Math-to-English Cheat Sheet

[nahoj]

Say you’ve learnt math in your native language which is not English. Since then you’ve also read math in English and you appreciate the near universality of mathematical notation. Then one day you want to discuss a formula in real life and you realize you don’t know how to pronunce “$a_n$”.

Status: I had little prior knowledge of the topic. This was mostly generated by ChatGPT4 and kindly reviewed by @TheManxLoiner.

General

Distinguishing case

• $F,\delta$

• “Big F” or “capital F”, “little delta”

Subscripts

• $a_n$

• “a sub n” or, in most cases, just “a n”

Calculus

Pythagorean Theorem

• $a^2+b^2=c^2$

• “a squared plus b squared equals c squared.”

Area of a Circle

• $A=\pi r^2$

• “Area equals pi [pronounced ‘pie’] r squared.”

Slope of a Line

• $m=\frac{y_2-y_1}{x_2-x_1}$

• “m equals y 2 minus y 1 over x 2 minus x 1.”

Quadratic Formula

• $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

• “x equals minus b [or ‘negative b’] plus or minus the square root of b squared minus four a c, all over two a.”

Sum of an Arithmetic Series

• $S=\frac{n}{2}(a_1+a_n)$

• “S equals n over two times a 1 plus a n.”

Euler’s Formula

• $e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)$

• “e to the i theta equals cos [pronounced ‘coss’ or ‘coz’] theta plus i sine theta.”

Law of Sines

• $\frac{\sin \left(A\right)}{a}=\frac{\sin \left(B\right)}{b}=\frac{\sin \left(C\right)}{c}$

• “Sine A over a equals sine B over b equals sine C over c.”

Area of a Triangle (Heron’s Formula)

• $A=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{a+b+c}{2}$

• “Area equals the square root of s times s minus a times s minus b times s minus c, where s equals a plus b plus c over two.”

Compound Interest Formula

• $A=P(1+\frac{r}{n}{)}^{nt}$

• “A equals P times one plus r over n to the power of n t.”

Logarithm Properties

• ${\log }_b\left(xy\right)={\log }_b\left(x\right)+{\log }_b\left(y\right)$

• Don’t state the base if clear from context: “Log of x y equals log of x plus log of y.”

• Otherwise “Log to the base b of x y equals log to the base b of x plus log to the base b of y.”

More advanced operations

Derivative of a Function

• $\frac{df}{dx}$ or $\frac{d}{dx}f\left(x\right)$ or $f^{\prime }\left(x\right)$

• “df by dx”, or “d dx of f of x”, or “f prime of x” or “f dash of x”.

Second Derivative

• $\frac{d^2}{d{x}^2}f\left(x\right)$ or $f^{\prime \prime }\left(x\right)$

• “d squared dx squared of f of x” or “f double prime of x.”

Partial Derivative (unreviewed)

• $\frac{\partial }{\partial x}f(x,y)$

• “Partial with respect to x of f of x, y.”

Definite Integral

• ${\int }_a^{b}f\left(x\right)dx$

• “Integral from a to b of f of x dx.”

Indefinite Integral (Antiderivative)

• $\int f\left(x\right)dx$

• “Integral of f of x dx.”

Line Integral (unreviewed)

• ${\int }_{C}f(x,y)ds$

• “Line integral over C of f of x, y ds.”

Double Integral

• ${\int }_a^b{\int }_c^{d}f(x,y)dxdy$

• “Double integral from a to b and c to d of f of x, y dx dy.”

Gradient of a Function

• $\nabla f$

• “Nabla f” or “del f”, or “gradient of f” or “grad f” to distinguish from other uses such as divergence or curl.

Divergence of a Vector Field

• $\nabla \cdot F$

• “Nabla dot F” or “div F”.

Curl of a Vector Field

• $\nabla \times F$

• “Nabla cross F” or “curl F”.

Laplace Operator

• $\Delta f$ or ${\nabla }^{2}f$

• “Delta f” or “Laplacian of f”, or “nabla squared f” or “del squared f”.

Limit of a Function

• ${lim}_{x\to a}f\left(x\right)$

• “Limit as x approaches a of f of x.”

Linear Algebra (vectors and matrices)

Vector Addition

• $v+w$

• “v plus w.”

Scalar Multiplication

• $cv$

• “c times v.”

Dot Product

• $v\cdot w$

• “v dot w.”

Cross Product

• $v\times w$

• “v cross w.”

