Which football team is bottom of the league on goal difference, and which bank balance is worse — owing a little or owing a lot? Answering questions like these means putting negative numbers in order.

Once a number line has negative numbers on it, putting numbers in order works exactly the same way as before — you just keep reading left to right. The number further to the left is always the smaller one, and the number further to the right is always the bigger one. Nothing new to learn — the line just keeps going past zero into the negatives. So, smallest first:

reads: negative seven, negative three, zero, two, six. Each number sits to the right of the one before it, so each is a little bigger than the last. Find them on the line and you can see it: -7 is way out on the left, and 6 is way out on the right.

The hook: how cold is it?

Temperature is the friendliest place to meet negative numbers. On a freezing day the thermometer dips below zero, and the colder it gets, the further below zero it goes. So ordering temperatures from coldest to warmest is exactly the same as ordering numbers from smallest to largest.

One frosty week a town recorded these temperatures, in degrees: 2, -6, 0, -3 and 4. Lining them up coldest first — that is, smallest first — gives:

The coldest day was -6 degrees (furthest left), and the warmest was 4 degrees (furthest right). Zero sits in the middle: it is warmer than every below-zero day and colder than every above-zero day.

Comparing two negatives

Here is the trick that catches people out. With two negative numbers, the one closer to zero is the bigger one. We know 5 > 2 — five is more than two. But flip both signs and the order reverses:

Why? On the line, -5 sits further to the left than -2, so it is the smaller one. Think of money: owing 5 dollars (-5) leaves you worse off than owing 2 dollars (-2). A bigger debt is a smaller balance.

You can write the comparison either way round — they say the same thing:

The wide-open end of the < or > always points at the bigger number, and the pointy end at the smaller one — just like when you were comparing with symbols before. It is the same idea as ordering positive numbers, just continued past zero.

Three worked examples

1. Put these in order, smallest first: -3, 4, -7, 0.

Walk the number line from the left. The furthest-left number is the most negative, -7, then -3, then 0, then 4:

2. Which is smaller, -8 or -2?

Both are negative. -8 is further from zero, so it is further left, so it is smaller. The 8 being a bigger digit does not make -8 bigger — the minus sign flips it:

3. Which day was coldest: -6 degrees, -9 degrees, or -2 degrees?

Coldest means smallest means furthest left. -9 is the most negative, so -9 degrees was the coldest:

See it: markers on the line

A few numbers — some negative, some positive — are dropped onto the line as coloured markers. Reading them off from left to right gives them in order, smallest to largest. Press Refresh for a fresh set, and check the order yourself before you read the line underneath.

Here is the same idea as an animation. Press play. A few numbers — some positive, some negative — appear out of order. We place each one on the line and then read them off from left to right, smallest to largest. Watch how the most negative number sits furthest to the left. Replay it: each time it shuffles a fresh set of numbers.

Khan Academy works through ordering negative numbers here: