Extensions of number systems

The natural numbers ${\mathbb{N} = \{0, 1, 2, \ldots \}}$ are often our first exposure to a number system: they arises “naturally” from counting and we can do addition and multiplication on them.

This can then be extended to the integers ${\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots \}}$ by adding negative numbers into our number system. The subtraction operation is then well-defined.

This can be further extended to the rational numbers ${\mathbb{Q} = \{\frac{a}{b}: a,b \in \mathbb{Z}, b \neq 0 \}}$ by allowing fractions which greatly facilitates division.

Unfortunately (much to the dismay of the Pythagoreans) quantities like $\sqrt{2}, \pi$ and $\mathrm{e}$ are irrational.

We thus further extend our number system to the real numbers $\mathbb{R}$, and visualize them on a number line.

The imaginary unit

While the real numbers are sufficient for many applications, a general solution to cubic equations eluded mathematicians until the discovery of the imaginary unit $\mathrm{i}$ representing $``\sqrt{-1}"$.

While seemingly bizarre (hence the rather derogatory “imaginary” moniker, even though, if we really think about it, all numbers are imaginary!), the addition of $\mathrm{i}$ into our number system leads to many beautiful mathematical results, and have found very real applications in fields like electrical engineering and quantum physics.

Extending the real numbers to the complex numbers

The complex numbers $\mathbb{C}$ is the set of numbers of the form $x+y\mathrm{i}$, where $x$ and $y$ are real numbers and ${i^2 = -1}$.

We consider this an extension of the real numbers because a real number is a complex number too. For example, $2$ can be written as $2+0\mathrm{i}$.