Per Alexandersson

Introduction

First and foremost, a mathematical text with equations, expressions and symbols is still a text. It should still be built with complete sentences, ending with a period. The text should have a natural flow when read. It is a good idea to read the text out loud, even all the symbols and equations!

Symbols

It is common to overuse symbols or being redundant.

Example.

This implies that $ \implies x^2=2+x. $ The implication arrow ($\implies$) is redundant.

Symbols should usually not replace words in inline text.

Example.

We have that $x=2 \wedge x=3$ are solutions.

It is better to just write We have that $x=2$ and $x=3$ are solutions.

Symbols are mainly used when making definitions, where it is important to avoid ambiguity.

Example.

An injective function is a function such that whenever $x$ and $y$ are different values in its domain, the function values at $x$ and $y$ are different.

Notice that this definition is rather difficult to parse. The following definition is easier to understand.

A function $f$ is injective if $x\neq y \implies f(x) \neq f(y)$ whenever $x,y \in D_f.$

Inline math expressions

Do not start sentences with a mathematical expression. This usually looks strange and more often than not, sound strange when read out loud.

Example.

$x=2$ is the only solution.

Better write The only solution is $x=2.$

Equation mode expressions

A centered equation should also be part of a sentence.

Example.

We add two to both sides.

\[ (x-2) + 2 = (x^2-4)+2 \]

Both sides are now simplified.

\[ x = x^2-2 \]

Notice that there is no real sentence structure here. Rewrite as follows instead and notice the period and comma in the equations.

We add $2$ to both sides and get

\[ (x-2) + 2 = (x^2-4)+2, \]

which then simplifies to the equation

\[ x = x^2-2. \]

Avoid really long multi-line equations. This is for the same reason that long sentences should be avoided. Another reason is that LaTeX might struggle to fit this equation on a page. Either you get an annoying page break in the middle, or LaTeX is forced to move the big calculation to a new page, leaving a big part of the previous page blank.

Usage of colon and semicolon

There is usually no need to use a colon, or a semicolon before an equation.

Example.

The identity:

\[ \sin^2(x) + \cos^2(x)=1 \]

is called the "Pythagorean trigonometric identity".

Here, the colon makes no sense, as the identity is part of the statement. However, for other sentence structures, it makes sense to use a colon or semicolon.

Consider the following identity, which is called the Pythagorean trigonometric identity:

\[ \sin^2(x) + \cos^2(x)=1. \]

In this case, the formula does not fit into the sentence as it is — we need the semicolon to start a new phrase.

Being specific and math terminology

I see many hand-ins where the words answer and problem is used.

Example (Incorrect).

The problem $x^2-5x+6=0$ has the answers $x=2$ and $x=3.$

Example (Correct).

The equation $x^2-5x+6=0$ has the solutions $x=2$ and $x=3.$

Make sure to call things by the correct name. Here are a few notions which seem to cause confusion.

Expression — Any mathematical expression or part of an equation, such as $x^2+4$ or $\frac{a+b}{c+e}.$ Basically, almost everything is an expression, it is the main building block which is not plain text.

Equation — An expression with an equality sign, $x^2+4x+6=0,$ with at least one unknown variable. The expression is true only for some values of $x,$ and one is interested in finding those values of $x.$ When the equation has a polynomial on one side, and a zero on the other, such as $x^3+5x-2=0,$ the solutions are called roots. In general, the zeroes of the polynomial $P(x)$ are the roots of the equation $P(x)=0.$ A system of equations is a collection of equations, for which one is interested in the common set of solutions.

Example.

The following is a perfectly valid expression:

\[ |\{ x \in \setR : x^2+5x+6=0 \}|^2 = |\{ (x,y) \in \setR^2 : (x^2+5x+6)^2 + (y^2+5y+6)^2 = 0 \}|. \]

As an exercise, compute the sum

\[ \sum_{k=1}^n |\{ x \in \setC : x^k=1 \}| \]

for every positive integer $n.$

Inequality — Expressions such as $x^2+2x \geq 3$ or $(x+y)/2 \geq \sqrt{xy}.$ There are two types of inequalities: (a) Inequalities for which we seek to find for what values $x$ the inequality holds. (b) Inequalities which are true for all values of $x$ and $y$ (with some reasonable restriction). There are many inequalities of type (b) which have names, such as the AM-GM inequality, Jensen's inequality, or Azuma's inequality. See Wikipedia for a long list.

