Math

Every circle's hidden number

What Is Pi and Why Does It Never End?

Fifteen decimal places would size a circle as wide as the solar system to within a finger's width, which is rather more than anyone has ever needed.

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Never running out

What Is Infinity?

Infinity is not a huge number — it is the fact that counting never has to stop. Infinitely many fractions fit between 0 and 1, with room to spare.

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A 50.7 per cent chance

In a Room of Twenty-Three, Two People Share a Birthday

With 23 people the odds of a shared birthday pass one half, because 23 people make 253 pairs — and it is the pairs, not the people, that get the chances.

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A number your eye reads first

Why Does Everything Cost £4.99?

People read the leftmost digit first and round the rest away, so a price of 4.99 gets quietly filed in memory as four. Shops have known this for a century.

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You are usually in the slow one

Why Is the Other Queue Always Faster?

With three tills, two queues are beating yours most of the time. One shared queue feeding every till makes the wait both fairer and shorter.

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Proved in 1976, by machine

Four Colours Are Always Enough

Four colours suffice for every possible map — a claim that resisted proof for over a century, then fell to a computer checking 1,936 cases no human could.

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Putting together

What Does Adding Do?

Adding only gathers — nothing is created. The order makes no difference, which halves what anyone has to memorise, and hard sums split into friendly steps.

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A recipe with no judgement

What Makes Something an Algorithm?

An algorithm is a list of steps precise enough to need no judgement: every decision definite, every step doable, and the whole thing finishes.

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Turn, not length

What Is an Angle Actually Measuring?

An angle measures how far you turn between two directions, which is why stretching the arms out longer does not make the angle any bigger.

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Counting squares, quickly

Why Do We Multiply to Find Area?

Area counts the squares covering a shape, and multiplying rows by columns is a fast way of counting them. Double both sides and you get four times as many.

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Counting without counting

Why Are Rows and Columns Useful?

Lay things out in rows and columns and counting one by one becomes multiplying. The grid also shows why the order of the two numbers cannot matter.

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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 out evenly. It is a fact about a group rather than about any one member of it.

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A thousand times, not a bit

How Much Bigger Is a Billion Than a Million?

A billion is a thousand millions. Count them in seconds and the gap stops being abstract: a million is eleven days, a billion is thirty-two years.

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On and off is enough

How Do Computers Count With Only Two Digits?

A switch is only ever off or on, so computers count with two digits. Place value that doubles each column can still write any number you like.

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The small things add up first

Why Does the Money Run Out Before the Month?

Budgets fail on the small, frequent spending rather than the big purchases you remember — so the number that matters is price times how often.

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Two clocks that do not match

Why Does a Year Have 365 Days?

A year is one orbit and a day is one spin, and neither divides neatly into the other. The leftover quarter of a day is what leap years exist to mop up.

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More orders than atoms nearby

Has Anyone Ever Had This Hand Before?

Adding the numbers one to fifty-two gives a few thousand; multiplying them fills sixty-eight digits. It takes seven riffles to reach that properly.

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Objects have no memory

Why Does a Coin Not Owe You Heads?

A coin has no memory, so a long run of tails changes nothing at all about the next throw. The coin does not owe you anything and never did.

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Counting up, not taking away

How Do You Work Out Your Change?

Your change is not a distance but a sum of hops counted forwards from the price. That is why the small coins come out first — the order the hops came in.

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Two hands, one set of numbers

Why Does the 3 on a Clock Mean Both 3 and 15?

The twelve printed numbers belong to the hour hand alone. The minute hand is counting sixty little marks in between, which is why 3 means 15.

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Interest on the interest

Why Does Saved Money Grow Faster Each Year?

Interest is added to your money, and next year you earn interest on that too. The curve accelerates — and it works exactly as hard against you on debt.

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Two numbers, one place

How Do You Describe Exactly One Square?

Two numbers in an agreed order, measured from a shared corner, name one exact place and no other — how far across, then how far up.

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One thing, one word

How Does Counting Work?

Counting has three rules: one word per object, always the same order, and the last word is the answer. Young children reliably miss the third one.

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Count a little, multiply a lot

How Do You Count a Huge Crowd?

Any count taking an hour is counting a different crowd by the end, so one square is counted and the rest is measured. All the error lives in that square.

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More ways to get there

Why Is Seven the Commonest Roll?

Two dice do not make every total equally likely. Six different pairs add up to seven, while two and twelve have exactly one route each.

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Taking a slice off the top

How Do Shops Work Out a Sale Price?

