How to Read Mathematical Expressions Out Loud

Basic

Expression	How to Read This
$$a + b$$	a plus b
$$a - b$$	a minus b
$$a \times b$$	a times b
$$a / b$$	a over b
$$a > b$$	a is greater than b
$$a < b$$	a is less than b
$$a = b$$	a is equal to b
$$a \neq b$$	a is not equal to b
$$a \geq b$$	a is greater than or equal to b
$$a \leq b$$	a is less than or equal to b
$$x^2$$	x squared
$$x^3$$	x cubed
$$x^n$$	x to the power of n
$$\sqrt{x}$$	the square root of x
$$\sqrt[n]{x}$$	the n-th root of x
$$\log x$$	the logarithm of x (base 10)
$$\ln x$$	the natural logarithm of x (base e)
$$e^x$$	e to the power of x
$$\|x\|$$	the modulus of x
$$s(t)$$	s of t(s as a function of t)
$$f(x)$$	f of x

Expression	How to Read This
$$\sin \theta$$	sine of theta
$$\cos \theta$$	cosine of theta
$$\tan \theta$$	tangent of theta
$$\csc \theta$$	cosecant of theta
$$\sec \theta$$	secant of theta
$$\cot \theta$$	cotangent of theta
$$\sin^{-1} x$$	arcsine of x or inverse sine of x
$$\cos^{-1} x$$	arccosine of x or inverse cosine of x
$$\tan^{-1} x$$	arctangent of x or inverse tangent of x
$$\sin(\alpha \pm \beta)$$	sine of (alpha plus or minus beta)
$$\cos(\alpha \pm \beta)$$	cosine of (alpha plus or minus beta)
$$\tan(\alpha \pm \beta)$$	tangent of (alpha plus or minus beta)
$$\theta \pm \phi$$	theta plus or minus phi
$$\theta \cdot \phi$$	theta times phi
$$\theta / \phi$$	theta over phi

Expression	How to Read This
$$a := b$$	- a is defined by(given as) b (colon equal)
$$\triangleq$$	delta equal, interchangeable with $:=$ (more of personal preference, tradition, or convention)
$$\doteq$$	is approximately equal to
$$\equiv$$	- is equivalent to - is identically equal to (identical under all circumstances)
$$\exists$$	there exists
$$\nexists$$	there does not exist
$$\forall$$	for all

Expression	How to Read This
$$S_1 \Rightarrow S_2$$	S sub one implies S sub two
$$S_1 \Leftrightarrow S_2$$	S sub one is equivalent to S sub two
$\{a, b, c\}$	the set of a, b, and c
$$a \in A$$	a is an element of set A
$\emptyset$ or $\{\}$	empty set
$$A \subset$$ B	A is a proper subset of
$$A \subseteq B$$	A is a subset of B
$$A \cup B$$	A union B
$$A \cap B$$	A intersection B
$$A \setminus B$$	A complement of B
$$\exists p \in \mathbb{P}, p = 2$$	there exists a prime number p such that p equals two.
$$\forall x \in \mathbb{R}, x + 0 = x$$	for all x in the set of real numbers, x plus zero equals x.
$$\mathbb{N}$$	the set of all natural numbers
$$\mathbb{Z}$$	the set of all integers
$$\mathbb{Q}$$	the set of all rational numbers
$$\mathbb{R}$$	the set of all real numbers
$$\mathbb{C}$$	the set of all complex numbers
$$\mathbb{P}$$	the set of all prime numbers

Expression	How to Read This
$$[a, b]$$	closed interval from a to b
$$(a, b)$$	open interval from a to b
$$[a, b)$$	the interval from a to b, inclusive of a and exclusive of b.
$$(a, b]$$	the interval from a to b, exclusive of a and inclusive of b.
$$(a, \infty)$$	the half line starting at a and extending to infinity, exclusive of a.
$$(-\infty, a)$$	the half line starting at negative infinity and extending to a, exclusive of a.

