We are now ready to make the final transition to the traditional expository proofs found in most textbooks and articles about mathematics.
Until now we have been using formal and semiformal proofs. These proofs satisfy the first goal of a proof - they are objectively verifiable by a computer.
In practice, formal proofs of all but the most trivial theorems can be very long and tedious, and not as easy to read or understand by a human as we might like. As a result, we introduced a number of shortcuts in along the way to eliminate tedious, repetitive steps, shorten the proof, and emphasize the aspects of the proof that would make it more readable to a human without sacrificing the objective correctness of the proof. This gave us what we refer to as semiformal proofs.
It is now time to make the final transition to the far right end of the proof spectrum illustrated by the bridge in Chapter 2. We will refer to these proofs as expository proofs or traditional proofs or informal proofs.
Because they are informal by design, and their goal is exposition, it is not easy to define precisely what constitutes a traditional proof as distinct from a formal or semiformal proof. Indeed, since the main goal of a traditional proof is exposition for human readers, defining it precisely is equivalent to defining what constitutes good expository writing in general.
That having been said, a traditional proof can be thought of as a proof obtained if you start with a semiformal proof, and modify it to conform to the following principles.
Mathematical writing has many features that distinguish it from other types of writing. The following is a list of guidelines to keep in mind that will help you to express your mathematical ideas in ways that will help others to more easily understand what you are trying to say.
An important part of making mathematical writing unambiguous and easy to understand is to choose good notation for the things you are writing about, and to carefully explain this notation to your readers. Some guidelines to keep in mind concerning mathematical notation are as follows.
Always clearly define new notation as it is introduced
Even if the notation seems obvious to you, it is not a good idea to assume that your reader knows what you mean by it. In particular, you should always declare new variables and constants when they are first used in the proof.
Bad. We can see that $n$ is odd, so $n = 2k + 1$.
Good. We can see that $n$ is odd, so $n = 2k + 1$ for some integer $k$.
Bad. If a continuous function $f$ satisfies $f(nx) = f(x)^n$ for all $x, n$, is it true that $f$ is an exponential function?
Good. If a continuous function $f$ satisfies $f(nx) = f(x)^n$ for all real $x$ and all positive integers $n$, is it true that $f$ is an exponential function?
Use standard notation for common types of objects
For instance, $m$ and $n$ are often used to denote integers, $p$ denotes a prime, $x$ can be a real number, $f$ and $g$ describe functions, and so on.
The standard notation may depend on the context of your writing. For instance, in complex analysis, $z$ is often used to denote a complex number, while in analytic number theory, $s$ is more common. Use the notation that your readers will most readily recognize.
Use consistent symbols for similar objects
For example, when choosing new symbols that have a similar meaning, we should choose them from the same alphabet, use the same case, and the same font. When naming several variables with a similar meaning at the same time it often helps to make them consecutive symbols in the alphabet, like, $a,b,c$ or $i,j,k$. This helps the reader remember the intended meaning of similar mathematical expressions.
Bad. Let $x, u, \xi$ be real numbers.
Good. Let $x, y, z$ be real numbers.
One exception to this guideline is that inconsistent symbols can be useful to denote objects that you want to distinguish in meaning to the reader:
Okay. Let $m, b$ be real numbers. Then the function $f\colon \bbR \to \bbR$ satisfying $f(x) = mx + b$ for all real numbers $x$ is linear.
Do not use the same notation for different objects in the same scope
Aside from being confusing to the reader, this usually will result in incorrect proofs.
Bad. To show that the product of any two even integers is divisible by 4, suppose $a$ and $b$ are even. Then $a = 2k$ and $b = 2k$ for some integer $k$. Thus $ab = 2k \cdot 2k = 4k^2$ is divisible by 4.
Good. To show that the product of any two even integers is divisible by 4, suppose $a$ and $b$ are even. Then $a = 2j$ for some integer $j$ and $b = 2k$ for some integer $k$. Thus $ab = 2j \cdot 2k = 4jk$ is divisible by 4.
Notice that the first “proof” could likewise be used to prove that the product of any two even integers is a square, a claim that is obviously not true.
Avoid convoluted or overloaded notation
Use several simple expressions rather than a single convoluted expression to increase clarity. Sometimes written prose can be more clear than symbolic expressions.
Bad. Let $0 < n \in \bbZ$.
Good. Let $n \in \bbZ$ such that $0 < n$.
Better. Let $n$ be a positive integer.
Mathematical symbols are a short and precise way to express mathematical ideas, but it’s important to use them in ways that don’t interfere with communication:
Don’t begin a sentence with a symbol
Every sentence in your proof should begin with an English word, even if that word is meaningful (e.g., Let or Declare) and should be capitalized just like any word at the start of a sentence. A sentence should not start with a variable, constant, number, or compound mathematical expression.
Bad. $x$ is a global maximum of $f$, so we have $f(y) \leq f(x)$ for every $y \in \bbR$.
