There are a lot of different sets of numbers. In this post we will look at some of the more important numbers. Each set of numbers is represented by a single letter in the Blackboard Bold font.
The real numbers (represented with a fancy R) are all of the numbers most people can easily think of. Anything that can be represented as a decimal is a real number even transcendental numbers like π or √2 that have an infinite number of places.
For example: 0, 2.719, -0.5, 1.333, 10006
Seriously, pick a number. Unless your familiar with some of the stranger realms of mathematics its going to a real number.
We describe as “real space” or “coordinate space” any position that can be described in terms of real real numbers. The numberline is R1 space. Cartesian coordinates are R2 space. Our universe has three dimensions of real space so it is R3 space.
The rational numbers (represented with a fancy Q) are all of the numbers which can be represented in terms of a ratio or fraction. All of the rational numbers are real numbers.
For example: 1/1, 5/7, 8/1, 0/1
The integers (represented with a fancy Z) are the whole numbers, both positive and negative.
For example: -5, 0, 11, 54, -10
All of the integers are real numbers and rational numbers (every integer can be represented as fractions of the form x/1).
The natural numbers (represented with a fancy N) are either the positive integers or the non-negative integers. That is, every whole number from 1 to infinity or from 0 to infinity.. Sometimes these are called the “counting numbers” because they are how we count objects. You can have two things or three things but you can’t have -1 things or 1.2 things (a fifth of an apple is a different thing than an apple in this mode of thinking).
All of the natural numbers are real numbers, integers, and rational numbers.
Since there are two popular definitions of the natural numbers they are usually specified. I will try, on this site, to use N0 and N1.
N1 is 1, 2, 3, 4 . . . (the positive integers)
N0 is 0, 1, 2, 3 . . . (the non-negative integers)
Natural numbers show up frequently when discussing discrete events. For example the Poisson distribution has values or “support” only at the N0.
Here’s where things really get weird.
The imaginary numbers (represented with a fancy I) are the numbers which, when squared, produce a negative. Why are these imaginary? Because they exist outside of the real numbers. There are formal proofs of this but feel free to try finding a real number that has a negative square.
The imaginary unit, which is the equivalent of 1 for the imaginary numbers, is written as i. All of the imaginaries can be represented in the form ri where r is a real number.
The imaginary number line exists perpendicular to the real number line. You might think that imaginaries and the reals never meet but consider the imaginary number 0i. An imaginary times zero is . . . yes, zero. The two lines must then intersect at zero.
Despite being called imaginary these numbers do exist, they just don’t represent any sorts physical things. Imaginary numbers see use in many different contexts.
The complex numbers (represented with a fancy C) are the numbers which can be represented as the sum of a real number and an imaginary number. Complex numbers do no exist on a line, their simplest form is in two dimensions which is referred to as the complex plane.
This set contains all of the real numbers. For every real number x there is a corresponding complex number of the form x+0i.
Intuitor 