#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 to be 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, sounds 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 \[x = x^2-2.\]
#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.
#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 the 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 for equations, where we mainly care about 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 values of \(x.\) The second equality sign \(3x^2+2+\sin(x)=0\) is an equation and is 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, \(\pm \infty,\) or undefined, so an equality sign should be used.
Example
\[\text{Incorrect: } \lim_{t\to 1} \frac{t^3-1}{t-1} = \frac{t^2+2t+1}{1} \qquad \text{Correct: } \lim_{t\to 1} \frac{t^3-1}{t-1} = \lim_{t\to 1} \frac{t^2+2t+1}{1}.\] 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 writer’s intention, one could alternatively express the identity as \[\lim_{t\to 1} \frac{t^3-1}{t-1} = \left. \frac{t^2+2t+1}{1} \right\vert_{t=1}.\] 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 terminology, this notation allows us to evaluate an anonymous function at a particular argument. We read \(t^2+4\vert_{t=1}\) out loud as
Tee-squared-plus-four evaluated at tee-equals-one.
Example
\[\text{Incorrect: } \frac{t^3-1}{t-1} = 3 \text{ when $t=1$}. \qquad \text{Correct: } \frac{t^3-1}{t-1} \to 3 \text{ as $t\to 1$}.\] 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: \[\text{Let } A \coloneqq \lim_{x\to 0} \frac{e^{2x}-1}{x}, \text{ then } \lim_{x\to 0} \sqrt{\frac{e^{2x}-1}{x}} = \sqrt{A}.\] Since \(A=2,\) the original limit is \(\sqrt{2}.\)