Matrix Multiplication

• $AB$

• “A B.”

Matrix Transpose

• $A^T$

• “A transpose.”

Determinant of a Matrix

• $|A|$ or $det\left(A\right)$

• “Determinant of A” or “det A”.

Inverse of a Matrix

• $A^{-1}$

• “A inverse.”

Eigenvalues and Eigenvectors

• $\lambda$ for eigenvalues, $v$ for eigenvectors

• “Lambda for eigenvalues; v for eigenvectors.”

Rank of a Matrix

• $\text{rank}\left(A\right)$

• “Rank of A.”

Trace of a Matrix

• $\text{tr}\left(A\right)$

• “Trace of A.”

Vector Norm

• $\parallel v\parallel$

• “Norm of v” or “length of v”.

Orthogonal Vectors

• $v\cdot w=0$

• “v dot w equals zero.”

With numerical values

Matrix Multiplication with Numerical Values

• Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)$, then $AB=\left(\begin{array}{cc}19& 22\\ 43& 50\end{array}\right)$.

• “A B equals nineteen, twenty-two; forty-three, fifty.”

Vector Dot Product

• Let $v=(1,2,3)$ and $w=(4,5,6)$, then $v\cdot w=32$.

• “v dot w equals thirty-two.”

Determinant of a Matrix

• For $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$, $|A|=-2$.

• “Determinant of A equals minus two.”

Eigenvalues and Eigenvectors with Numerical Values

• Given $A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$, it has eigenvalues ${\lambda }_1=3$ and ${\lambda }_2=1$, with corresponding eigenvectors $v_1=\left(\begin{array}{c}1\\ 1\end{array}\right)$ and $v_2=\left(\begin{array}{c}-1\\ 1\end{array}\right)$.

• “Lambda one equals three with v one equals one, one; lambda two equals one with v two equals minus one, one.”

Solving a System of Linear Equations

• For the system given by $Ax=b$, where $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $b=\left(\begin{array}{c}5\\ 11\end{array}\right)$, the solution $x$ can be found using $A^{-1}b$.

• “x equals A inverse b, solving the system.”

Probabilities and Statistics

Probability of an Event

• $P\left(A\right)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

• “P of A equals the number of favorable outcomes over the total number of outcomes.”

• “P A” and “probability of A” are also common.

Mean of a Dataset

• $\mu =\frac{1}{N}{\sum }_{i=1}^N{x}_i$

• “Mu equals one over N times the sum from i equals one to N of x i.”

Sample Mean

• $\overline{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i$

• “x bar equals one over n times the sum from i equals one to n of x i.”

Standard Deviation of a Population

• $\sigma =\sqrt{\frac{1}{N}{\sum }_{i=1}^N(x_i-\mu {)}^2}$

• “Sigma equals the square root of one over N times the sum from i equals one to N of x i minus mu squared.”

Sample Standard Deviation

• $s=\sqrt{\frac{1}{n-1}{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}$

• “s equals the square root of one over n minus one times the sum from i equals one to n of x i minus x bar squared.”

Covariance of Two Variables

• $\text{Cov}(X,Y)=\frac{1}{n-1}{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})$

• “Covariance of X and Y equals one over n minus one times the sum from i equals one to n of x i minus x bar times y i minus y bar.”

Correlation Coefficient

• $r=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$

• “r equals Covariance of X and Y over sigma X times sigma Y.”

Binomial Probability Formula

• $P(X=k)=(\frac{n}{k})p^k(1-p{)}^{n-k}$

• “P of X equals k equals n choose k times p to the k times one minus p to the n minus k.”

Central Limit Theorem Approximation

• $P(a\le X\le b)\approx \Phi \left(\frac{b-\mu }{\sigma /\sqrt{n}}\right)-\Phi \left(\frac{a-\mu }{\sigma /\sqrt{n}}\right)$

• “Probability of X between a and b approximately equals Phi of b minus mu over sigma divided by the square root of n minus Phi of a minus mu over sigma divided by the square root of n.”

• or “P of a less than or equal to X less than or equal to b …”

Logic

Propositional Logic

Negation

• Notation: $\neg P$

• English: “Not P.”

Conjunction

• Notation: $P\wedge Q$

• English: “P and Q.”

Disjunction

• Notation: $P\vee Q$

• English: “P or Q.”

Implication (Conditional)

• Notation: $P\to Q$

• English: “P implies Q” or “If P then Q.”

