In the language of Mathematics, there are many strange symbols that can complicate papers and proofs for those without knowledge of them (take the Standard Model Lagrangian equation, for example). In this page I will collect every strange Mathematical or Physics-related symbol that I come across in my studies. NOTE: This is not inclusive of electrical engineering circuitry symbols or those of any other discipline. E.E. symbols are found here.

In the unlikely event there are any egregious errors in this page, contact me at jdlacabe@gmail.com.

+

The plus sign. A symbol for addition in arithmetic.
Example: 5 + 906 = 911

-

The minus sign. The symbol for subtraction in arithmetic. It is the opposite of the plus sign and can also denote negative numbers.
Examples: 20 - 5 = 15
-7 - 12 = -19

×, •

The multiplication symbol. The symbol denoting multiplying two numbers together. Can also be represented using a dot.
Examples: 3 × 5 = 15
-1 • 0 = 0

÷, /

The division symbol. The symbol denoting division, the opposite of multiplication. It can also be represented using a slash.
Example: 21 ÷ 3 = 7

±

Plus-Minus. Denotes that a number can be either positive or negative. It can also be used to denote a range of values, such as 5±3 having a range from 2 to 8.
Example: 2 ± 1 = 1, 3

∓

Minus-Plus. A symbol used in conjunction with the plus-minus sign to denote the opposite sign of plus-minus.
Example: 1 ± 2 ∓ 3 = 1 + 2 - 3 OR 1 - 2 + 3, not anything else.

=

Equals. A symbol denoting equality between two expressions.
Example: 1 + 2 = 3

≠

The not equals sign. The opposite of the equals sign, indicating that two expressions are not equal.
Example: 6 + 2 ≠ 3

≈

The approximately equals sign. Used when 2 values are not eactly equal but are close enough. A tilde can also be used, but it has other specialized purposes as well.
Example: 12.00910239012931 + 1.999923891273 ≈ 14

~

Tilde. A symbol that indicates proportionality/similarity between two geometric figures. Image.
Example: ABC ~ DEF

∝

The proportional symbol. Can be used for more explicit reference to proportionality in place of ~.
Example: y = k × x, and thus y ∝ x.

≡

The Triple Bar or Equivalence Bar. Used to denote an identity or definition, commonly used in modular arithmetic (see Cryptography).
Example: 22 ≡ 2(mod 10)

√

The square root. Denotes the square root of a number, e.g. what number multiplied by itself gives x. Additional roots exist as well, with an integer greater than two serving as n as the superscript to the left of the root symbol: ⁿ√x. This denotes the nᵗʰ root of x.
Example: 1 + 2 = 3

<

The less-than symbol. Used to indicate that one quantity is smaller than another.
Example: 0.99999 < 1

>

The greater than sign. Indicates that one quantity is larger than the other. It is the opposite of the less-than sign.
Example: 8 > 5

≤

The less than or equal to symbol. Indicates that one value is smaller or equal to another.
Example: 2 ≤ 9
2 ≤ 2

≥

The greater than or equal to symbol. Indicates that one value is greater or equal to another.
Example: 13 ≥ 1
13 ≥ 13

≪

The much less than sign. Denotes one value is much smaller than the other.
Example: 0.02 ≪ 10000

≫

The much greater than sign. Denotes one value is much greater than the other.
Example: 20349823 ≫ 3

||x||

The Magnitude. Simply represents the magnitude of a particular value, the directionless 'amount' of something.
Example: ||a|| = -9.81 m/s²

θ

Theta. Used as the primary variable to represent an angle.
Example: In a right triangle, θ₁ + θ₂ + θ₃ = 180°.

Φ

Phi. An alternate angle symbol, oftentimes used when theta is already taken by another angle.
Example: tan(Φ) = (sin(Φ) / cos(Φ))

∅

An empty set. Denotes a set that contains no elements.
Example: ∅ = {}

#

The number sign, also known as the pound sign, octothorpe, or hashtag. This denotes the cardinality of the set, e.g. the number of elements in the set.
Example: P = {2, 4, 5}
#P = 3

∈

The element symbol, also known as the member symbol. Used to denote membership of an object (or objects) in a set.
Example: 2 ∈ P

∉

The not element of symbol, or not member of symbol. It denotes that an element is not a member of a set.
Example: 8 ∉ P

⊂

The set inclusion sign. Denotes that one set is a subset of another set.
Example: P = {2, 4, 5}
Q = {4, 5}
R = {4, 5}
Q ⊂ P

⊆

The set inclusion sign (but more specific and used than the one above). Denotes set inclusion, in addition to emphasizing that the sets can be equal.
Example: P = {2, 4, 5}
Q = {4, 5}
R = {4, 5}
Q ⊆ P

