# How to Read Math Expressions

## Learn to Pronounce Math Symbols and Formulas

## Basic

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$a + b$$ | a plus b | Translate | |

$$a - b$$ | a minus b | Translate | |

$$a \cdot b$$ | a times b | Translate | |

$$a / b$$ | a over b | Translate | |

$$a > b$$ | a is greater than b | Translate | |

$$a < b$$ | a is less than b | Translate | |

$$a = b$$ | a is equal to b | Translate | |

$$a \neq b$$ | a is not equal to b | Translate | |

$$a \geq b$$ | a is greater than or equal to b | Translate | |

$$a \leq b$$ | a is less than or equal to b | Translate | |

$$x^2$$ | x squared | Translate | |

$$x^3$$ | x cubed | Translate | |

$$x^n$$ | x to the power of n | Translate | |

$$\sqrt{x}$$ | the square root of x | Translate | |

$$\sqrt[n]{x}$$ | the n-th root of x | Translate | |

$$\log x$$ | the logarithm of x (base 10) | Translate | |

$$\ln x$$ | the natural logarithm of x (base e) | Translate | |

$$e^x$$ | e to the power of x | Translate | |

$$|x|$$ | the modulus of x | Translate | |

$$\overline{x}$$ | x bar (the average of x) | Translate | |

$$a \pm b$$ | a plus or minus b | Translate | |

$$\frac{a}{b}$$ | a over b (fraction) | Translate | |

$$a!$$ | a factorial | Translate |

## Numbers

## Fractions

## Powers and Roots

## Angle Operations

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$\sin \theta$$ | sine of theta | Translate | |

$$\cos \theta$$ | cosine of theta | Translate | |

$$\tan \theta$$ | tangent of theta | Translate | |

$$\csc \theta$$ | cosecant of theta | Translate | |

$$\sec \theta$$ | secant of theta | Translate | |

$$\cot \theta$$ | cotangent of theta | Translate | |

$$\sin^{-1} x$$ | arcsine of x or inverse sine of x | Translate | |

$$\cos^{-1} x$$ | arccosine of x or inverse cosine of x | Translate | |

$$\tan^{-1} x$$ | arctangent of x or inverse tangent of x | Translate | |

$$\sin(\alpha \pm \beta)$$ | sine of (alpha plus or minus beta) | Translate | |

$$\cos(\alpha \pm \beta)$$ | cosine of (alpha plus or minus beta) | Translate | |

$$\tan(\alpha \pm \beta)$$ | tangent of (alpha plus or minus beta) | Translate | |

$$\theta \pm \phi$$ | theta plus or minus phi | Translate | |

$$\theta \cdot \phi$$ | theta times phi | Translate | |

$$\theta / \phi$$ | theta over phi | Translate | |

$$\sinh x$$ | hyperbolic sine of x | Translate | |

$$\cosh x$$ | hyperbolic cosine of x | Translate | |

$$\tanh x$$ | hyperbolic tangent of x | Translate |

## Logic and Sets

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$a := b$$ | a is defined by(given as) b | Translate | |

$$S_1 \Rightarrow S_2$$ | S sub one implies S sub two | Translate | |

$$S_1 \Leftrightarrow S_2$$ | S sub one is equivalent to S sub two | Translate | |

$$\{a, b, c\}$$ | the set of a, b, and c | Translate | |

$$a \in A$$ | a is an element of set A | Translate | |

$$\emptyset \text{ or } \{\}$$ | empty set | Translate | |

$$A \subset B$$ | A is a proper subset of B | Translate | |

$$A \subseteq B$$ | A is a subset of B | Translate | |

$$A \cup B$$ | A union B | Translate | |

$$A \cap B$$ | A intersection B | Translate | |

$$A \setminus B$$ | A minus B | Translate | |

$$\equiv$$ | is equivalent to | Translate | |

$$\exists$$ | there exists | Translate | |

$$\nexists$$ | there does not exist | Translate | |

$$\forall$$ | for all | Translate | |

$$\exists p \in \mathbb{P}, p = 2$$ | there exists a prime number p such that p equals two. | Translate | |

$$\forall x \in \mathbb{R}, x + 0 = x$$ | for all x in the set of real numbers, x plus zero equals x. | Translate | |

$$\mathbb{N}$$ | the set of all natural numbers | Translate | |

$$\mathbb{Z}$$ | the set of all integers | Translate | |

$$\mathbb{Q}$$ | the set of all rational numbers | Translate | |

$$\mathbb{R}$$ | the set of all real numbers | Translate | |

$$\mathbb{C}$$ | the set of all complex numbers | Translate | |

$$\mathbb{P}$$ | the set of all prime numbers | Translate | |

$$A \triangle B$$ | symmetric difference of A and B | Translate | |

$$A^c$$ | complement of A | Translate | |

$$\overline{A}$$ | closure of A | Translate |

## Intervals

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$[a, b]$$ | closed interval from a to b | Translate | |

$$(a, b)$$ | open interval from a to b | Translate | |

$$[a, b)$$ | the interval from a to b, inclusive of a and exclusive of b. | Translate | |

$$(a, b]$$ | the interval from a to b, exclusive of a and inclusive of b. | Translate | |

$$(a, \infty)$$ | the half line starting at a and extending to infinity, exclusive of a. | Translate | |

$$(-\infty, a)$$ | the half line starting at negative infinity and extending to a, exclusive of a. | Translate |