Biconditional

• Notation: $P\leftrightarrow Q$

• English: “P if and only if Q.”

Predicate Logic

Universal Quantification

• Notation: $\forall xP\left(x\right)$

• English: “For all x, P of x.”

Existential Quantification

• Notation: $\exists xP\left(x\right)$

• English: “There exists [an] x such that P of x.”

Modal Logic

Necessity

• Notation: $\square P$

• English: “Necessarily P.”

Possibility

• Notation: $◊P$

• English: “Possibly P.”

Set Theory

Union and Intersection of Sets with Set Builder Notation

• $A\cup B=\{x\mid x\in A\text{or}x\in B\}$

• “A union B equals the set of x such that x is in A or x is in B.”

Function Definition

• $f:R\to R\text{defined by}f\left(x\right)=x^2-2x+1$

• “f from R to R defined by f of x equals x squared minus two x plus one.”

Cartesian Product and Relations

• $R=\left\{\right(x,y)\in Z\times Z\mid x^2+y^2=25\}$

• “R equals the set of ordered pairs x, y in Z cross Z such that x squared plus y squared equals twenty-five.”

Group Operation

• $(G,\ast )\text{where}\forall a,b\in G,a\ast b=ab+a+b$

• “G[, star], where for all a, b in G, a star b equals a b plus a plus b.”

Monoid Example with Identity

• $(M,\cdot )\text{is a monoid if}\exists e\in M\text{such that}\forall a\in M,a\cdot e=e\cdot a=a$

• “M[, dot], is a monoid if there exists e in M such that for all a in M, a dot e equals e dot a equals a.”

Power Set and Subset

• $P\left(S\right)=\{T\mid T\subseteq S\}$

• “Power set of S equals the set of T such that T is a subset of S.”

Equivalence Relation

• $a\sim b\iff a-b\in Z$

• “a equivalent to b [or ‘a twiddles b’] if and only if a minus b is in Z.”

Direct Product of Groups

• $G\times H=\left\{\right(g,h)\mid g\in G\text{and}h\in H\}$

• “G cross H equals the set of ordered pairs g, h such that g is in G and h is in H.”

Vector Space Over a Field

• $V\text{over}F\text{with addition}+\text{and scalar multiplication}\cdot$

• (spelled out rather than written with symbols)

[gjm]

There are a few things in the list that I would say differently, which I mention not because the versions in the post are _wrong_ but because if you’re using a crib-sheet like this then you might get confused when other people say it differently:

• I say “grad f”, “div f”, “curl f” for $\nabla f$, $\nabla \cdot f$, $\nabla \times f$. I more often say “del” than “nabla” and for the Laplacian I would likely say either “del squared f” or “Laplacian of f”.

• I pronounce “cos” as “coss” not as “coz”.

• For derivatives I will say “dash” at least as often as “prime”.

The selection of things in the list feels kinda strange (if it was mostly produced by GPT-4 then that may be why) -- if the goal is to teach you how to say various things then some of the entries aren’t really pulling their weight (e.g., the one about the z-score, or the example of how to read out loud an explicit matrix transpose, when we’ve already been told how to say “transpose” and how to read out the numbers in a matrix). It feels as if whoever-or-whatever generated the list sometimes forgot whether they were making a list of bits of mathematical notation that you might not know how to say out loud or a list of things in early undergraduate mathematics that you might not know about.

It always makes me just a little bit sad when I see Heron’s formula for the area of a triangle. Not because there’s anything wrong with it or because it isn’t a beautiful formula—but because it’s a special case of something even nicer. If you have a cyclic quadrilateral with sides $a,b,c,d$ then (writing $s=\frac{1}{2}(a+b+c+d)$) its area is $\sqrt{}(s-a)(s-b)(s-c)(s-d)$. Heron’s formula is just the special case where two vertices coincide so $d=0$. The more general formula (due to Brahmagupta) is also more symmetrical and at least as easy to remember.

[nahoj]

Thanks, I have applied most suggestions.

Indeed I didn’t choose the formulas myself but just told GPT to produce some, and then removed a few that seemed dubious or irrelevant.

[Viliam]

in English, π is pronounced “pie” (not “pee”)

[nahoj]

Added

[Archimedes]

Some of these depend on how concise you want to be. For example,

“Partial derivative with respect to x of f of x, y” may be shortened to “partial x, f of x, y”.

Conciseness is more common when speaking if the written form is also visible (as opposed to purely vocal communication).