⊊

The proper subset symbol. Denotes set inclusion where the sets are not equal.
Example: P = {2, 4, 5}
Q = {4, 5}
R = {4, 5}
R ⊂ P

∪

The Union symbol. Denotes an operation of combining two sets, which results in another set containing all unique elements from both sets. Oftentimes, it is used to combine intervals on a graph for some purpose, such as where x is increasing or decreasing.
Example: A = {2, 3, 4, 5}
B = {3, 5, 6, 7}
A ∪ B = {2, 3, 4, 5, 6, 7}

∩

The intersection symbol. Denotes an operation of combining 2 sets, the result of intersection is a set which contains elements that are common in both sets.
Example: A = {2, 3, 4, 5}
B = {3, 5, 6, 7}
A ∩ B = {3, 5}

\

The set difference symbol. Denoted by a backslash, the result of this operation is a set which contains all elements of the first set that are not in the second set.
Example: A = {2, 3, 4, 5}
B = {3, 5, 6, 7}
A \ B = {2, 4}

Δ

The delta symbol, represents 'the change in' some value. For example: Δx represents the displacement, or change in position of some value - see here for a more detailed explanation. It is occasionally used interchangeably with ⊖ for symmetric differences.
Example: y = x + 5 has a Δx of 5.

ω

The angular velocity in rotational motion.
Example: ω_avg = (∆θ / ∆t)

α

Alpha. Can be used to represent angular acceleration.
Example: α = ((ω_F - ω_i) / ∆t)

⊖

The symmetric difference symbol. Denotes an operation which creates a set of all elements that belong to exactly one of the 2 sets. Can also be represented using Δ.
Example: A = {2, 3, 4, 5}
B = {3, 5, 6, 7}
A ⊖ B = {2, 4, 6, 7}

¬

The negation symbol. Used in logic to indicate the opposite of a statement.
Example: P = false
¬P = true

∨

The "or" operator. Returns true if at least one of the operands is true. It works akin to the || operator in programming.
Example: P = false, Q = true
P ∨ Q = true

∧

The "and" operator. Returns true only if both operands are true. It works akin to the && operator in programming.
Example: P = false, Q = true
P ∨ Q = false

⊕

The "Exclusive or" operator. Returns true if exactly one of the operands is true. Is also known as an XOR gate.
Example: P = false, Q = true
P ⊕ Q = true

⊤

The "Tee". Denotes the logical constant for a true value or a statement that is always true.
Example: R ∨ ¬R = ⊤

⊥

Uptack. Denotes the logical constant for a false value or a statement that is always false. Furthermore, this symbol can also be used to denote Perpendicularity, or that 2 numbers are coprime ([[[).
Example: R ∧ ¬R = ⊥ (logical constant)
AB⊥EF (perpendicular)
a⊥b (coprime)

∀

The Universal Quantifier. Asserts that a statement is true for all elements in a given domain. Also, this symbol can occasionally be used in place of V, such as in reference to volume when an equation has already used velocity.
Example: ∀n ∈ ℕ (2n ≥ n)
"For all natural numbers n, 2n is greater or equal to n"

∃

The Existential Quantifier. Asserts that there exists at least one element in a given domain for which a particular statement holds true.
Example: ∃x (x > 5)
"There exists at least one number x that is bigger than 5"

∃!

The Uniqueness Quantifier. Used to assert that there is exactly one element in a given domain for which a particular statement holds true.
Example: ∃!n ∈ ℕ (n = 5)
"There exists exactly one natural number n that is equal to 5"

⇒

The conditional operator. Denote an implication between 2 statements. If the first element is true, then the second is also true
Example: x > 2 ⇒ x >1
"If x is bigger than 2, then x is also bigger than 1"

⟺

The logical equivalence symbol. Indicates that 2 statements have the same logical value.
Example: x = 0 ⟺ x² = 0
"x is equal to 0 if and only if x² is also equal to 0"

⨍′ / f'

The Lagrangian notation for the derivative of a function. By adding additional apostrophes, further level derivatives can be denoted. For example, three apostrophes denotes the third-level derivative.
Example: ⨍(x) = x²
⨍′(x) = 2x

ẋ

The Newtonian notation for the derivative of a function. Similar to the apostrophe of the Lagrangian notation, adding more dots wil increase the level of the derivative.
Example: x(t) = x²
ẋ = 2x

df/dx

The Leibniz notation for the derivative of a function. Differing from the Lagrangian and Newtonian systems, the notation specifically represents the derivation of the function/variable on top with respect to the variable at the bottom. Adding exponents to the upper d and the lower variable will specific the specific level derivative.
Example: y² + 4x³ = 9y - 3x²
df/dx = (-12x² - 6x) / (2y - 9)

∂f/∂x

The Leibniz notation for the partial derivative of a function. Partial derivatives are used for functions of several variables (particularly in Multivariable Calculus), such as f(x, y, z, ...).

