Math Symbols

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:

  • xy (aka superscript)
    • Exponents, which are multiple iterations of multiplication
    • e.g., 23 = 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:

  • ¬
  • ∃!

Number theory (aka blackboard bold):

  • N
  • Z
  • Zp
  • Q
  • Qp
  • R
  • C
  • H
  • Fq
  • O

Calculus:

  • □’
  • ◌̇
  • ◌̈
  • d □/d □
  • ∂ □/∂ □
  • ? □/? □
  • ∬, ∯
  • ∇ or ∇→
  • 2 or ∇⋅∇
  • Δ
  • ∂ or ∂μ
  • ◻or ◻2

Linear and multilinear algebra:

Advanced group theory:

Infinite numbers:

  • ?
  • 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:

  • ■ , □
  • !
  • *
  • |