Speaking Mathematical Equations

A common difficulty encountered by math students is the difficulty of translating equations into words. Such translations are unavoidable when verbally discussing mathematics, and being a fluent translator is also a powerful aid to a student’s understanding. Equations are, of course, ubiquitous in mathematics (although this was not the case in antiquity; cf. Euclid’s Elements). Compared to an English sentence, an equation is often more compact, less ambiguous, and allows information to be arranged on the page in ways that aids clarity.

Even simple equations can be difficult to put into words accurately. One example is the use of parentheses to indicate grouping. Consider the equation,

\[a (b + c) = 1 / (b + c).\]

A naive translation into English would lose the groupings imposed by the parentheses: “a times b plus c equals one over b plus c”. Indeed, this sentence erroneous translation indicates a different equation:

\[ab + c = 1/b + c.\]

This example illustrates how a speaker must be careful to indicate the groupings in their speech. It is not self-evident how to do this and the remainder of this document describes how to translate specific mathematical expressions into English. For this example, a precise translation is “a times the sum of b plus c equals one over the sum of b plus c”.

Math to English Translation Recipes

This section contains recipes for how to translate from various mathematical expressions into English. Since any mathematical expression can be translated multiple ways, several translations are given in the right column of each table.

Parentheses and Grouping

	Mathematical Expression	English Translation
Arithmetic Grouping	$a(b+c)$	"a times open parentheses b plus c close parentheses." (avoid this!) "a times the sum of b and c." "a times the quantity b plus c." "The product of a and the quantity b plus c."
Arithmetic Grouping	$(a+b)c$	"a plus b all times c." "The product of a plus b times c." "The quantity a plus b multiplied by c."
Division	$(a + b)/c$	"a plus b all divided by c" "The sum of a and b all divided by c" "The fraction a plus b over c"

Sets

	Equation	English Translation
Set	$\{1, 2, 3\}$	"The set containing one, two, and three." "The set of one, two, and three."
Set-builder Notation	$\{x \in \mathbb{R} \mid P(x)\}$	"The set of all 'x' in 'R' such that 'P' of 'x' is satisfied." "The set of 'x' in 'R' such that 'P' of 'x'."
Element of	$x \in A$	"'x' is an element of 'A'." "'x' is in 'A'."
Subset of	$A \subset B$	"(The set) 'A' is a subset of 'B'." "(The set) 'A' is contained in 'B'."
Superset of	$A \supset B$	"(The set) 'A' is a super set of 'B'." "(The set) 'A' contains 'B'."
Strict Subset of	$A \subsetneq B$	"(The set) 'A' is a strict subset of 'B'." "(The set) 'A' is a proper subset of 'B'." "(The set) 'A' is a subset of 'B', but not equal to 'B'."
Union	$A \cup B$	"The union of 'A' and 'B'." "'A' union 'B'."
Intersection	$A \cap B$	"The intersection of 'A' and 'B'." "'A' intersection 'B'." "'A' intersect 'B'."

Calculus

	Equation	English Translation
Integrals	$$\int_a^b f(x) dx$$	"The integral of f of x over x from a to b." "The integral of f of x, dee-x with lower limit a and upper limit b."
Total Derivatives	$$\frac{df}{dx}$$	"dee-f, dee-x." "The derivative of f with respect to x."
Second Derivatives	$$\frac{d^2f}{dx^2}$$	"The second derivative of f with respect to x." (If you have a shorter or alternative way to say this, please let me know).
Second Derivatives	$$\frac{d^2f}{dxdy}$$	"The second derivative of f with respect to x and y." (If you have a shorter or alternative way to say this, please let me know).
Total Derivatives (Evaluated)	$$\frac{df}{dx}\Big\rvert_{x_0}$$	"dee-f, dee-x at x-zero." "The derivative of f with respect to x evaluated at x-zero."
Total Derivatives (Evaluated)	$$f'(x)$$	"f-prime of x." "The derivative of f at x."
Partial Derivative	$$\frac{\partial f}{\partial x}$$	"partial f, partial x." "The partial derivative of f with respect to x."

Linear Algebra

	Equation	English Translation
Matrix Element	$$A_{ij}$$	"The $i$,$j$ element of $A$."
Matrix Multiplication	$$Ax$$	"$A$ times $x$." "$A$ $x$."
Transpose	$$A^\top$$	"$A$ transpose." "The transpose of $A$."
Inverse	$$A^{-1}$$	"$A$ inverse." "The inverse of $A$."
Dot Product	$$x \cdot y$$	"$x$ dot $y$." "The dot product of $x$ and $y$."
Inner Product	$$\langle x, y\rangle$$	"The inner product of $x$ and $y$."
Null space	$$\operatorname{null} A$$	"The null space of matrix $A$."
Column space	$$\operatorname{col} A$$	"The column space of matrix $A$."
Coordinate	$$[x]_{\mathcal{B}}$$	"The coordinate vector of (vector) $x$ relative to (basis) $\mathcal{B}$." "The $\mathcal{B}$-coordinate vector of $x$."

This is document is a work in progress that will continue to grow as I find examples that are worthy of inclusion. If you have suggestions, please contact me.

For more on this topic, see “How can we speak math?” by Richard Fateman.