Measuring

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 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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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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Pressure from below

Why Does Water Push Up?

Water pressure is stronger the deeper you go, so the push on an object's underside beats the push on its top. That leftover push is what holds it up.

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Heavy for its size

What Is Density?

Density is not weight but how much weight is packed into a given space. It is what decides whether a thing floats, sinks, or rises through the air.

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Waves bunched and stretched

Why Does a Siren Change Pitch as It Passes?

To the driver the siren never changes at all. Light does the same thing, which is how a receding galaxy shifts red and betrays an expanding universe.

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Water pushes back

Why Do Things Float?

You float when the water you shove aside weighs more than you do. That is why a steel hull floats where a solid steel lump sinks straight to the bottom.

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The pull you never notice

What Is Gravity?

Every object with mass pulls on every other one. Earth simply has enough mass for its pull to dominate everything you have ever felt.

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Everything is shaking

What Is Heat?

Heat is not a substance you add. It is how fast the atoms are jiggling, which is why a thermometer is really a gauge of atomic speed.

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Rømer, 1676

How We First Timed Light

Light was thought to be instant until Jupiter's moon kept running late, arriving behind schedule whenever Earth was further away.

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Cavendish, 1798

How Anyone Weighed the Earth

Henry Cavendish weighed the Earth by measuring the faint pull between four lead balls, in a sealed shed he watched through a telescope from outside.

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The answer is: nothing

What If Everyone Jumped at Once?

If everyone on Earth jumped together, the planet would shift by less than the width of an atom — and then shift straight back.

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You would come back

What If You Fell Through a Hole in the Earth?

A hole through the Earth is a pendulum. You speed up to the centre, slow down to the far side, and it takes 42 minutes each way.

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Trading distance for force

How Do Levers Work?

A lever lets a small push move a big load, by making you push a long way to shift it a short one. Force and distance always trade off against each other.

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How big the push is

What Makes a Sound Loud?

Loudness is how hard the air gets shoved; pitch is how often. The two are entirely independent, which is why a whisper and a shout can hit the same note.

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Mass times speed

Why Are Heavy Things Hard to Stop?

Nothing is lost when things collide: what one object loses another gains. Airbags, crash mats and bent knees simply make the stop take longer.

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Counting wiggles

What Makes a Sound High or Low?

Fast vibrations sound high and slow ones sound low. That is the whole of pitch — a count of wiggles per second, and nothing more complicated.

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Force, spread out

What Is Pressure?

The same push feels utterly different depending on the area it lands on. Shrink the area and you have a drawing pin; spread it out and you have a snowshoe.

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Sharing the load

How Do Pulleys Help?

Run the rope through more pulleys and the load is shared across more sections, so each pull is lighter. You pay for it by pulling a great deal more rope.

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Fifteen times faster through steel

Sound Goes Faster Through a Wall Than Through Air

Sound travels by particles shoving their neighbours, so the stiffer the material the faster the shove passes — steel beats water, and water beats air.

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A skin made of pulling

Why Does a Paperclip Float on Water?

Water molecules cling to one another so strongly that the surface behaves like a stretched skin, and a paperclip simply rests in the dent.

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Closer to the sun, and colder

Why Is It Colder Up a Mountain?

Climbing takes you nearer the Sun and makes you colder, because air is warmed from the ground up and cools again as it rises and expands.

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Weighing the sky

How Does a Barometer Work?

A barometer weighs the column of air above it, and that weight drops before a storm and rises before fine weather — hours ahead of the sky.

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An argument about very large numbers

Every Breath You Take Has Been Breathed Before

A single breath holds more molecules than the atmosphere holds breaths — so an old one, well mixed, is almost certainly inside the breath you just took.

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One day, or thirty years

What's the Difference Between Weather and Climate?

Weather is what happened today; climate is what usually happens over decades — which is why one cold week proves nothing either way.

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The temperature where air gives up

What Is the Dew Point?

The dew point is the temperature at which air is completely full of water. Chill anything below it and the invisible vapour has to become visible drops.

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A bump of one part in seven hundred

Shrink Earth to a Marble and You Could Not Feel Everest

The tallest mountain on Earth is a rounding error against the planet's radius, which is why a scale model of Earth would feel smooth in your hand.

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The line the sunlight hits squarely

What Is the Equator?