## Factorials and Binomial Coefficients

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$n!$$ | n factorial (the product of all positive integers up to n) | Translate | |

$$\binom{n}{k} \text{or} nC_k$$ | n choose k (the number of ways to choose k elements from a set of n elements) | Translate | |

$$P(n,k)$$ | permutation of n things taken k at a time | Translate |

## Sum and Product

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$\sum_{n=1}^{\infty} a_n$$ | the sum of a sub n from n equals 1 to infinity | Translate | |

$$\sum_{k=0}^n \binom{n}{k}$$ | the sum from k equals zero to n of n choose k | Translate | |

$$( \sum_{k=1}^n a_k b_k )^2$$ | the square of the sum of the products of a sub k and b sub k from k equals 1 to n | Translate | |

$$\prod_{k=1}^n k$$ | the product from k equals one to n of k (n factorial) | Translate | |

$$\bigcup_{i=1}^n A_i$$ | union of sets A sub i from i equals 1 to n | Translate | |

$$\bigcap_{i=1}^n A_i$$ | intersection of sets A sub i from i equals 1 to n | Translate |

## Functions

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$\varphi$$ | phi (variant) | Translate | |

$$\varphi(0) = 0$$ | phi of zero equals zero | Translate | |

$$f : A \to B$$ | f is a function that maps from the set A to the set B | Translate | |

$$x \mapsto f(x)$$ | x is mapped to f(x) | Translate | |

$$g: \mathbb{R} \to \mathbb{C}, \theta \mapsto g(\theta) := e^{i\theta}$$ | g from R(real numbers) to C(complex numbers), theta maps to g of theta, defined as(given by) e to the i theta | Translate | |

$$f: \mathbb{R} \to \mathbb{R}, x \mapsto f(x) := 1 + x^2$$ | f from R(real numbers) to R, x maps to f of x, defined as(given by) one plus x squared | Translate | |

$$h: \mathbb{C} \to [0, \infty), z \mapsto h(z) := |z|$$ | 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 | Translate | |

$$\text{im}(f)$$ | image or range of f | Translate | |

$$\text{im}(f) = B$$ | the image(range) of f equals B | Translate | |

$$f: A \to \text{im}(f)$$ | f from A to the image of f | Translate | |

$$f^{-1}$$ | inverse function of f | Translate | |

$$f^{-1}(b) := a$$ | f inverse of b is defined as(given by) a | Translate | |

$$f_{i,j}$$ | f sub i comma j | Translate | |

$$f \circ g$$ | f composed with g | Translate | |

$$\ker(f)$$ | kernel of f | Translate | |

$$f(x)$$ | f of x | Translate | |

$$\lim_{x \to 0}$$ | x approaches 0 | Translate |

## Calculus Operation Notations

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$\frac{d}{dx}$$ | d by dx (the derivation with respect to x) | Translate | |

$$\frac{d}{dx} f(x)$$ | the derivative of f with respect to x | Translate | |

$$f'(x)$$ | f prime of x (the derivative of f with respect to x) | Translate | |

$$\int f(x) dx$$ | the integral of f with respect to x | Translate | |

$$\int_a^b f(x) dx$$ | the definite integral of f from a to b with respect to x | Translate | |

$$\lim_{x \to a} f(x)$$ | the limit of f as x approaches a | Translate | |

$$\oint f(x) dx$$ | the line integral of f with respect to x | Translate |

## Vector calculus and physics

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$\nabla$$ | nabla or del | Translate | |

$$\nabla f$$ | gradient of(nabla) f | Translate | |

$$\nabla \cdot \mathbf{F}$$ | divergence of(nabla dot) F | Translate | |

$$\nabla \times \mathbf{F}$$ | curl of(nabla cross) F | Translate | |

$$\partial$$ | partial | Translate | |

$$\frac{\partial}{\partial x} f(x)$$ | the partial derivative of f with respect to x | Translate | |

$$\frac{\partial f}{\partial x}$$ | partial of f with respect to x | Translate | |

$$\frac{\partial^2 f}{\partial x^2}$$ | the second partial derivative of f with respect to x squared | Translate | |

$$\frac{\partial}{\partial x_j} ( \frac{\partial f}{\partial x_i} )$$ | The partial derivative with respect to x sub j of the partial derivative of f with respect to x sub i | Translate | |

$$\frac{\partial^2 f}{\partial x_j \partial x_i}$$ | The second partial derivative of f with respect to x sub j and x sub i | Translate | |

$$\mathbf{a} \cdot \mathbf{b}$$ | dot product of vectors a and b | Translate | |

$$\mathbf{a} \times \mathbf{b}$$ | cross product of vectors a and b | Translate |

## Matrix Operations

## Complex Numbers

## Differential Equations

Expression | How to Read | Translate | Pronounce |
---|---|---|---|

$$\frac{dy}{dx} = f(x,y)$$ | dy by dx equals f of x and y (first-order differential equation) | Translate | |

$$\frac{d^2y}{dx^2} + p(x)\frac{dy}{dx} + q(x)y = r(x)$$ | 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) | Translate |

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