Good. Since $x$ is a global maximum of $f$, we have $f(y) \leq f(x)$ for every $y \in \bbR$.
Bad. Let $a$ be a quadratic residue. $a = b^2$ for some $b \in \bbZ_n$.
Good. Let $a$ be a quadratic residue. Then $a = b^2$ for some $b \in \bbZ_n$.
Don’t needlessly mix symbols with prose
Many mathematical symbols have a spoken English equivalent, so it can be tempting to cleverly substitute the symbol for the corresponding English word in mathematical writing. However, this usually distracts from the meaning of the sentence and makes writing less understandable.
Bad. Clearly, $0 < x^2 + y^2 \implies x\neq 0 \vee y \neq 0$.
Good. Clearly, if $0 < x^2 + y^2$ then either $x\neq 0$ or $y \neq 0$.
Bad. There are exactly eight primes $< 20$.
Good. There are exactly eight primes less than 20.
The logical connectors and quantifiers $\vee$, $\wedge$, $\neg$, $\implies$, $\iff$, $\forall$, $\exists$ are very easy to misuse in this way. In general, use these only in contexts where you are discussing mathematical logic, or as part of longer symbolic expressions.
Bad. It is true $\forall$ integers that $\exists$ an even larger integer.
Good. It is true for all integers that there exists an even larger integer.
Okay. Thus we can express this in formal logic as $\forall x. P(x) \wedge \neg Q(x)$.
Good. Thus, for any $x$ both $P(x)$ and $\neg Q(x)$ are true.
Okay. Let $Z = \Set{ n\in\bbZ : \forall k \in \bbZ,k \mid n }$.
Good. Let $Z$ be the set of all integers that are divisible by every integer.
Don’t use unnecessary variable names in theorem statements
If an object needs a name in a proof, declare it in the body of the proof rather than in the theorem statement.
Bad. Any continuous function $f$ on the interval $[0, 1]$ is uniformly continuous.
Good. Any continuous function on the interval $[0, 1]$ is uniformly continuous.
This also applies for claims made in the middle of a proof.
When possible, use words to separate symbols which are not in a list
This can often make statements more readable.
Bad. If the congruence equation $n^2 \xcong{p} a$ has a solution $n = \overline{b}$, $n = \overline{p}-\overline{b}$ is the only other solution of the equation.
Good. If the congruence equation $n^2 \xcong{p} a$ has a solution $n = \overline{b}$, then $n = \overline{p}-\overline{b}$ is the only other solution of the equation.
Write out integers used as adjectives, and use Arabic numerals to write integers describing numerical values
The rule of thumb is adjectives are written out but nouns are not. If you have a number that is excessively long to write out as an adjective, consider rewording the sentence to make it a noun.
Good. There are exactly twenty-four elements in the symmetric group on four symbols.
Good. The first three positive powers of $2$ are $2$, $4$, and $8$.
Good. The set of prime numbers less than $20$ has eight elements.
Equations and formulas play an important role in many types of mathematical writing, so it is a good idea to present them in as clear a manner as possible. Some considerations to keep in mind are to:
Place important equations or formulas on their own line
This makes the expression more visible, and indicates its importance in the writing. Such typesetting is sometimes called a display style. If you need to reference the expression later in the text, give it a numbered label, as in \begin{equation} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \tag{12.1} \end{equation} In general, don’t label an expression like this if you don’t plan to reference it later in the writing. If you only reference an expression in the few preceding or following lines, consider instead using language such as “in the following expression” or “by the above equation”.
When writing out extended computations in a display style, use transitive chain notation
This will avoid lengthy inline expressions that are difficult to read, and the clutter introduced by repeating the same expression unnecessarily.
Bad. $(a+b)^3 = (a+b)(a+b)^2 = (a+b)(a^2 + 2ab + b^2) = a^3 + 3a^2 b + 3a b^2 + b^3$
Bad. $$\begin{align*} (a+b)^3 &= (a+b)(a+b)^2 \\ (a+b)^3 &= (a+b)(a^2 + 2ab + b^2) \\ (a+b)^3 &= x^3 + 3x^2 + 3x + 1 \end{align*}$$
Bad. $$\begin{align*} (a+b)^3 &= (a+b)(a+b)^2 \\ (a+b)(a+b)^2 &= (a+b)(a^2 + 2ab + b^2) \\ (a+b)(a^2 + 2ab + b^2) &= x^3 + 3x^2 + 3x + 1 \end{align*}$$
Good. $$\begin{align*} (a+b)^3 &= (a+b)(a+b)^2 \\ &= (a+b)(a^2 + 2ab + b^2) \\ &= a^3 + 3a^2 b + 3a b^2 + b^3 \end{align*}$$
Use consistent punctuation after display style expressions
Some mathematicians treat display style expressions as part of the surrounding sentence structure, ending with a comma or a period as appropriate in the writing, while others omit all punctuation after display style expressions. In these notes, for instance, we chose to omit punctuation. Either way is acceptable, but be consistent with your choice in a given piece of writing.
