Real Numbers and Their Properties

This course is about the real numbers and the rules they follow. It covers how numbers are sorted into sets, how to work with them, and the properties behind each step.

It starts by sorting. The first lesson names six sets: natural, whole, integers, rational, irrational and real numbers. It shows how five of them fit one inside the next, and where the irrational numbers sit. It also teaches the symbol $\in$, which is short for belongs to.

The next two lessons work through the four operations. You add and subtract counts of the same root, such as $2\sqrt{3} + 5\sqrt{3}$. You also see when a sum like $4 + \sqrt{5}$ is already finished. Then you multiply and divide, using the rule $\sqrt{a} \times \sqrt{a} = a$. You share a number out over a bracket, and you move a root off the bottom of a fraction.

The last two lessons give the properties their names. Closure asks whether the answer is still a real number. The commutative property asks whether you may swap the two numbers round. Grouping, the identities $0$ and $1$, and the partners that undo a number come last.

By the end you will be able to:

• name the six sets, and say which of them a number such as $-4$ belongs to;
• write statements like $-3 \in Z$ using the belongs to symbol;
• add and subtract counts of the same root, and leave $\sqrt{2} + \sqrt{3}$ as it stands;
• multiply roots using $\sqrt{a} \times \sqrt{a} = a$, and multiply out a bracket such as $4(\sqrt{2} + 3)$;
• rewrite $\displaystyle \frac{6}{\sqrt{3}}$ with no root on the bottom;
• say which operations are closed, commutative and associative, and which are not;
• name $0$ and $1$ as the identities, and find the partner that undoes a number;
• explain why you may never divide by zero.