Using different characters in the notation result in different variables being taken the partial different in lines with the rules as taught in multivariable calculus. ([[[) Adding exponents to the upper curved d and the lower variable will specific the specific level derivative. ([[[)

Example: f(x, y) = 2x² + 3xy + 6x + 7y
∂f/∂x = 4x + 3y + 6

∫

The integral symbol. Denotes an antiderivative, which is the opposite of the derivative. When adding a subscript and superscript, it denotes a definite integral, which represents the area under a curve or the accumulation of a quantity over an interval. Used in Calculus and all its applications.
Example: $$\int 1 \, dx = x + c$$ $$\int_{x_i}^{x_f} 1 \, dx = x_f - x_i$$

↦

The "Maps to", "Maplet", or simply the Arrow symbol. Used to define a function without having to name it.
Example: x ↦ 3x² - 1

∘

The function composition symbol - an operation that combines two functions, with one function, forming the internal of another.
Example: (f ∘ g)(x) = f(g(x))

log

The Logarithm symbol. Denotes the inverse operation of exponentiation. Subscript denotes the base of the logarithm, while the log without a subscript represents the "Common Logarithm", with base 10. The "Natural Logarithm" has its own denotation, seen below.
Example: log_ba = c → bᶜ = a
log x = log₁₀x

ln

The Natural Logarithm symbol. Denotes a logarithm with base e
Example: ln e = 1

lim

The limit symbol. Used to denote the behavior of a function or an expression as its input approaches a certain value.
Example: $$\lim_{x \to 3} \frac{x + 5}{2} = 4$$

ℜ

Fraktur R, or Fraktur-R. Represented by the calligraphic/fancy R, it denotes the real part of a complex number.
Example: x = 7 - 3i
ℜ(x) = 7

ℑ

Fraktur I, or Fraktur-I. Represented by the calligraphic/fancy I, it denotes the imaginary part of a complex number.
Example: x = 7 - 3i
ℑ(x) = -3

x̄

The bar symbol. When placed above a complex number, it denotes a complex conjugate of that number, which just changes the sign of the imaginary part of the expression.
Example: x = 7 - 3i
x̄ = 7 + 3i

Σ

The capital greek letter Sigma. Denotes the summation of a series of terms.
Example: $$\sum_{i=a}^{b} i^2 = a^2 + (a+1)^2 + (a+2)^2 + \dots + b^2$$

σ

Lowercase sigma. Can be used to represent Surface Mass Density.
Example: σ = m / A.

∏

Capital Pi, used as the Product Operator. Works similar to the summation symbol (sigma), but denotes a product instead, meaning every term is being multiplied instead of added.
Example: $$\prod_{i=a}^{b} i^2 = a^2 \times (a+1)^2 \times (a+2)^2 \times \dots \times b^2$$

τ

The 'Tau' symbol. As a variable, it can be used to represent Torque.
Example: τ = r × F × sinθ

∞

The infinity symbol. Denotes the unreachable concept of 'unlimitedness', a value that is greater than any finite quantity. There is no number above it - infinity plus one is balderdash.
Example: ∞ > 10^99999999999999999999
∞ = ∞ + 1
∞ = ∞^∞

ℵ

The Aleph symbol. Used to represent the 'cardinality' of infinite sets, e.g. the number of numbers with a set. Aleph-null, for example, represents the cardinality of the set of natural numbers.
Example: #ℕ = ℵ₀

𝔠

Fraktur C, or Fraktur-C. Represented by the calligraphic/fancy C, it denotes a type of infinity, akin to the aleph - it represents the cardinality of the set of real numbers.
Example: #ℝ = 𝔠

!

The Factorial symbol (an exclamation mark). Denotes the Factorial operation, in which a number is multiplied by all positive integers smaller than that number.
Example: n! = 1 × 2 × 3 × ... × n

$(n; k)$

The Binomial Coefficient symbol/notation. Denotes the number of ways to choose k elements from a set of n elements.
Example: $$\binom{n}{k} = \frac{n!}{k! (n - k)!}$$

|x|

The Absolute Value symbol. Denotes the distance of a number from zero on the number line, essentially serving to return the positive version of whichever value. Thus,
|x| = x if x ≥ 0
|x| = -x if x < 0

Example: |3| = 3
|-4| = 4

⌊x⌋

The Floor function. Returns the greatest integer less than or equal to the given value:
n ∈ ℕ
⌊x⌋ = n if n ≤ x < (n + 1)

