Number Systems: Real, Rational, Integer

When we're working in mathematics, we often want to make reference to different types of numbers.

We'll often see and use set notation as a shorthand, with key sets written as $\mathbb {B}$lackboard $\mathbb {B}$old letters.

Useful types of numbers to know and recognise:

1. $\mathbb {R}$ - the Real Numbers
2. $\mathbb {Q}$ - the Rational Numbers
3. $\mathbb {N}$ - the Natural Numbers
4. $\mathbb {Z}$ - the Integers
5. $\mathbb {Z} ^{+}$ - the Positive Integers
6. $\mathbb {Z} _{0}^{+}$ - the Non-Negative Integers
7. $\mathbb {C}$ - the Complex Numbers

Real Numbers

$\mathbb {R}$

This refers to all numbers that can be found on a normal number line. So, it includes $0,-3,7.2,-\pi ,$..., i.e. all of the numbers.

The Real Numbers likely includes all numbers that you have studied, unless you have reached imaginary and complex numbers.

To talk about the real numbers, we'll write $x\in \mathbb {R}$, and read this as "$x$ is a member of the Real Numbers" or "$x$ belongs to the Real Numbers".

Rational Numbers

$\mathbb {Q}$

The letter $\mathbb {Q}$ refers to the word 'quotient', the result of a division - in other words, a fraction. Rational numbers are numbers that can be written as a fraction.

But be careful about this! Only whole numbers are allowed in the fraction.

Therefore, irrational numbers like $\pi$ won't count. You can't write $\pi$ as a whole number over a whole number.

Note that integers do count as rational. $-3={\frac {-3}{1}}$, so any whole number can be written as a fraction.

Note that zero does also count as rational! $0={\frac {0}{1}}$. We are always allowed $0$ on the top of a fraction.

All numbers in $\mathbb {Q}$ can be written as ${\frac {p}{q}}$ where $p,q\in \mathbb {Z}$ and $q\neq 0$.

We'll write $x\in \mathbb {Q}$, and say "$x$ is a rational number".

Natural Numbers

$\mathbb {N}$

Warning: Natural Numbers and $\mathbb {N}$ have more than one possible definition, which can be confusing.

The "Natural" numbers are also called counting numbers: $1,2,3,4,...$

We'll write $n\in \mathbb {N}$, often using $n$ in preference $x$, to help remind us that we are talking about a natural number or integer.

Many mathematicians (including teachers) may prefer to start the Natural numbers with 0 instead of starting with 1, or this may depend on context. So, this can sometimes be unclear.
If you see $\mathbb {N}$ in an official A-Level question, they are starting with 1, the same as $\mathbb {Z} ^{+}$.

I recommend avoiding this language and notation, where possible, and using a variation of $\mathbb {Z}$ instead.

Integers

$\mathbb {Z}$

This refers to all whole numbers, including positive integers, negative integers and zero.

So, we have all the numbers: $...,-3,-2,-1,0,1,2,3,...$

We'll write $n\in \mathbb {Z}$ and say "$n$ is an integer" or "$n$ belongs to the integers". It's nice to use $n$ to remind us that we're talking only about an integer, and not any other real number.

Positive Integers

$\mathbb {Z} ^{+}$

This refers clearly to only positive integers - the numbers $1,2,3,4,5,...$

Zero is in its own category: neither positive nor negative, so it doesn't belong here in $\mathbb {Z} ^{+}$.

If it's especially important to remember that zero doesn't belong, saying "strictly positive integers" can act as a reminder to exclude zero.

Non-Negative Integers

$\mathbb {Z} _{0}^{+}$

Warning: The notation $\mathbb {Z} _{0}^{+}$ can be unclear, but non-negative does have a clear meaning.

$\mathbb {Z} _{0}^{+}$ means the positive integers together with zero. So, the numbers: $0,1,2,3,...$

We can write $n\in \mathbb {Z} _{0}^{+}$ and say "$n$ is a non-negative integer".

I find it clearer to write this as two statements with a comma: $n\in \mathbb {Z} ,n\geq 0$.

Complex Numbers

$\mathbb {C}$

Complex Numbers are an extension of the number line into a two dimensional set of axes.

Their existence follows from the definition of the imaginary number, $i$, which is first seen in Further Maths A-Level.

We'll write $z\in \mathbb {C}$, usually using $z$ in place of $x$, to help remind us that we are taking about complex numbers.

We can also say that a complex number $z=x+yi$, where $x,y\in \mathbb {R}$

If we need a second complex number, we'll often use $w$ and write $w\in \mathbb {C}$, or: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w = u + vi} , where $u,v\in \mathbb {R}$