Natural numbers, also called counting numbers, are the first numbers that we learn. They have immediate use in keeping count of objects.
When accounting needs became more sophiscated, negative numbers and the number zero became necessary. For example, they allow one to solve equations such as \(x+3 = 1\) that have no solutions in natural numbers. The set of integers thus came into existence.
But there are equations that have no integer solution; e.g. \(x+x=1\). One can take the view that fractions (or rational numbers) were invented to solve such equations.
Of course, the quest could go on to invent numbers that solve equations such as \(x\cdot x = 2\). Inventing the number \(\sqrt{2}\) would solve this equation. There are many proofs that \(\sqrt{2}\) is not a rational number and this fact has been known to the ancient Greeks. More generally, for every prime number \(p\), the equation \(x\cdot x = p\) has no rational solution.
Then there are numbers that denote various quantities such as the area of the unit circle which are not solutions to equations with integer coefficients. Thus we have the set of real numbers which are normally represented by points on a line that extends indefinitely in both directions, or numerically as infinite decimals. For example, writing the first few digits of the number \(\pi\) gives \(3.14159\ldots\).
However, an equation as simple as \(x\cdot x +1 = 0\) still has no solution even with the very large set of real numbers. So the number \(i\) was invented. That is, \(i\) is a number such that \(i^2 = -1\). This gave rise to the set of complex numbers which we will soon study.
The question one might want to ask now is, “Do we need to invent any more numbers to solve equations with complex coefficients?”
An important result in mathematics is the Fundamental Theorem of Algebra which states that
every equation of the form \(a_nx^n + a_{n-1}x^{n-1} + \cdots a_1 x + a_0 = 0\) where \(a_0,\ldots,a_n\) are complex numbers has at least one complex number as solution.
In a sense, the theorem tells us that there is no need to invent any more numbers. This is not to say that other sets of numbers cannot exist. In fact they do but such sets either look in almost every aspect like the complex numbers or exhibit very different properties.
Does the equation \(x^2 - 3x + 1 = 0\) have a rational number as a solution?