Avoid pointless parentheses in mathematical expressions
In general, if you can make use of the precedence of operation to eliminate extra parentheses, and omit other parts of a math expression when they are clear from context, you should do so to keep things simple and clean.
Bad. $(x-1)^2 = (x^2 - 2x + 1)$
Good. $(x-1)^2 = x^2 - 2x + 1$
Extra parentheses are fine if they are serving to emphasize a part of the expression to the reader as being special or grouped together. For instance, the following example indicates to the reader that the grouping of the terms on the right side deserves special attention:
Okay. $(a+b)^3 = (a^3 + b^3) + 3(a^2 b + a b^2)$
Use parentheses to clarify between subtraction and negative signs
Also consider whether using concatenation for multiplication is clear from context or whether it should be indicated explicitly.
Bad. $(x+y) \cdot -z = -xz - yz$
Good. $(x+y)(-z) = -xz - yz$
Good. $(x+y)\cdot (-z) = -x\cdot z - y\cdot z$
Writing mathematics is similar in many ways to any other type of writing you might do! It’s about distilling your ideas down into a simple and organized form, and communicating these ideas to your readers clearly and efficiently. This is, by its nature, a somewhat messy process, but some things to keep in mind include:
Tell the reader where you are going
As you are explaining the steps of your proof, include a few words to describe the bigger picture of your argument or the strategy you are using. This allows the reader to anticipate the specifics of your proof more accurately, and can greatly increase their understanding of how the pieces of your proof fit together to form a cohesive argument.
Good. We will prove this claim by induction on $n$.
Good. We now consider the converse direction.
Good. The following will make use of compactness of the space $X$.
Four common situations where such comments can be valuable are as follows.
contradiction.induction, announcing the
base case, and announcing the
inductive hypothesis or step.if and only if statement),cases, and announcing the start of the proof of
each case.Use key phrases to explain your reasoning
Expressions like since, because,
on the other hand, observe, and
note help guide the reader’s attention and
elaborate on the relations between different statements. Vary
your word choices to avoid monotonous writing. Notice that
unlike semiformal or formal proofs, we generally do not
usually cite the rules of inference for logic in traditional
expository proofs, although we may refer to the overall
strategy being used (induction, proof by contradiction, proof
by cases, etc.).
Bad. Suppose 5 divides $b^3$ and 3 divides $b^3$. We showed that if a prime $p$ divides a power $a^k$, then $p$ divides $a$. Thus, 5 divides $b$. Thus, 3 divides $b$. Thus, 15 divides $b$.
Good. Suppose $3$ divides $b^3$ and $5$ divides $b^5$. We showed that any prime that divides a power of a natural number, $n$, must also divide $n$. Thus, as $3$ and $5$ are both prime, they both divide $b$. Therefore their product, $15$, must also divide $b$.
Plan enough time to write
Any form of writing takes time, and the rigor and attention to detail needed in mathematical writing only magnifies this requirement.
Outline your ideas before you begin writing
When writing down a mathematical proof or mathematical exposition in general, it helps to have a clear idea of what you want to say, and in what order. This gives you an opportunity to look at the big picture and make changes in the structure before you spend a lot of time hammering out the little details. Although it might feel better to just dive right into the writing phase, you’ll save time and produce significantly better exposition by planning ahead.
Proofread!
When time is short, it may be tempting to finish the last line of your proof, throw on a quick Q.E.D., and hand in your writing, but this is a terrible idea. Math is hard to write, and it is nearly impossible to write it without at least a few typos. When you look back on what you’ve written, you’ll be able to correct any small errors that you made, and you might also gain some additional insight into the math you were writing about, or come up with a better way of expressing the solution. Writing is an iterative process, so make sure to give your work at least a second look over to improve its quality.
Check your spelling, grammar, and punctuation
People reading mathematical proofs are usually intelligent and easily bothered by spelling errors, improper use of contractions, incorrect homophones, and general grammatical sloppiness. These sorts of flaws can distract from the content of your proof, so make sure you pay attention to these kinds of errors as well when you’re proofreading your work!
In addition to the special rules for mathematical writing mentioned in the previous section, the requirements of correctly typesetting mathematical expressions are a challenge for most modern word processors and text editors. As a result, the de facto standard for mathematical typesetting that all mathematicians use is based on a language called $\LaTeX$.
There are many great $\LaTeX$ tutorials on the internet, and in most cases doing a search for the thing you are trying to do will immediately answer any question that comes up. (See our course website for details.)
There are many do’s and don’ts when using $\LaTeX$ itself, which you will learn with time and experience (usually because what you typed ends up looking terrible). But there is one beginner mistake that we will mention for you to be aware of. Resist the urge to manually format your compiled document by hand.