The equator is the line halfway between the poles, where sunlight lands most directly and days stay the same length all year.

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Atoms that tick

How We Know How Old a Rock Is

Certain atoms turn into other atoms at an utterly fixed rate, so the ratio between them in a rock says how long it has been sitting there.

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Nobody has been below 12 km

How We Know What Is Inside the Earth

Earthquake waves travel through the planet, and one kind of wave never comes out the far side — which is how we know the core is liquid.

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One-ninth showing

Why Do Icebergs Float So Low?

Ice is only slightly less dense than seawater, so an iceberg floats with roughly eight-ninths of itself hidden below the surface and out of sight.

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Most of it would change nothing

What If All the Ice Melted?

Melting sea ice raises the sea by nothing at all. Everything depends on the ice sitting on land — and almost all of that is in one place.

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Two numbers for anywhere on Earth

How Do You Say Exactly Where You Are?

Any point on the planet can be named with two angles, and longitude was far harder than latitude: it had to wait for an accurate clock.

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Everything that drains to one place

Where Does a River's Water Come From?

A river basin is all the land draining into one river, and every point on Earth belongs to exactly one — decided by ridges, not by borders.

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Counting between flash and bang

How Far Away Is the Storm?

Lightning and thunder leave together, but light arrives instantly and sound takes three seconds a kilometre. Count the gap and divide by three.

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Lines, triangles and half-circles

How Do You Read a Weather Map?

The lines join places of equal pressure. Crowded lines mean wind, highs mean settled air, lows mean rain, and fronts mark where air masses collide.

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Named for where it starts

Where Does the Wind Come From?

A north wind blows from the north, and it brings whatever the north was like. Naming wind for its origin is really naming the air that is arriving.

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Watching for a tiny dip

How Do We Find Planets Around Other Stars?

Almost no planet round another star has been seen. They are found by the tiny, precisely repeating dip in starlight as the planet crosses in front.

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An island of stars

What Is a Galaxy?

We see the Milky Way as a band across the sky because we are sitting inside its flat disc, looking the long way along it from out on one arm.

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Yes, lots

Is There Gravity in Space?

Gravity never switches off, it only gets gentler with distance, and it reaches everywhere. Space is full of it — which is why everything there is orbiting.

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The missing wobble

How We Proved the Earth Moves

The best argument against a moving Earth was that the stars never shifted — a good argument, until telescopes got precise enough to catch the shift.

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A distance, not a time

What Is a Light Year?

A light year sounds like a duration and is really a distance: the 9.5 trillion kilometres light covers in a year. Every glance up looks back in time.

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Further than you think

How Far Away Is the Moon?

The Moon sits about 384,000 km away — a gap wide enough to line up every other planet in the solar system, side by side, with room to spare.

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Too far away to fall behind

Why Does the Moon Follow the Car?

A hedge sweeps past in a second and a far hill barely stirs. The Sun, the planets and every star hold their place on a journey for the same reason.

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A city that weighs a sun

What Is Left After the Explosion?

A supernova leaves its crushed core behind: a ball the width of a city, holding more material than the Sun, spinning hundreds of times a second.

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A definition, not a demotion

Why Is Pluto Not a Planet Any More?

Finding Eris, heavier than Pluto, forced a choice: name a tenth planet or redraw the line. Pluto now leads a category instead of trailing one.

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Nine times wider than Earth

Saturn Would Float in a Big Enough Bath

Saturn is enormously heavy and enormously light at the same time, and the difference between those two words is the whole of this page.

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The Sun is not the big one

How Big Do Stars Get?

Stars run from smaller than Earth to wide enough to swallow Jupiter's orbit. Our Sun, enormous next to our planet, sits unremarkably in the middle.

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Bigger than your brain wants

How Big Is the Sun?

109 Earths would span the Sun's face and 1.3 million would fill it. It holds 99.8% of everything in the solar system, and among stars it is ordinary.

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Pre-built, then inflated

How Does Bamboo Grow So Fast?

Bamboo grows a metre a day because the whole stem is built underground in advance, then simply pumped full of water and stretched out.

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Normal, but not this fast

Is Extinction Normal?

Nearly every kind of animal that has ever lived is already extinct, so the alarming measurement today is the rate of it, not the fact of it.

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The food is the small part

What Gets Wasted When Food Is Thrown Away?