Identity — An equality between two expressions; $\sin^2(x)+\cos^2(x)=1$ and $a^2+b^2=c^2$ are examples of identities. It is common that there are some restrictions on the variables involved, the Pythagorean identity only holds if $a,$ $b$ and $c$ are appropriate side lengths in a right triangle.

Function — A (named) expression whose value depends on the variables. For example, $h(x,y) = a \sin(x) + by+5 $ is a function of $x$ and $y.$ The $a,$ and $b$ are called constants, and thought of as being fixed, while $x$ varies. Note that the equality sign here denotes assignment or definition, so that whenever we write $h(x,y),$ we refer to the expression $a \sin(x) + by+5.$

Statement — Something which can be true or false. For example, "It is raining", or "Almost all real numbers are nice" and "$x^2+2=54$" are statements. All equations and inequalities are statements. Moreover, all identities are statements, and there is a proof that the statement is true.

Proof — A sequence of arguments and perhaps calculations, which shows that a statement is true. We do not prove equations, we solve equations. However, we can provide a proof that a given equation has a certain set of solutions. Both examples below are correct. In the first example, we have a statement which can be either true or false, while in the second example, the problem text is not a statement.

Example (Correct).

The equation $x^2-5x+6=0$ has the solutions $x=2$ and $x=3.$

Proof. By plugging in $x=2$ and $x=3$ in the left hand side, a simple calculation shows that this is indeed 0.

Example (Correct).

Solve the equation $x^2-5x+6=0.$

Solution. The $pq$-formula gives that $x = \frac{5}{2} \pm \sqrt{25/4-6},$ so $x=2$ and $x=3$ are the solutions to the equation.

Problems with types

Confusing notation

We are exposed to different types (as in programming) in mathematics. Think of these as units in physics. Some common types are numbers, statements, functions, matrices, etc. Note that for example, $3$ can be seen as both a number and a function. We have different notation for dealing with equality between objects of different types.

Make sure that equivalence arrow ($\iff$) is only between statements (things that can be true or false). Arrows $(\to)$ are used for various purposes, for example when talking about limits. Equality $(=)$ is for things which have the same (numerical, usually) value. It is unfortunate that we use $=$ both for identically equal to (i.e, equal for all values of parameters), and when talking about equations, where we mainly are interested in for which values equality holds. For example, the identity $\sin^2(x)+\cos^2(x)=1$ is an equality between functions, while $\sin(x)+\cos(x)=1$ is interpreted as an equation and thus $=$ is now equality between numbers.

Equality is also used for introducing new notation, or making substitutions. For example, in the middle of

\[ \lim_{x\to 1} \frac{(\sqrt{x}-1)^2}{x(\sqrt{x}-1)} = \left[ \begin{smallmatrix} t = \sqrt{x} \\ t^2 = x \end{smallmatrix} \right] = \lim_{t\to 1} \frac{(t-1)^2}{t^2(t-1)} \]

we make a substitution by introducing new notation. In this context, $t = \sqrt{x}$ can be seen as introducing the function $t(x) = \sqrt{x}.$

When introducing new functions or notation, mathematicians sometimes write $\coloneqq.$ This is read as defined as. For example,

\[ |x| \coloneqq \begin{cases} x \text{ if } x\geq 0 \\ -x \text{ otherwise}. \end{cases} \]

This is read out loud as

Let the absolute value of x be defined as x, if x is greater-than-or-equal-to zero, and minus x otherwise.

Example with equality signs

A common source of confusion is the mixing of equality-as-expression with equation-equality.

Example.

Consider the following fragment from a solution:

In order to find extremal points of $f(x)=x^3+2x-\cos(x),$ we set the derivative to 0. That is, $f'(x) = 3x^2+2+\sin(x)=0.$

The problem here is that the first equality sign is the identity $f'(x) = 3x^2+2+\sin(x).$ This is true for all value of $x.$ The second equality sign $3x^2+2+\sin(x)=0$ is an equation and only true for some particular values of $x.$ What is not true, is that $f'(x)=0$ for all $x.$

We can instead write as follows in order to avoid mixing types of equality.