A discount is a percentage of the old price, so the same thirty per cent off saves wildly different amounts — and two discounts never simply add up.

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Fair shares

How Does Division Work?

Division takes one pile and shares it into equal groups, and the answer it hands back is the size of a single group rather than the number of groups.

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Two of the same

What Does Doubling Mean?

Doubling adds a copy rather than a fixed amount, so it grows far faster than anyone expects: ten doublings of 1 reach 1,024, and thirty pass a billion.

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Balance, not 'the answer'

What Does the Equals Sign Really Mean?

Equals does not mean an answer comes next. It means both sides are worth the same, which is why 7 = 4 + 3 is perfectly correct written that way round.

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Close enough on purpose

What Is Estimating?

An estimate trades precision for speed, and it is often the better answer — as well as the check that catches an exact calculation gone badly wrong.

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Bigger bottom, smaller piece

Why Is a Third Bigger Than a Quarter?

The bottom number counts how many pieces the whole was cut into, so a bigger bottom means smaller pieces, and that is why a third beats a quarter.

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Seeing beats reading

Why Draw Numbers as a Picture?

Eyes compare lengths far faster than they read columns, so a graph turns numbers into heights — as long as the axis honestly starts at zero.

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Comparing

Which Number Is Bigger?

A number is bigger if you reach it later while counting — further right on the number line, whatever the number happens to look like.

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Small steps that stop being small

Why Does Doubling Get Out of Hand?

Ten doublings multiply anything by roughly a thousand, which is why one grain per chessboard square ends as more rice than the world has grown.

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Two equal shares

What Does Half Mean?

A half is one of two pieces that match exactly. The shape can be anything — straight, wiggly, striped — as long as the amount is the same.

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Eratosthenes, about 240 BC

How We Measured the Earth With a Stick

Two noon shadows and one long walk gave Eratosthenes the size of the planet, to within a few per cent, without his ever leaving Egypt.

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It reaches past the Sun

What If You Folded Paper Fifty Times?

Every fold doubles the thickness, so fifty folds of ordinary paper would reach three quarters of the way from here to the Sun.

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The catalogue is not the shelf

How Does a Library Find One Book?

Shelves cannot be sorted by subject and by author at once, so a library keeps a small separate list of pointers that sends you to one exact spot.

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Everything divided by the same number

How Does a Map Fit a Country on a Page?

Halve a map's scale and it takes a quarter of the paper, not half — and the wording runs backwards: a small-scale map is the one covering a country.

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Everyone agreeing

Why Do We Measure With Units?

Measuring is counting agreed, equal-sized steps. The agreement is the whole point: without it, the number you end up with tells nobody anything.

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One giant number spoils it

Why Is the Average Sometimes a Lie?

One extreme value drags an average a long way and leaves the middle value untouched. Which of the two is honest depends on the question being asked.

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Making any amount

Why Do Coins Have Different Values?

Coin values follow a 1-2-5 pattern so that a handful can build any price out of very few coins. Their worth comes entirely from agreement.

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Many equal groups

What Does Multiplying Do?

Multiplication is adding the same number over and over, compressed into a single move — which is exactly why the times tables are worth knowing by heart.

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The line carries on

How Can a Number Be Less Than Nothing?

Negative numbers are just the number line carrying on past zero instead of stopping there. The minus sign says which side of zero you are standing on.

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Numbers as places

What Is a Number Line?

Lay the numbers out in order and arithmetic turns into walking. Adding steps right, subtracting steps left, and suddenly negatives have somewhere to live.

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Can everyone find a partner?

Odd and Even Numbers

Pairing is the whole test: what pairs up is even, what strands one is odd. It is also why odd plus odd is even — the two lonely ones find each other.

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Something that repeats

What Is a Pattern?

A pattern is a rule that keeps going. Name the rule and you can say what comes next without doing any of the work of getting there.

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Out of a hundred, always

What Does Per Cent Actually Mean?

Half a class of eight and half a school of eight hundred are the same share and wildly different numbers of children. The hundred is just a yardstick.

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The fence and the field

How Are Area and Perimeter Different?

One is a length in plain metres and the other a count of squares, so the very same fence can enclose wildly different amounts of field.

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Position carries meaning

Why Does 25 Mean Twenty-Five?

A digit's worth depends on where it sits, not on what it looks like. That one idea is why ten symbols are enough to write any number at all.

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No rectangle but a line

Which Numbers Refuse to Split?

Twelve always breaks down to two, two and three, however you start. Primes thin out as numbers grow but never stop — proved two thousand years ago.

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One of four

What Is a Quarter?