Expression	How to Read This
$$n!$$	n factorial (the product of all positive integers up to n)
$\binom{n}{k}$ or $nC_k$	n choose k (the number of ways to choose k elements from a set of n elements)

Expression	How to Read This
$$\overline{x}$$	x bar (the average of x)
$$\sigma$$	sigma (standard deviation)
$$\mu$$	mu (mean, sound ends with an "ew" as in "new")
$$r$$	r (correlation coefficient)
$$p$$	p-value (probability)
$$\chi^2$$	chi-squared (a statistical test)
$$\hat{p}$$	p-hat (sample proportion)
$$H_0$$	H-naught(null hypothesis)
$$H_a$$	H-a(alternative hypothesis)

Expression	How to Read This
$$\sum_{n=1}^{\infty} a_n$$	the sum of a sub n from n equals 1 to infinity
$$\sum_{k=0}^n \binom{n}{k}$$	the sum from k equals zero to n of n choose k
$$\left(\sum_{k=1}^n a_k b_k\right)^2$$	the square of the sum of the products of a sub k and b sub k from k equals 1 to n
$$\prod_{k=1}^n k$$	the product from k equals one to n of k (n factorial)

Expression	How to Read This
$$\varphi$$	phi (variant)
$$\varphi(0) = 0$$	phi of zero equals zero
$$f : A \to B$$	- f is a function that maps from the set A to the set B - f is a function from A to B(more succinct and standard way) - the function f maps each element of set A to an element in set B(more explanatory)
$$x \mapsto f(x)$$	x is mapped to f(x)
$$g: \mathbb{R} \to \mathbb{C}, \quad \theta \mapsto g(\theta) := e^{i\theta}$$	g from R(real numbers) to C(complex numbers), theta maps to g of theta, defined as(given by) e to the i theta
$$f: \mathbb{R} \to \mathbb{R}, \quad x \mapsto f(x) := 1 + x^2$$	f from R(real numbers) to R, x maps to f of x, defined as(given by) one plus x squared
$$h: \mathbb{C} \to [0, \infty), \quad z \mapsto h(z) := \|z\|$$	function h maps complex numbers C to the set of non-negative real number from zero to infinity, z maps to function h of z equals the modulus of z
$$\text{im}(f)$$	image or range of f
$$\text{im}(f) = B$$	the image(range) of f equals B (the image (or range) of the function f is equal to the set B)
$$f: A \to \text{im}(f)$$	f from A to the image of f (the image of the function, includes all the outputs the function can produce as its input varies over the set A.)
$$f^{-1}$$	inverse function of f
$$f^{-1}(b) := a$$	f inverse of b is defined as(given by) a
$$f_{i,j}$$	f sub i comma j

Expression	How to Read This
$$\frac{d}{dx}$$	d by dx (the derivation with respect to x)
$$\frac{d}{dx} f(x)$$	the derivative of f with respect to x
$$f'(x)$$	f prime of x (the derivative of f with respect to x)
$$\int f(x) dx$$	the integral of f with respect to x
$$\int_a^b f(x) dx$$	the definite integral of f from a to b with respect to x
$$\lim_{x \to a} f(x)$$	the limit of f as x approaches a

Expression	How to Read This
$$\nabla$$	nabla or del
$$\nabla f$$	gradient of(nabla) f
$$\nabla \cdot \mathbf{F}$$	divergence of(nabla dot) F
$$\nabla \times \mathbf{F}$$	curl of(nabla cross) F
$$\partial$$	partial
$$\frac{\partial}{\partial x} f(x)$$	the partial derivative of f with respect to x
$$\frac{\partial f}{\partial x}$$	partial of f with respect to x
$$\frac{\partial^2 f}{\partial x^2}$$	- the second partial derivative of f with respect to x squared - the second partial of f with respect to x
$$\frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right)$$	The partial derivative with respect to x sub j of the partial derivative of f with respect to x sub i
$$\frac{\partial^2 f}{\partial x_j \partial x_i}$$	The second partial derivative of f with respect to x sub j and x sub i