In order to find extremal points of $f(x)=x^3+2x-\cos(x),$ we want to find zeros of the derivative. The derivative is given by $f'(x) = 3x^2+2+\sin(x),$ so we need to solve the equation $3x^2+2+\sin(x) = 0.$

This is a bit more verbose, but it has the advantage of being correct.

Examples with limits

We shall now see a few common mistakes in the context of limits.

Example. \[ \text{Incorrect: } \lim_{t\to 1} \frac{t^3-1}{t-1} \to 3 \qquad \text{Correct: } \lim_{t\to 1} \frac{t^3-1}{t-1} = 3. \]

Explanation: A limit is either a number, (or $\pm \infty$ or undefined), so equality sign should be used.

Example.

Sometimes, a function is named in the context of limits. The intention is clear, but sometimes incorrect, as here:

\[ \text{Incorrect: } f(x) = \lim_{x \to 1} \frac{x^3-1}{x-1} \qquad \text{Correct: } \lim_{x\to 1} f(x), \text{ where } f(x)=\frac{x^3-1}{x-1}. \]

The first expression is not strictly syntactically incorrect, but probably not what is intendend. Note that it is perfectly fine to define a function $f(t)$ as

\[ f(t) = \lim_{x \to 1} \frac{x^t-1}{x-1}. \]

In fact, computing the limit (by say l'Hospitals rule) shows that $f(t) = t.$

Example. \[ \text{Incorrect: } \lim_{t\to 0} \frac{t^3-1}{t-1} = \frac{t^2+2t+1}{1} \qquad \text{Correct: } \lim_{t\to 0} \frac{t^3-1}{t-1} = \lim_{t\to 0} \frac{t^2+2t+1}{1}. \]

Explanation: The first one is incorrect, as it states that the number $3$ is equal to the expression $\frac{t^2+2t+1}{1}.$

Perhaps more true to the writers intention, one could alternatively express the identity as

\[ \lim_{t\to 0} \frac{t^3-1}{t-1} = \left. \frac{t^2+2t+1}{1} \right\vert_{t=1}. \]

Here, $\left. \frac{t^2+2t+1}{1} \right\vert_{t=1}$ is a commonly established shorthand for

\[ f(1) \text{ where } f(t) = \frac{t^2+2t+1}{1}. \]

In computer science lingo, this notation allows us to evaluate an anonymous function with a particular argument. We read $t^2+4\vert_{t=1}$ out loud as

Tee-squared-plus-four evaluated at tee-equals-one.

Example. \[ \text{Incorrect: } \frac{t^3-1}{t-1} = 3 \text{ when $t=1$}. \qquad \text{Correct: } \frac{t^3-1}{t-1} \to 3 \text{ as $t\to1$}. \]

Explanation: The expression is not defined at $t=1.$ However, it is true that the limit is $3,$ and we are justified to use the arrows to express this. The latter statement is read out loud as

Tee-cubed minus one, over tee-minus-one approaches three, as tee approaches one.

Example.

The following is incorrect:

\[ \lim_{x\to 0} \frac{e^x-1}{x} \frac{\sin(x)}{x} \implies \lim_{x\to 0} \frac{e^x-1}{x} \lim_{x\to 0} \frac{\sin(x)}{x}. \]

We do not use implication between values. To make it correct, put an equality sign instead. What is meant is perhaps that the equality is a consequence of the following implication, correctly stated as:

\[ \lim_{x\to 0} \frac{e^x-1}{x} = A \text{ and } \lim_{x\to 0} \frac{\sin(x)}{x}=B \implies \lim_{x\to 0} \frac{e^x-1}{x} \frac{\sin(x)}{x} = AB. \]

Note that this implication in general has some restrictions on $A$ and $B.$

Example.

The following is incorrect, for the same reason as above.

\[ \lim_{x\to 0} \sqrt{ \frac{e^{2x}-1}{x} } \implies \sqrt{\lim_{x\to 0} \frac{e^{2x}-1}{x} }. \]

Here, it would be clearer to give the limit a name:

\[ \text{Let } A \coloneqq \lim_{x\to 0} \sqrt{ \frac{e^{2x}-1}{x} }, \text{ then } \lim_{x\to 0} \frac{e^{2x}-1}{x} = \sqrt{A}. \]

Since $\sqrt{A}=2,$ we find that $A=4.$