Dumb Data: 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) 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

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

∞

?

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

■ , □

☡

∴

∵

∋

∝

!

*

|

∤

∥

∦

⟂

⊙