Mathematical Notations

List of the most used mathematical notations that are used in expressions and formulas.

The list includes:

Essential Notations

$$\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n$$

Summation notation or Sigma notation

$$\prod_{i=1}^{n} a_i = a_1 \cdot a_2 \cdot \cdots \cdot a_n$$

Product notation or Pi notation

$$\int_a^b f(x) \, dx$$ Integral notation

$$\lim_{x \to a} f(x) = L$$ Limit notation

$$\frac{d}{dx} f(x) = f'(x)$$ Derivative notation [Leibniz’s notation]

$$\frac{\partial}{\partial x} f(x, y)$$ Partial derivative notation

$$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$$ Vector notation

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$ Matrix notation

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ Binomial coefficient notation

$$A = \{x \in \mathbb{R} : x^2 < 4\}$$ Set notation

Conditionals and Operators

  • $ r /; c$ : Relation $ r $ holds under the condition $ c$ .
  • $ a=b$ : The expression $a$ is mathematically identical to $ b$ .
  • $ a \ne b$ : The expression a is mathematically different from $ b$ .
  • $ x > y$ : The quantity $ x$ is greater than quantity $ y$ .
  • $ x \ge y$ : The quantity $ x$ is greater than or equal to the quantity $ y$ .
  • $ x < y$ : The quantity $ x$ is less than quantity $ y$ .
  • $ x \le y$ : The quantity $ x$ is less than or equal to quantity $ y$ .
  • $ P := Q$ : Statement $ P$ defines statement $ Q$ .
  • $ a \wedge b$ : a and b.
  • $ a \vee b$ : a or b.
  • $ \forall a$ : for all $ a$ .
  • $ \exists$ : [there] exists.
  • $ \iff$ : If and only if.

Sets & Domains

  • $ \{ a_1, a_2, \ldots, a_n \}$ : A finite set with some elements $ a_1, a_2, \ldots, a_n$ .
  • $ \{ a_1, a_2, \ldots, a_n \ldots \}$ : An infinite set with elements $ a_1, a_2, \ldots$
  • $ \mathrm{\{ listElement /; domainSpecification\}}$ : A sequence of elements `listElement` with some `domainSpecifications` in the set. For example, $ \{ x : x=\frac{p}{q} /; p \in \mathbb{Z}, q \in \mathbb{N^+}\}$ $ a \in A$ : $ a$ is an element of the set A.
  • $ a \notin A$ : a is not an element of the set A.
  • $ x \in (a,b)$ : The number x lies within the specified interval $ (a,b)$ .
  • $ x \notin (a,b)$ : The number x does not belong to the specified interval $ (a,b)$.

Standard Set Notations

  • $ \mathbb{N}$ : the set of natural numbers $ \{0, 1, 2, \ldots \}$
  • $ \mathbb{N}^+$ : The set of positive natural numbers: $ \{1, 2, 3, \ldots \}$
  • $ \mathbb{Z}$ : The set of integers $ \{ 0, \pm 1, \pm 2, \ldots\}$
  • $ \mathbb{Q}$ : The set of rational numbers
  • $ \mathbb{R}$ : The set of real numbers
  • $ \mathbb{C}$ : The set of complex numbers
  • $ \mathbb{P}$ : The set of prime numbers.
  • $ \{ \}$ : The empty set.
  • $ \{ A \otimes B \} $ : The ordered set of sets $ A$ and $ B$ .
  • $ n!$ : Factorial of n: $ n!=1\cdot 2 \cdot 3 \ldots (n-1) n /; n \in \mathbb{N}$