Sets of Numbers

The numbers we work with in Mathematics can be represented using set notation. In CSEC General Mathematics, we were limited to the set of real numbers ($\mathbb{R}$) which is the set of all positive and negative whole and decimal numbers and zero ($0$). Simply put, the real numbers are every number you have come across so far:

• Positive whole numbers e.g. $2$ and $5$
• Negative whole numbers e.g. $-13$ and $-27$
• Positive decimal numbers e.g. $2.4$ and $0.5$
• Negative decimal numbers e.g. $-1.6$ and $-40.002$
• Zero ($0$)

The set of real numbers has subsets (a set within a set) that have special properties.

Rational and irrational numbers

If a number can be written as the ratio/quotient of any two whole numbers then it is rational. For example, $0.5$ can be written as: $$ \begin{equation}\begin{aligned} 0.5=\frac{1}{2}\\ \end{aligned}\end{equation} $$ $1$ and $2$ are clearly whole numbers so $0.5$ is rational. Pi ($\pi$) on the other hand, cannot be written as one whole number divided by another thus it is irrational (not rational).

On lower levels, you may have been taught that $\frac{22}{7}$ is the same as $\pi$ but this has just an approximation for it. An approximation is a value that is close to but not exactly the same as something else.

Using the approximation was just to make computation easier but simply evaluating $\pi$ and $\frac{22}{7}$ in your calculator will show that after a few decimal places, the two numbers start to vary.

The symbol used to represent rational numbers is $\mathbb{Q}$ and the irrational numbers, being the complement of the set of rational numbers, is represented by $\mathbb{Q}’$.

Integers

The integers ($\mathbb{Z}$) are the set of positive and negative whole numbers and zero ($0$). Any integer qualifies as a rational number because it can be written as a ratio of itself and one ($1$): $$ \begin{equation}\begin{aligned} 1&=\frac{1}{1}\\ 2&=\frac{2}{1}\\ 0&=\frac{0}{1}\\ -4&=\frac{-4}{1}\\ \end{aligned}\end{equation} $$

Natural numbers

This is the set of counting numbers ($1,2,3,…$). The set is represented by the symbol $\mathbb{N}$. Every natural number qualifies as an integer because it is a positive whole number.

These are subsets

We should know by now that when a set $B$ is a subset of another set $A$, we write $B\subset A$. Thus, since the natural numbers are a subset of integers, we write: $$ \begin{equation}\begin{aligned} \mathbb{N}\subset \mathbb{Z}\\ \end{aligned}\end{equation} $$

And integers are a subset of rational numbers: $$ \begin{equation}\begin{aligned} \mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\\ \end{aligned}\end{equation} $$

And rational numbers are a subset of real numbers: $$ \begin{equation}\begin{aligned} \mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\\ \end{aligned}\end{equation} $$

Complex numbers

This pattern of subsets might bring up the question of what the real numbers are a subset of. The answer to this question involves the possibility of the square root of a negative number. From the laws of indices we know that the product of two numbers raised to a power is the product of the individual numbers raised to the power: $$ \begin{equation}\begin{aligned} (ab)^n=a^n b^n\\ \end{aligned}\end{equation} $$

Applying this to the square root of two numbers we get: $$ \begin{equation}\begin{aligned} \sqrt{ab}&=(ab)^\frac{1}{2}\\ &=a^\frac{1}{2} b^\frac{1}{2}\\ &=\sqrt{a}\times \sqrt{b}\\ \therefore \sqrt{ab}&=\sqrt{a}\times \sqrt{b}\\ \end{aligned}\end{equation} $$

If we consider the square root of a negative number like $-5$: $$ \begin{equation}\begin{aligned} \sqrt{-5}=\sqrt{5}\times \sqrt{-1}\\ \end{aligned}\end{equation} $$

Or $-13$: $$ \begin{equation}\begin{aligned} \sqrt{-13}=\sqrt{13}\times \sqrt{-1}\\ \end{aligned}\end{equation} $$

Or in general: $$ \begin{equation}\begin{aligned} \sqrt{-a}=\sqrt{a}\times \sqrt{-1}\\ \end{aligned}\end{equation} $$

Notice how each square root can be written as the square root of a positive number and the square root of $-1$. If we let $\sqrt{-1}=i$, we get: $$ \begin{equation}\begin{aligned} \sqrt{-5}&=\sqrt{5}\times \sqrt{-1}\\ &=\sqrt{5}i\\ \sqrt{-13}&=\sqrt{13}\times \sqrt{-1}\\ &=\sqrt{13}i\\ \sqrt{-a}&=\sqrt{a}\times \sqrt{-1}\\ &=\sqrt{a}i\\ \end{aligned}\end{equation} $$

We now have a way to represent the square root of negative numbers. Because these square roots do not actually exist, we refer to them as imaginary numbers. The combinations of real and imaginary numbers is referred to as the complex numbers ($\mathbb{C}$): $$ \begin{equation}\begin{aligned} \mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\subset \mathbb{C}\\ \end{aligned}\end{equation} $$