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 generaluse 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:
 With #S as the number of elements, may alternatively represent as S
 With n# as the sum of prime numbers up to n (i.e., primorial)
 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 □⟩
Nonmathematical symbols frequently used for reasoning or communication
■ , □
☡
∴
∵
∋
∝
!
*

∤
∥
∦
⊙