The fuck are real numbers?

Real numbers are the only numbers that most people ever come into contact with. What the fuck are they? Why are they important? Why are they called “real?” Are there “unreal” numbers? We’ll get into all that bullshit right now.

Definition

What they be

Real numbers–represented in mathematics with the boojie symbol $\mathbb{R}$–are essentially all numbers that can be plotted on a number line. 

This is a pretty simplistic explanation of real numbers but in all honesty it’s pretty fucking accurate. It’s not nearly rigorous enough a definition for big baller mathematicians, but for scrubs like me it’s a-ok.

What they ain’t be

There’s a whole expanded set of numbers called the complex numbers–$\mathbb{C}$–which are basically any number that have the tricky fucking “imaginary unit” $i$ aka $\sqrt{-1}$. These numbers are not included in $\mathbb{R}$. Also, $\mathbb{C}$ contains $\mathbb{R}$, so all real numbers are also complex numbers. Not all complex numbers are real, though.

Quick note here to hopefully avoid some commonplace–ass ignorance. Complex or “imaginary” numbers are just as valid and real as “real” numbers. They exist. They are only called “real” and “imaginary” because
René “I think therefore I am” Descartes decided to call them that. Fucker couldn’t wrap his mind around complex numbers so he just called them “imaginary” because whatever you can’t understand must not be real, amirite? And now high school kids think complex numbers are just made up because of these the dumbass names “real” and “imaginary.”

So don’t go around thinking that complex numbers aren’t a real thing–they are. Fucker. They’re just generally reserved for next-level shit like electronic circuits and quantum mechanics.

Subsets of $\mathbb{R}$

The set of real numbers $\mathbb{R}$ contains several of the most important sets in mathematics:

• $\mathbb{Q}$, the rational numbers. Anything you can write as a fraction. This includes obvious shit like $\frac{1}{2}$, but also integers like 10 because you can express it as a fraction: $\frac{10}{1}$
• $\mathbb{Z}$, the integers. Basically any whole number–positive, negative, or zero: 0, 52, -8008, etc.
• $\mathbb{N}$, the natural numbers. These are your standard counting numbers: 1, 2, 3, etc. Depending on who you ask, it may or may not include zero, but there are definitely no negative numbers.
• $\mathbb{R}-\mathbb{Q}$, the irrational numbers. These fuckers are the wild cards. Anything that can not be expressed as a fraction. If you take $\mathbb{R}$ and remove all of $\mathbb{Q}$, you’re left with the irrationals. This set includes weirdos like $\sqrt{2}$, $\pi$, and $e$. These are the numbers that have decimals that go on forever in unpredictable patterns.

So there you go. If you can place it on a number line, it’s a real number. They contains pretty much every number that most people will ever come into contact with.  And don’t let the name “real number” trick your silly ass into thinking complex numbers aren’t real things. Thanks a lot René.