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!*

It is common to overuse symbols or being redundant.

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

Symbols should usually not replace words in inline text.

*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.

*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 $x,y \in D_f.$
*

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

*$x=2$ is the only solution.*

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

A centered equation should also be part of a sentence.

*We add two to both sides.*

*Both sides are now simplified.*

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*

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.

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

*The identity:*

*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:*

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

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

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

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.

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.

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.

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.

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,

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.

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

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.

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

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

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 $f(t) = t.$

*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.

*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.

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 $B.$

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 $A=4.$