The different types of numbers

In maths we split numbers into different groups. Each group is built from the one before it by adding a new kind of number, so the groups sit inside each other like a set of boxes. Here they are, from the smallest box outwards.

The natural numbers

The natural numbers are the numbers we can count on our fingers: 1, 2, 3, 4 and so on. We write them with the letter N:

$$\mathbb{N} = \{1, 2, 3, 4, 5, ...\}$$

In some places you will see that 0 is included in the natural numbers, but then that is shown with a small 0 next to the N. Different books do it differently, and both conventions are in use.

The integers

Then there are the integers, which is what we get when we add the negative numbers and 0 to the natural numbers. We write the integers with the letter Z:

$$\mathbb{Z} = \{..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...\}$$

So the integers are the natural numbers, plus zero, plus a mirror image of the natural numbers on the negative side. Every natural number is also an integer; we have only added to the group, not taken anything away.

The rational numbers: fractions

Next there are halves and quarters and all sorts of fractions. When we add the numbers that can be written as fractions to the integers, we get the rational numbers. These are fractions where the numerator and the denominator are integers, and where the denominator is not 0. The rational numbers are written with the letter Q, and every number that can be described with a fraction is in there:

$$\mathbb{Q}$$

Notice that the integers are in this group too. A whole number is a fraction with 1 as its denominator, so 2 is $\tfrac{2}{1}$, and that makes it rational. Fractions are not a different kind of thing from the integers; they are the integers plus more.

The irrational numbers

There are, however, also numbers that cannot be described as a fraction of integers, namely the irrational numbers. These are numbers like pi, the square root of 2 and Euler's number:

$$\pi, \quad \sqrt{2}, \quad e$$

You will meet pi in everything to do with circles, and the square root of 2 in the indices and roots glossary and in Pythagoras' theorem. Euler's number e turns up later than GCSE, but it belongs in the list, so it is named here.

The real numbers

When we add numbers like these to the rational numbers, we get the real numbers, which we write with the letter R:

$$\mathbb{R}$$

The real numbers are all the numbers on the number line: every integer, every fraction and every irrational number. This is the group of numbers that all the maths in these articles lives in.

Complex numbers

There are also complex numbers, written with the letter C, which are an extension of the real numbers. A complex number is a real number put together with an imaginary number, where an imaginary number is the square root of a negative number, such as $\sqrt{-1}$. We do not use complex numbers in any of these articles, and they are not part of GCSE, but it is worth knowing that the list does not stop at the real numbers.

Common misunderstandings

• "0 is definitely a natural number" or "0 is definitely not a natural number". Both conventions exist. When 0 is included, it is shown with a small 0 next to the N.
• "Fractions are a different kind of number from whole numbers." They are not. The rational numbers contain the integers, because every integer can be written as a fraction with 1 on the bottom.
• "Pi is the only irrational number." It is the most famous, but the square root of 2 and Euler's number e are irrational too, and there are many more.
• "Every number is either a whole number or a fraction." No. The irrational numbers, like pi and the square root of 2, cannot be written as a fraction of integers at all, and they are real numbers just the same.

Related

• What is a fraction?
• What are indices (powers) and roots?