into fives so you can add up groups instead of recounting one by one.

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Sharing between three

What Is a Third?

A third is one of three exactly equal shares — and it never divides evenly in decimals.

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Shortcuts, not memory

What Patterns Hide in the Times Tables?

The times tables hide patterns that turn dozens of separate facts into a handful of shortcuts.

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Ten per cent, then adjust

How Do People Work Out a Tip in Their Head?

A tip is worked out by finding ten per cent and adjusting up or down, not by exact sums.

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The number under the price

Which Packet Is Actually Cheaper?

Unit price divides cost by amount so packets of different sizes can be compared.

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Size and heaviness are separate

Why Is a Big Thing Not Always Heavy?

Big does not mean heavy, because weight depends on how tightly matter is packed, not on size.

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A number your eye reads first

Why Does Everything Cost £4.99?

Prices end in 99 because people read the first digit and round the rest away.

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A number for nothing

What Is Zero?

Zero is the number that counts nothing, and it makes big numbers possible.

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One-ninth showing

Why Do Icebergs Float So Low?

Ice is only slightly lighter than water, so about nine-tenths of an iceberg is submerged.

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Notes that share their wobbles

Why Does Singing Together Sound Good?

Notes sound good together when their vibrations fit in simple whole-number ratios, so the waves line up rather than clash.

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Millions of tiny corrections

How Does a Computer Learn?

A computer learns by guessing, being told how wrong it was, and nudging its own numbers — millions of times over.

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Guessing is done by machine

Why Do Passwords Have to Be Long?

Passwords must be long because computers guess billions per second, and each extra character multiplies the work.

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