Primary Number Systems

There are a few common number systems we have to deal with in Mathematics with their own associated symbols.

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The Natural Numbers

$$\mathbb{N}=\{1,2,3,\dots \}$$

The natural numbers are the counting numbers. They are used for counting discrete objects. The set of natural numbers is infinite, meaning there is no largest natural number. This set is considered a Countably Infinite set.

The Integers

$$\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$$

The integers include all whole numbers, both positive and negative, as well as zero. Like the natural numbers, the set of integers is infinite. This set is also considered a Countably Infinite set.

The Rational Numbers

$$\mathbb{Q}=\left\{\frac{a}{b}\mid a,b\in \mathbb{Z},b\ne 0\right\}$$

The rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This includes integers (which can be expressed as a fraction with a denominator of 1) and fractions. The set of rational numbers is also infinite and dense, meaning that between any two rational numbers, there are infinitely many other rational numbers. This set is considered a Countably Infinite set.

The Real Numbers

$$\mathbb{R}=\{x\mid x\text{ is a point on the number line}\}$$

The real numbers include all rational numbers as well as irrational numbers (numbers that cannot be expressed as a fraction of two integers, such as √2 and π). The set of real numbers is also infinite and can be visualized as all points on an infinite number line. This is considered an Uncountably Infinite set.

The Complex Numbers

$$\mathbb{C}=\{a+bi\mid a,b\in \mathbb{R},i^2=-1\}$$

The complex numbers include all real numbers as well as imaginary numbers (numbers that involve the square root of a negative number). A complex number is expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The set of complex numbers is also infinite and can be visualized as points on a two-dimensional plane, known as the complex plane. This set is considered an Uncountably Infinite set.

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At this point we can contruct extensions of the Complex numbers, such as the Quaternions and Octonions, but these are less commonly used in basic mathematics and more often found in advanced fields such as theoretical physics and computer graphics. Some of these that might be usefull are the dual numbers, which are used in automatic differentiation, and the p-adic numbers, which are used in number theory. This field of study is known as Ring Theory.