The Number Line

Take an ordinary school ruler. It measures from 0 up to 30 and then it just… stops. Now imagine a magical ruler that never stops: it runs on to the right past a hundred, past a million, forever — and it runs on to the left too, past zero, into the negative numbers. That endless ruler is the number line, one of the most useful pictures in all of mathematics.

You have already met it in disguise. The thermometer outside your window on a frosty morning is a number line standing on end — and when it dips below zero, you can see negative numbers with your own eyes. The timeline on a classroom wall is a number line of years. The lift buttons in a big building are one too: floors 1, 2, 3 going up, and -1, -2 for the car park hiding below the ground.

What makes the number line so special is this: every number gets its own address. Whole numbers, fractions, decimals, negatives — each lives at exactly one spot, and no two ever share. Find a number's address, and comparing, ordering and measuring distance all become things you can simply see.

What a number line is

Once we can count, we can lay the numbers out in a row. A number line is exactly that: the numbers written in order, side by side, evenly spaced — the same gap from 0 to 1 as from 1 to 2, and so on. That even spacing is the whole trick: it is what turns the line from a mere list of numbers into a ruler that can measure.

The order is always the same: smaller numbers sit on the left, and bigger numbers sit on the right. So as you walk along the line to the right, the numbers get bigger one step at a time:

\dots,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ \dots

The little arrowheads you often see drawn at both ends are a promise: the line keeps going. There is no biggest number waiting at the right-hand end, and no smallest one lurking at the left. Whatever number you name, the line has already saved it a seat.

To find a number's address, you start at 0 — home base — and step right until you reach it. Press play, then replay it: a marker lands on a few different spots, and each time we read aloud the number it is sitting on. Notice that every jump to the right lands on a bigger number — that is not a coincidence, it is the number line's one unbreakable rule.

A messy list, sorted in one glance

Suppose a friend scribbles down five numbers in a jumble: 4,\ -3,\ 0,\ -1,\ 2 — and asks: which is the biggest? Which is the smallest? Is -3 more than -1? You could puzzle over each pair… or you could let the line do the work. Place each number at its address, then read them off left to right. Step through it:

The line settles every argument instantly. Which is bigger, -3 or -1? On the line, -1 sits to the right of -3 — and right always means bigger, so -1 > -3. If that feels odd, think of temperature: a day at -1 degree is warmer than one at -3.

Squeezing in between: fractions and decimals

The whole numbers are like lampposts along a street — but the street doesn't disappear between the lampposts! The stretch of line between 2 and 3 is packed with numbers, each at its own spot. The most famous one is 2.5: it lives exactly halfway between 2 and 3, the way half past two sits between two o'clock and three o'clock.

Grab a magnifying glass and zoom in on just that one gap:

Fractions get addresses the same way: \tfrac{1}{2} lives halfway between 0 and 1, and 2\tfrac{3}{4} is the same spot as 2.75. Worked example: where does 2.75 live? First find the right gap: it is bigger than 2 and smaller than 3, so it sits between those lampposts. Then place it within the gap: past the halfway mark 2.5, three quarters of the way along. Two moves, every time: which gap, then where in the gap.

How far apart? Count the hops

Because the gaps are all the same size, the number line doesn't just order numbers — it measures the distance between them: simply count the one-step hops from one to the other. From 3 to 8? Hop 3 \to 4 \to 5 \to 6 \to 7 \to 8 — five hops, so the distance is 5. (Notice 8 - 3 = 5: subtraction is secretly a distance-measurer.)

The line really shows off when the journey crosses zero. How far is it from -2 to 3? Don't guess — count: two hops bring you from -2 up to 0, then three more carry you on to 3. Five hops in all. Watch:

This is exactly what the weather forecast does. If the temperature climbs from -2 degrees at dawn to 3 degrees by lunch, the day warmed by 5 degrees — five hops up the thermometer's number line.

"−3 must be bigger than −1, because 3 is bigger than 1." No! On the number line, -3 sits further to the left than -1, and left always means less — no exceptions, ever. So -3 < -1. Think of pocket money: owing your friend 3 coins leaves you worse off than owing 1. With negative numbers, a bigger digit means you are further below zero.
"There's nothing between 2 and 3." There is more between them than you could ever list! 2.5, 2.25, 2.999, 2.0001… Between any two different numbers — however close — there is always another number (halfway between them, for a start). The number line has no gaps: you can zoom in forever and it never turns dotty.

Once you know the shape, you spot number lines everywhere:

A thermometer is a number line turned on its end — up means bigger, and the negatives sit below zero, exactly where winter keeps them.
A history timeline is a number line of years, with BC years on the left of a middle point and AD years on the right. Here's a secret that annoys historians: the calendar has no year zero — it jumps straight from 1 BC to AD 1. It is a number line with one address missing!
In golf, scores are counted against "par", so the leaderboard is a number line where smaller wins: a player on -4 is beating a player on -1, who is beating everyone on +2.
Sea level works the same way: mountains have positive heights, submarines cruise at negative ones, and the beach is everybody's zero.

Different clothes, same idea: one straight line, every value at its own address.

Want to hear it once more, from another teacher? Khan Academy shows how to use a number line here: