What Are Numbers?

How This Course Works

This course starts from the very first ideas in mathematics and builds, one small lesson at a time, all the way to advanced topics like calculus. You don't need to remember any formulas or words from school. If you have ever thought "I was never good at maths", you are exactly who this course is written for.

Our approach is simple: build intuition first. You'll understand why something works before being asked to remember a rule. For example, much later you'll meet a sum like this:

Instead of memorising a method for it, we'll picture half a pizza and a quarter of a pizza, and see that together they make three quarters. Every topic in this course works that way. But first, let's start with the most basic question of all.

Counting Things

Imagine you have these apples:

How many apples are there? You probably counted them without even thinking: one, two, three. There are

That symbol, $3$, is a number. It isn't an apple, and it isn't the word "three". It's a short way of writing down how many — a quantity. The same symbol works for three apples, three coins or three days. That is what makes numbers so useful: they let us talk about amounts of anything.

Putting Numbers Together

Now suppose someone gives you two more apples:

Count them all together and you get five. In mathematics we write that like this:

Let's read that one piece at a time:

• $3$ and $2$ are numbers — the two quantities we started with.
• $+$ means addition: putting the quantities together.
• $=$ means "is equal to": the quantity on the left is exactly the same as the quantity on the right.
• $5$ is the result — how many we have altogether.

So the whole line says: three and two together are the same amount as five. Nothing more mysterious than that.

Natural Numbers

When you count apples, you say "one, two, three, four…" and you could keep going for as long as you like. The numbers you use for counting are called the natural numbers:

The three dots mean "and so on, forever". There is no biggest natural number: whatever number you think of, you can always add one more.

Zero and Whole Numbers

Counting starts at one, but sometimes there is nothing to count. Suppose you have 5 chocolates and you eat all 5 of them:

How many are left? None. We write that as:

The $-$ sign means taking away — we'll explore it properly in the next lesson. The important new number here is $0$, zero. For now, zero simply means none: no chocolates, no apples, nothing at all.

When we add zero to the start of the counting numbers, we get the whole numbers:

Recap

Your First Mini Exercise