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|

- The absolute value of a number (i.e., how far it is from 0), which always gives a positive number

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

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

- Approximately equal, where two things are
**≡**(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)

- √ (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 indicate also cube roots (∛), fourth roots (∜), and so on

Set theory:

- ∅
- #
- ∈
- ∉
- ⊂
- ⊆
- ⊊
- ⊃, ⊇, ⊋
- ∪
- ∩
- ∖
- ⊖ 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 that are used frequently for reasoning or communication: