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How to Read Math Expressions

Learn to Pronounce Math Symbols and Formulas

Basic

Expression	How to Read	Pronounce
$$a + b$$	a plus b
$$a - b$$	a minus b
$$a \cdot b$$	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
$$\overline{x}$$	x bar (the average of x)
$$a \pm b$$	a plus or minus b
$$\frac{a}{b}$$	a over b (fraction)
$$a!$$	a factorial

Numbers

Expression	How to Read	Pronounce
$$-5$$	negative five
$$0.25$$	zero point two five
$$1.5$$	one point five

Fractions

Expression	How to Read	Pronounce
$$\frac{1}{2}$$	one half
$$\frac{3}{4}$$	three quarters
$$\frac{1}{7}$$	one seventh

Powers and Roots

Expression	How to Read	Pronounce
$$3^2$$	three squared
$$5^4$$	five to the power of four
$$\sqrt{4}$$	the square root of four
$$\sqrt[3]{8}$$	the cube root of eight

Angle Operations

Expression	How to Read	Pronounce
$$\sin \theta$$	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
$$\sinh x$$	hyperbolic sine of x
$$\cosh x$$	hyperbolic cosine of x
$$\tanh x$$	hyperbolic tangent of x

Logic and Sets

Expression	How to Read	Pronounce
$$a := b$$	a is defined by(given as) b
$$S_1 \Rightarrow S_2$$	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
$$\emptyset \text{ or } \{\}$$	empty set
$$A \subset B$$	A is a proper subset of B
$$A \subseteq B$$	A is a subset of B
$$A \cup B$$	A union B
$$A \cap B$$	A intersection B
$$A \setminus B$$	A minus B
$$\equiv$$	is equivalent to
$$\exists$$	there exists
$$\nexists$$	there does not exist
$$\forall$$	for all
$$\exists p \in \mathbb{P}, p = 2$$	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
$$A \triangle B$$	symmetric difference of A and B
$$A^c$$	complement of A
$$\overline{A}$$	closure of A

Intervals

Expression	How to Read	Pronounce
$$[a, b]$$	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.

Factorials and Binomial Coefficients

Expression	How to Read	Pronounce
$$n!$$	n factorial (the product of all positive integers up to n)
$$\binom{n}{k} \text{or} nC_k$$	n choose k (the number of ways to choose k elements from a set of n elements)
$$P(n,k)$$	permutation of n things taken k at a time

Sum and Product

Expression	How to Read	Pronounce
$$\sum_{n=1}^{\infty} a_n$$	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
$$( \sum_{k=1}^n a_k b_k )^2$$	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)
$$\bigcup_{i=1}^n A_i$$	union of sets A sub i from i equals 1 to n
$$\bigcap_{i=1}^n A_i$$	intersection of sets A sub i from i equals 1 to n

Functions

Expression	How to Read	Pronounce
$$\varphi$$	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
$$x \mapsto f(x)$$	x is mapped to f(x)
$$g: \mathbb{R} \to \mathbb{C}, \theta \mapsto g(\theta) := e^{i\theta}$$	g from R(real numbers) to C(complex numbers), theta maps to g of theta, defined as(given by) e to the i theta
$$f: \mathbb{R} \to \mathbb{R}, x \mapsto f(x) := 1 + x^2$$	f from R(real numbers) to R, x maps to f of x, defined as(given by) one plus x squared
$$h: \mathbb{C} \to [0, \infty), z \mapsto h(z) := \|z\|$$	function h maps complex numbers C to the set of non-negative real numbers 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
$$f: A \to \text{im}(f)$$	f from A to the image of f
$$f^{-1}$$	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
$$f \circ g$$	f composed with g
$$\ker(f)$$	kernel of f
$$f(x)$$	f of x
$$\lim_{x \to 0}$$	x approaches 0

Calculus Operation Notations

Expression	How to Read	Pronounce
$$\frac{d}{dx}$$	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
$$\oint f(x) dx$$	the line integral of f with respect to x

Vector calculus and physics

Expression	How to Read	Pronounce
$$\nabla$$	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
$$\frac{\partial}{\partial x_j} ( \frac{\partial f}{\partial x_i} )$$	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
$$\mathbf{a} \cdot \mathbf{b}$$	dot product of vectors a and b
$$\mathbf{a} \times \mathbf{b}$$	cross product of vectors a and b

Matrix Operations

Expression	How to Read	Pronounce
$$A^T$$	transpose of matrix A
$$A^{-1}$$	inverse of matrix A

Complex Numbers

Expression	How to Read	Pronounce
$$z = a + bi$$	z equals a plus b i (complex number)
$$\|z\|$$	modulus of z
$$\arg(z)$$	argument of z
$$\overline{z}$$	complex conjugate of z

Differential Equations

Expression	How to Read	Pronounce
$$\frac{dy}{dx} = f(x,y)$$	dy by dx equals f of x and y (first-order differential equation)
$$\frac{d^2y}{dx^2} + p(x)\frac{dy}{dx} + q(x)y = r(x)$$	d squared y by dx squared plus p of x times dy by dx plus q of x times y equals r of x (second-order linear differential equation)

Source

What is the Math Expression Reader?

The Math Expression Reader is a powerful yet user-friendly tool designed to help you understand and pronounce mathematical symbols and formulas. Whether you're dealing with basic arithmetic, complex algebra, or advanced calculus, our tool provides clear and professional guidance on how to read each expression correctly.

This tool is perfect for students, educators, and anyone looking to improve their mathematical literacy. With easy-to-follow instructions and examples, you can confidently navigate through various mathematical expressions and enhance your learning experience.

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