\begin{align} A &= \left \{ m,n,o \right \} && \text{A is the set of members m, n, and o} \\ o &\in A && \text{In. Object o is in set A.} \\ \\ A &\subset B && \text{Subset. Set A is a subset of set B.} \\ B &\supset A && \text{Superset. B is a superset of A.} \\ \\ A &\cup B && \text{union of sets A and B} \\ A &\cap B && \text{intersection of sets A and B} \\ A &\setminus B && \text{set difference. in A, but not in B.} \\ A &\triangle B && \text{symetric difference. in A or B, but not in both.} \\ A &\times B && \text{Cartesian product. All possible ordered pairs.} \\ \end{align}

\begin{align} \mathbb{R} &= \text{real numbers} \\ \mathbb{Q} &= \text{rational numbers (q for quotient)} \\ \mathbb{Z} &= \text{integers } (\cdots,-3,-2,-1,0,1,2,3,\cdots) \\ \mathbb{N} &= \text{natural numbers, positive integers } (1,2,3,\cdots) \\ \mathbb{W} &= \text{whole numbers, counting numbers, positive integers and zero } (0,1,2,3,\cdots) \\ \mathbb{P} &= \text{prime numbers } (1,3,5,7,11,\cdots) \\ \mathbb{R} &\supset \mathbb{Q} \supset \mathbb{Z} \supset \mathbb{N} \supset \mathbb{W} \supset \mathbb{P} \\ \\ \mathbb{C} &= \text{complex numbers} \\ \mathbb{C} &\not\subset \mathbb{R} \\ \end{align}

Either boldface $\mathbf{R}$ or “blackboard bold” $\mathbb{R}$ is used to symbolize a set.

$\mathbb{R} $

All numbers except imaginary and complex numbers.

Real numbers include integers, fractions, decimals, and zero.

A rational number can be written as a quotient, that is, a fraction, a ratio of two integers, like for example $\frac{3}{1}$ or $\frac{1}{3}$

When written in decimal form, a rational number is terminating or repeating.

$\mathbb{Q}$ represents the set of rational numbers.

Irrational numbers cannot be written as a quotient.

When written in decimal form, an irrational number never terminates and does not repeat.

The following are irrational numbers.

\begin{align} \pi &= 3.14159265359\cdots && \text{pi}\\ e &= 2.7182818284\cdots && \text{Euler's number}\\ \varphi &= 1.6180339887\cdots && \text{golden ratio} \\ \sqrt{2} &= 1.4142135237\cdots && \text{square root of 2} \\ \end{align}

There is no standard-accepted symbol for the set of irrational numbers, though sometimes you see $\mathbb{I}$ or $\mathbb{Q}\prime$ or $(\mathbb{R} - \mathbb{Q})$.

An imaginary number is a negative number that can have a square root.

It is not a real number.

It is written as $bi$ such that $b$ is a real number and $i$ is the imaginary unit number.

For example:

• The square root of $9$ is $3$ or $-3$.
• The square root of $-9$ is impossible.
• The square root of $9i$ is $3$ or $-3$.

In that example, $9$, $-9$, $3$, and $-3$ are real numbers, and $9i$ is an imaginary number.

The imaginary Unit Number $i$ is a special number that exists for the purpose of creating imaginary numbers.

\begin{align} i^2 &= -1 \\ i &= \sqrt{-1} \\ \end{align}

A complex number includes a real part and an imaginary part, written in the form $(a + bi)$.

\begin{align} (a + bi) &\in \mathbb{C} \\ a &= \text{the real part} \\ bi &= \text{the imaginary part} \\ i &= \text{the imaginary unit number} \\ a,b &\in \mathbb{R} \\ i, bi &\not\in \mathbb{R} \\ \end{align}

\begin{align} A &\Rightarrow B && \text{if A then B; or, A implies B} &\text{\Rightarrow} \\ A &\iff B && \text{if and only if A, then B} &\text{\iff} \\ A &\therefore B && \text{A is true, therefore B is true} &\text{\therefore} \\ A &\mid B && \text{A, such that B} &\text{\mid} \\ \end{align}

Number Theory - a branch of mathematics devoted to the study of integers, especially of prime numbers.

Number System - aka numeral system. - a writing system for expressing numbers. Examples include:

• base-10 or decimal system,
• base-2 or binary system,
• base-16 or hexadecimal system
• base-1 or unary numeral system, used in tallying.