Negative numbers — introduction

Negative numbers — introduction

Until now every number you've worked with has been zero or above: 0, 1, 2, 3, … all the way up. But you may have already noticed that some everyday situations want a number that's below zero. A winter thermometer reads −7 °C. A submarine sits at "depth 200 m" — below the surface of the sea. A bank account is "overdrawn by £50".

For all of these, mathematicians invented a clean way to write them: negative numbers.

What a negative number is

A negative number is a number that is less than zero. We write it with a minus sign in front:

−1, −2, −3, −4, …

The little dash on the left isn't a subtraction — it's part of the number, telling us "this number lives below zero".

The numbers we already know — 1, 2, 3, … — are sometimes called positive to make the contrast clear, but we don't usually bother to write the plus sign. 5 and +5 mean the same thing.

Together, the whole numbers below zero, zero, and the whole numbers above zero are called the integers:

…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …

The number line stretches both ways

The cleanest picture for negative numbers is the number line. The number line you already know just keeps going past 0 to the left:

Two key things to spot:

  • 0 is in the middle. It's the "border" between positive and negative.
  • Going left = getting smaller. −3 is to the left of −1, so −3 is less than −1. The numbers don't get bigger when you go further from zero on the left — they get smaller.

That second rule trips up almost every learner the first few times. Keep checking against the number line until it feels natural.

Where you meet negative numbers in real life

  • Temperature. −7 °C means seven degrees below freezing point.
  • Money. A bank account that owes £20 is sometimes shown as −£20.
  • Lift / elevator floors. Underground floors are often labelled −1, −2, −3.
  • Depth below sea level. A point in Death Valley sits at about −86 m. Mount Everest is at +8849 m. Sea level is 0.
  • Score in games. Some board games let your score go below zero if you mess up badly enough.

In every case, negative means "below the natural zero point" and positive means "above it".

A first reflex: negative numbers are smaller

If you remember nothing else, remember this:

Every negative number is smaller than every positive number.

So −1000 is smaller than +1 — even though "1000" looks bigger than "1". The minus sign flips the whole thing.

This is so unexpected at first that we'll have a whole exercise about it later — comparing two integers and choosing the right `<`, `=`, `>`.

What you will learn next