Example: ⌊3⌋ = 3
⌊4.88⌋ = 4

⌈x⌉

The Ceiling function. Returns the smallest integer larger than or equal to the given value.
n ∈ ℕ
⌈x⌉ = n if (n - 1) < x ≤ n

Example: ⌈3⌉ = 3
⌈3.11⌉ = 4

⌊x⌉

The Nearest Integer Function. Returns the nearest integer to a given value.
n ∈ ℕ
⌊x⌉ = n if (n - 0.5) ≤ x < (n + 0.5)

Example: ⌊3.22⌉ = 3
⌊3.63⌉ = 4

|

The divisibility symbol. A single line, it is used to denote divisibility between two variables.
Example: a|b = a divides b, or a is a factor of b. Thus, b ÷ a produces an integer.

∤

The Non-divisibility symbol. Denotes that two variables do not hold divisibility (that one is not a factor of the other).
Example: a∤b = a does not divide b, or a is a NOT a factor of b.

∥

The parallel symbol. Denotes that two lines/line segments are parallel. When representing two lines as being 'antiparallel' is of any significance (determined by the 'sense' of the lines, see math Rule [[[[), then the bars can be swapped out for arrows: ↑↑ for parallel, and ↑↓ for antiparallel.
Example: AB||CD

∦

The Non-Parallel symbol. Denotes that two lines/line segments are NOT parallel.
Example: AB∦EF

AB

The Line Segment symbol. A bar over two letters (representing two points) represents a line segments between those points.
Example: AB ∥ BC

AB

The Ray symbol. An arrow over two letters denotes a ray, starting at the first point and ending at the second. Also used to denote a vector.
Example: AB · CD = |AB| times |CD| times cosϕ

AB

The "Infinite Line" symbol. Denotes an infinite line passing through both points.
Example: The domain and range of AB, as long as it is not in a unit direction, is -∞, ∞.

Blackboard Bold Typeface

These symbols typically denote the basic number systems. Generally, you would not consider ∞ to be a member of any of these sets, as they are all defined to not have a largest element and ∞ is defined as being greater than any element.

ℕ

The set of a natural numbers. All positive whole numbers.
Example: 1 ∈ ℕ
1.01 ∉ ℕ
0 ∉ ℕ

ℤ

The set of integers. Any whole number, positive or negative.
Example: 1 ∈ ℤ
-4 ∈ ℤ
-0.1 ∉ ℤ

ℚ

The set of rational numbers. Any number that can be represented as a fraction of whole numbers, positive or negative.
Example: 1 ∈ ℚ
-1/12 ∈ ℚ
e ∉ ℚ

ℝ

The set of real numbers. The set of numbers that includes both rational and irrational numbers, without regard for whole numbers.
Example: 0 ∈ ℝ
π ∈ ℝ
√-1 ∉ ℝ

ℂ

The set of complex numbers. Incorporates real numbers and imaginary numbers into one set.
Example: e ∈ ℕ
√-1 ∈ ℕ
∞ ∉ ℕ

ℍ

The set of quaternions. Numbers that extend complex numbers by introducing three imaginary units, 𝑖, 𝑗, 𝑘, such that 𝑖² = 𝑗² = 𝑘² = 𝑖 × 𝑗 × 𝑘 = -1. Quaternions are used in 3D rotations and transformations.
Example: 1 + 2i + 3j + 4k ∈ ℍ
0 ∈ ℍ
√-1 ∉ ℍ

𝕆

The set of octonions. Numbers that generalize quaternions by introducing additional dimensions, with seven imaginary units. Octonions are non-commutative and non-associative but are still used in advanced mathematics and physics.
Example: 1 + 2i + 3j + 4k + 5e₁ + 6e₂ + 7e₃ + 8e₄ ∈ 𝕆
0 ∈ 𝕆
1 + i ∉ 𝕆 (only part of a quaternion subset)

𝕌

The Universal set, the set that contains all possible values.
Example: ℕ ⊂ 𝕌
ℚ ⊂ 𝕌
𝕌 ∉ 𝕌

Constants.

NOTE: MANY OF WHAT ARE DESCRIBED AS CONSTANTS BELOW, ARE ONLY "CONSTANTS" WHEN USED IN THEIR SPECIFIC CONTEXT. THE INDIVIDUAL VARIABLES (if any) THAT REPRESENT THE CONSTANT ARE OFTEN REPURPOSED IN PROBLEMS TO SERVE AS GENERIC VARIABLE TERMS.

π

The ratio of a circle's circumference to its diameter.
Approximate Value: 3.14159265358...

τ

The ratio of a circle's circumference to its radius. This literally just 2 pi.
Approximate Value: 6.28318531...

e