A quarter is one of four equal shares, which makes it half of a half. Four unequal pieces contain no quarter at all, whatever anyone calls them.

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Two out of a million is still nearly none

Why Does Buying Two Tickets Barely Help?

Doubling a tiny chance leaves you with a tiny chance, twice over. And somebody winning is an entirely different claim from you winning.

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Position is not distance

Why Do Rankings Change When Nothing Does?

A ranking keeps the order and throws the gaps away, so a hair's difference can flip two positions while an enormous gap disappears without trace.

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Not everything scales

How Do You Double a Recipe?

Doubling a recipe doubles the ingredients and almost nothing else. The pan, the time and the temperature all have to be reconsidered separately.

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The bit left over

What Is a Remainder?

Make as many equal groups as you can and the bit that will not fit is the remainder. It is not a mistake — it is the honest answer to an uneven split.

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Balance beat arithmetic

Why Do Clocks Sometimes Say IIII?

Clock faces often use IIII rather than the correct IV, because four heavy strokes balance the VIII sitting opposite. On a dial, looks beat the rule.

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Close enough, on purpose

Why Do We Round Numbers?

Rounding deliberately gives up a little accuracy to get a number you can actually hold in your head. Do it at the end of a sum, never at the start.

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Length halves, weight drops by eight

Why Is a Model Never Quite Right?

Halving a length quarters the area and cuts the volume to an eighth, which is why a model never quite behaves like the thing it copies.

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Sorted by rules

Why Do Shapes Have Names?

A shape's name is a checklist, not a look. Any closed figure with three straight sides is a triangle, and each extra rule shrinks the family.

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Faster than one at a time

Counting in Jumps

Counting in twos, fives or tens moves in equal jumps instead of single steps. It gets you there faster, and it quietly builds the times tables.

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A third direction

What Makes a Shape 3D?

Depth is what lets a shape hold space rather than merely cover it, and with that third direction come faces, edges and corners.

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Order makes finding easy

Why Do We Sort Things?

Unsorted, finding one thing among a thousand costs 500 checks on average. Sorted, halving the problem at every look gets you there in ten.

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One triangle, three questions

How Long Will the Journey Take?

Speed, distance and time are one relationship asked three different ways. Rearrange it for whichever you are missing, and make sure the units agree first.

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Sharing, with an awkward remainder

How Do You Split a Bill Fairly?

Splitting a bill is straightforward division until the remainder appears. At that point it stops being arithmetic and becomes a question about people.

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Two matching halves

What Is Symmetry?

A shape is symmetrical if folding or turning it lands it back on itself. A square has four lines of symmetry, a circle infinitely many, a real face none.

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What is left

What Does Taking Away Do?

Subtraction does two jobs with one operation. It finds what is left after some is taken away, and it measures the gap between two numbers.

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Counting without losing your place

Why Count in Groups of Five?

Tally marks bundle counts into fives so that you record one event at a time and add up groups at the end, instead of recounting from the beginning.

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A scale built around water

Why Does Water Freeze at Zero?

A scale needs two events anyone can reproduce, so Celsius took water freezing and boiling and split the gap into a hundred convenient steps.

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Sharing between three

What Is a Third?

A third is one of three exactly equal shares: bigger than a quarter, and the first fraction most people meet that cannot be written exactly as a decimal.

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Noon cannot be everywhere at once

Why Is It a Different Day Somewhere Else?

Clocks are set by the Sun and the Sun is only overhead in one place at a time, so the planet's turn is cut into stripes an hour wide.

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Shortcuts, not memory

What Patterns Hide in the Times Tables?

Four times six is a rectangle of counters, not a phrase to recite — and every answer in the nine row has digits that add up to nine.

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Ten per cent, then adjust

How Do People Work Out a Tip in Their Head?

Nobody calculates a tip exactly. You move the decimal point one place to get ten per cent, build the rest from that, and round to something friendly.

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The number under the price

Which Packet Is Actually Cheaper?

Divide the price by the amount and two packets become comparable at last. The catch is that the cheaper unit price only counts if you actually use it all.

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Stacking layers of squares

How Much Does a Box Actually Hold?

Volume counts the cubes that would fill a space. Tile one layer, then stack that layer as many times as the height allows, and you have the answer.

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Size and heaviness are separate

Why Is a Big Thing Not Always Heavy?

Big does not mean heavy. Weight depends on how tightly matter is packed into a space, not on how much space the thing happens to take up.

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A number for nothing

What Is Zero?

Zero counts what is not there, and it also holds an empty place in a written number. Without that second job, 105 and 15 would look exactly the same.

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