Math-to-English Cheat Sheet

post by nahoj · 2024-04-08T09:19:40.814Z · LW · GW · 5 comments

Contents

  General
        Distinguishing case
        Subscripts
  Calculus
        Pythagorean Theorem
        Area of a Circle
        Slope of a Line
        Quadratic Formula
        Sum of an Arithmetic Series
        Euler's Formula
        Law of Sines
        Area of a Triangle (Heron's Formula)
        Compound Interest Formula
        Logarithm Properties
    More advanced operations
        Derivative of a Function
        Second Derivative
        Partial Derivative (unreviewed)
        Definite Integral
        Indefinite Integral (Antiderivative)
        Line Integral (unreviewed)
        Double Integral
        Gradient of a Function
        Divergence of a Vector Field
        Curl of a Vector Field
        Laplace Operator
        Limit of a Function
  Linear Algebra (vectors and matrices)
        Vector Addition
        Scalar Multiplication
        Dot Product
        Cross Product
        Matrix Multiplication
        Matrix Transpose
        Determinant of a Matrix
        Inverse of a Matrix
        Eigenvalues and Eigenvectors
        Rank of a Matrix
        Trace of a Matrix
        Vector Norm
        Orthogonal Vectors
    With numerical values
        Matrix Multiplication with Numerical Values
        Vector Dot Product
        Determinant of a Matrix
        Eigenvalues and Eigenvectors with Numerical Values
        Solving a System of Linear Equations
  Probabilities and Statistics
        Probability of an Event
        Mean of a Dataset
        Sample Mean
        Standard Deviation of a Population
        Sample Standard Deviation
        Covariance of Two Variables
        Correlation Coefficient
        Binomial Probability Formula
        Central Limit Theorem Approximation
  Logic
      Propositional Logic
        Negation
        Conjunction
        Disjunction
        Implication (Conditional)
        Biconditional
      Predicate Logic
        Universal Quantification
        Existential Quantification
      Modal Logic
        Necessity
        Possibility
  Set Theory
        Union and Intersection of Sets with Set Builder Notation
        Function Definition
        Cartesian Product and Relations
        Group Operation
        Monoid Example with Identity
        Power Set and Subset
        Equivalence Relation
        Direct Product of Groups
        Vector Space Over a Field
None
5 comments

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 [LW · GW].

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)

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