# Math symbols list

Most of the challenges with math come from the complexities of the symbols. It's effectively a different language, with its own logical rules that define all of its symbols.

Each of the symbols denote a concept, and learning math creates a cumulative codex that draws from most of the previous symbols to communicate an idea.

## Mathematical state

±

- Indicates the number is either positive or negative
- Can often indicate a range of values
- e.g., 5 ± 2 is an unknown value somewhere between 3 and 7

∓

- Indicates the inverse sign of ±, where it's + when ± is - and - when ± is +

|□|

- The absolute value of a number (i.e., how far it is from 0), which always gives a positive number
  - e.g., |-5| = 5
- Can also refer to the number of elements in a set (i.e., "cardinality")
  - e.g., |5,4,2| = 3
- Can also refer to the length of a line segment along with d(A,B)
  - e.g., |AB|

## Comparison between values

=

- Equality, where two things are effectively the same quantities
- e.g., 2 + 2 = 4

≠

- Inequality, where two things are *not* the same quantities
- e.g., 2 + 2 ≠ 5

≈

- Approximately equal, where two things are *almost* the same quantities
- e.g., 22 / 3 ≈ 7

≡ (aka triple bar)

- Identity, an indication of "if an only if"
- e.g., "Hat **≡** Hat", but not "Hat **≡** hat"

≅

- Isomorphism, where two things are effectively equal, but in different ways

<

- Less than, represents that it's certainly unequal (strict inequality) and the former is less than the latter
- e.g., 2 < 3

>

- Greater than, represents that it's certainly unequal and the former is more than the latter
- e.g., 3 > 2

≤

- Less than or equal to, sometimes uses ≦
- can also be *much* less than (≪), which isn't always clearly defined

≥

- Greater than or equal to
- can also be *much* greater than (≫), which isn't always clearly defined

~

- A general-use symbol that can mean "approximately equal" or "same order of magnitude"

≺ and ≻

- Indicates an order or preorder (in order theory)

□:□

- A ratio between two numbers

%

- A per cent (□/100) amount relative to another value

‰

- A per mille (□/1000) amount relative to another value

## Basic arithmetic

+

- Addition, which is combining two numbers together
- e.g., 1 + 2 = 3

- Subtraction, which is removing the following number from the preceding
- e.g., 3 - 1 = 2

x, - or *

- Multiplication, which is adding a number over and over a certain number of times
- e.g., 2 x 4 = 8

/ or **÷**

- Division, which is indicating how many times the latter number fits into the former number
- e.g., 7 / 2 = 3.5

**:**

- Indicates a ratio of quantities, which is a relationship of relative size
- e.g., 2:3

## Advanced arithmetic

x^y (aka superscript)

- Exponents, which are multiple iterations of multiplication
- e.g., 2^3^ = 2 x 2 x 2 = 8
- Can also be represented by ^ symbol (e.g., 2^3) when superscript isn't easily available

√ (aka radical symbol)

- Square root, which is the value that will become the source number when multiplied together
- e.g., √9 = 3
- Similar to exponents, superscript can also indicate cube roots (∛), fourth roots (∜), and so on

## Set theory

∅

- An empty set
- Can also be represented by { }

#

- Number sign, with 3 possible indications:
  1. With #*S* as the number of elements, may alternatively represent as |S|
  2. With *n*# as the sum of prime numbers up to *n* (i.e., primorial)
  3. With *M#N*, the topological connected sum of two manifolds or knots

∈

- Set membership
- Can read as "is in", "belongs to", or "is a member of"

∉

- Not a member of a set
- Reads as "is not in" (e.g., *x* ∉ *S* means ¬(*x* ∈ *S*)

⊂

⊆

⊊

⊃, ⊇, ⊋

∪

∩

∖

⊖ or △

∁

×

⊔

∐

## Logic

¬

∧

∨

⊻

Ɐ

∃

∃!

⇒

⇔

⊤

⊥

## Number theory (aka blackboard bold)

**N**

**Z**

**Z~p~**

**Q**

**Q~p~**

**R**

**C**

**H**

**F~q~**

**O**

## Calculus

□'

◌̇

◌̈

d □/d □

∂ □/∂ □

? □/? □

▭

→

↦

○

∂

∫

∮

∬, ∯

∇ or ∇→

∇^2^ or ∇⋅∇

Δ

∂ or ∂~μ~

◻or ◻^2^

## Linear and multilinear algebra

∑

∏

⊕

⊗

□^⊤^

□^⊥^

## Advanced group theory

⋉

⋊

≀

## Infinite numbers

∞

?

ℵ

ℶ

ω

## Brackets: parentheses

(□)

□(□, ..., □)

(□, □)

(□, □, □)

(□, ..., □)

(□, □, ...)

(matrix)

(□/□)

## Brackets: square brackets

- [□]
- □[□]
- [□, □]
- [□ : □]
- [□, □, □]
- [matrix]

## Brackets: braces

{□}

{□, ..., □}

{□ : □}

{□ | □}

{ (single brace)

## Brackets: other

|□:□|

||□||

⌊□⌋

⌈□⌉

⌊□⌉

]□, □[

(□, □] and ]□, □]

[□, □) and [□, □[

⟨□⟩

⟨□, □⟩ and ⟨□ | □⟩

⟨□| and |□⟩

## Non-mathematical symbols frequently used for reasoning or communication

■ , □

☡

∴

∵

∋

∝

!

*

|

∤

∥

∦

⟂

⊙