Reference and Synthesis
Mathematical Notation and Symbols: a Reference
Every symbol used across this collection, grouped by area, with its reading and a note on where it is easily confused.
What this page covers
- Read any symbol used in the collection
- Distinguish symbols that are easily confused
- Use set and interval notation correctly
- Follow the conventions for upright and italic type
Number sets
| Symbol | Name | Members |
|---|---|---|
| ℕ | Counting (natural) numbers | 1, 2, 3, … |
| ℤ | Integers | …, -1, 0, 1, … |
| ℚ | Rational numbers | All p/q, q ≠ 0 |
| ℝ | Real numbers | Every point on the line |
| ℂ | Complex numbers | All a + bi |
The containment ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ holds, and each is strict.
Sets, intervals and logic
| Symbol | Reading | Example |
|---|---|---|
| { } | The set of | {-3, 3} |
| ∈ | is an element of | x ∈ ℝ |
| : or | | such that | {x : x ≥ 5} |
| ⊂ | is a subset of | ℕ ⊂ ℤ |
| ∪ | union | (-∞, 3) ∪ (7, ∞) |
| ∩ | intersection | A ∩ B |
| ∅ | the empty set | No solutions |
| (a, b) | open interval | Both endpoints excluded |
| [a, b] | closed interval | Both included |
| (a, b] | half-open | a excluded, b included |
| ∞ | infinity | Never an endpoint value |
| ⇒ | implies | ab = 0 ⇒ a = 0 or b = 0 |
| ⇔ | if and only if | Equivalence |
| ∴ | therefore | Concluding a chain |
[3, ∞) is correct; [3, ∞] is not. Infinity is a direction, not a value, and cannot be attained.
Operations and relations
| Symbol | Reading | Note |
|---|---|---|
| ± | plus or minus | Two values, not one |
| ≠ | is not equal to | |
| ≤, ≥ | less/greater than or equal to | Endpoints included |
| ≈ | is approximately equal to | For rounded values |
| |x| | the absolute value of x | Distance from zero; never negative |
| √x | the square root of x | The principal root, non-negative |
| n√x | the nth root of x | = x1/n |
| x-1 | the reciprocal of x | = 1x |
| Δ | the discriminant | = b2 - 4ac |
| · | times | Preferred over × in algebra |
The radical denotes one number, the principal root. The ± appears when solving x2 = 4, because that equation has two solutions.
Functions
| Symbol | Reading | Warning |
|---|---|---|
| f(x) | f of x | Not f times x |
| f : x → y | f sends x to y | Arrow notation |
| f(-x) | f evaluated at -x | Used to test parity |
| ° f | the degree of f | For polynomials |
| sin, cos, tan | the trigonometric functions | Upright type; they are names, not products |
| sin2t | (sin t)2 | Not sin(t2) |
| sin-1t | the inverse sine | Not 1sin t, which is csc t |
sin2t means the square of the sine, but sin-1t means the inverse function, not the reciprocal. The exponent notation breaks its own rule at -1. The source flags this explicitly in Week 8.
Greek letters in use
| Letter | Name | Typical use |
|---|---|---|
| π | pi | The circle constant; also π radians = 180° |
| θ | theta | An angle |
| φ | phi | A direction angle for a vector |
| α, β | alpha, beta | Roots of a quadratic; reference angles |
| Δ | capital delta | The discriminant |
| Σ | capital sigma | Summation |
Vectors and matrices
| Symbol | Reading | Note |
|---|---|---|
| v or v | a vector | The source underlines vectors by hand |
| (v1, v2) | components | Also written with angle brackets |
| |v| | the magnitude of v | = √v12 + v22 |
| u · v | the dot product | A scalar, not a vector |
| A = (aij) | a matrix with entries aij | Row index first |
| m × n | order: m rows, n columns | Rows first |
| I | the identity matrix | AI = IA = A |
| O | the zero matrix | A + O = A |
| A-1 | the inverse of A | AA-1 = I |
| |A| | the determinant of A | A number, not a magnitude |
| Mij | the minor of aij | A determinant |
| [A | b] | an augmented matrix | The rule marks the equals signs |
|v| and |A| use the same bars for different things: a vector magnitude and a matrix determinant. Context distinguishes them, but the collision is worth being aware of.
Type conventions
| Set in | Used for | Example |
|---|---|---|
| Italic | Variables and constants that vary | x, y, a, n |
| Upright | Function names | sin, cos, log |
| Upright | Digits and units | 5, ° |
| Upright | Mathematical constants | e, i in some conventions |
| Blackboard bold | Number sets | ℝ, ℂ |
The distinction matters. Italic sin would read as three variables s, i, n multiplied together; upright sin is unambiguously the function.
Frequently asked questions
Why are variables italic and function names upright?
It is the international convention (ISO 80000-2), and it removes ambiguity: upright sin is the sine function while italic s, i, n would be three variables multiplied together.
What is the difference between (a, b) and [a, b]?
Round brackets exclude the endpoint, square brackets include it. So (1, 5] excludes 1 and includes 5. The same round-bracket notation is also used for an ordered pair, and context distinguishes them.
Does sin-1x mean 1sin x?
No. It means the inverse sine function. The reciprocal is csc x. This is the one place where the exponent notation breaks its own pattern, and the source flags it.
What is the difference between = and ≡?
= asserts equality for particular values; ≡ asserts it identically, for all values. The distinction matches conditional equations against identities.
Source. Compiled from the whole source: twelve weeks of teaching notes and the supplementary sheets.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.