Throwing away a sandwich wastes the field, the water, the fuel and the year behind it — and about a third of all food grown is never eaten.

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Cartilage turning into bone

How Do You Grow Taller?

Bones lengthen at soft cartilage zones near their ends, mostly during deep sleep, and when those zones harden for good your height is settled for life.

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A name that is probably wrong

Do Growing Pains Really Come From Growing?

Bones grow a few micrometres a day, which nobody could possibly feel — so growing pains are real aches with a badly misleading name.

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Two wards, two death rates

How We Found Out Germs Exist

One ward killed five times as many mothers as the other, and handwashing fixed it twenty years before anyone could name what was washed off.

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Harvey, 1628

How We Found Out Blood Goes Round

William Harvey proved blood circulates by multiplying: the heart pumps far more blood in an hour than the body could possibly make.

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It does not, and that is the problem

How Does a Pill Know Where to Go?

A swallowed pill goes everywhere in your body and works only where its shape fits something, which is also exactly why side effects happen.

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A saltiness detector

How Does Your Body Know You Need a Drink?

Thirst is not a dry mouth. A small patch of brain is measuring how salty your blood has become, and it switches off the moment water is on its way down.

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One line a year

Why Does a Tree Have Rings?

A tree adds one ring a year, with a dark line for the slow late growth. The widths record what the weather was doing long before anyone wrote it down.

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A bag that reports on itself

Why Can't You Just Hold It Forever?

Two tubes trickle into it continuously, the first polite signal arrives at about a third full, and cold weather really does make you produce more.

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A grain that drinks

Why Does Rice Need Exactly the Right Water?

Rice does not boil soft. Each grain absorbs a fixed amount of water and then stops, which is why the measure matters far more than the time does.

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Thickness, tightness, length

Why Do Guitar Strings Sound Different?

Only three things about a string decide the note it makes — how thick it is, how tight, and how long — and a guitar lets you change all three.

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An honestly unfinished answer

Hot Water Sometimes Freezes First, and Nobody Is Sure Why

The Mpemba effect — hot water sometimes freezing before cold — keeps being reported, keeps being disputed, and after two thousand years is unexplained.

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Blocking light, close and far

How Do Shadow Puppets Work?

Light travels in straight lines and spreads out as it goes, so a hand held near the lamp blocks a far wider cone and throws an enormous shadow.

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The same note, a different mix

Why Does a Piano Not Sound Like a Violin?

A flute and a violin can play exactly the same note and never be confused, because each adds its own mix of quieter higher notes above it.

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A compass painted on the ground

Why Are Runways Numbered?

A runway's number is its compass heading divided by ten, so a pilot lining up can check the direction against the instruments.

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Three feet always land

Why Does a Chair Wobble When a Stool Never Does?

Any three feet reach the floor together, because three points settle a flat surface between them. A fourth has to match a surface decided without it.

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A see-saw with a concrete end

Why Doesn't a Crane Tip Over?

A tower crane is a see-saw about its mast: a block of concrete on the short arm balances whatever hangs from the long one, distance included.

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Four clocks and some arithmetic

How Does a Phone Know Where It Is?

Nothing tracks your phone. It times signals from several satellites, turns each delay into a distance, and solves for the only point that fits.

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Steel on steel, barely gripping

How Can One Engine Pull a Two-Kilometre Train?

Steel wheels on steel rails barely rub, so one engine can pull an enormous train — and barely grip, so it cannot stop quickly.

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Shout, listen, measure

How Do You See With Sound?

Echolocation is measuring distance by timing an echo — and bats, dolphins, ships and hospital scanners all do the same arithmetic.

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If the head fits, the mouse fits

How Does a Mouse Fit Through a Gap?

A mouse pushes through a hole the width of a pencil because nothing behind its skull is locked at a fixed width, not even the shoulders.

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Measuring the dark

What Are Whiskers For?

A cat's whiskers are not for stroking. They are levers in nerve-packed follicles, measuring gaps and air movement in total darkness.

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Pieces too small to see

What Is Everything Made Of?

Everything around you is built from about a hundred kinds of atom, so small that a sheet of paper is half a million of them thick.

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How hard the atoms hold on

Why Do You Need a Furnace to Melt Iron?

The tipping point is sharp: a degree below it the lump holds its shape. Tungsten atoms grip hardest of all, which is why old bulbs used tungsten wire.

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