This chapter followed the equals sign from a line of calculation to a whole proof. The guides below gather the questions that help read both.

Five questions for an equals sign

  1. Does it ask, state or name? If it asks, it is an equation, and the task is to find the solutions. If it states something for all values, it is an identity, and the task is to prove it. If it names, it is a definition, and the task is to accept it and remember it.
  2. Do the two sides have the same sense or only the same reference? If the sense is the same, the equality is empty. If it is different, the equality teaches something, and it is worth asking what.
  3. Is it a true equality or an equivalence? $\frac12=\frac24$ is an equality between numbers, but an equivalence between fractions. $17\equiv2\pmod5$ says that two different numbers fall into the same drawer.
  4. Can I replace one side by the other here? Inside a calculation, always. Inside a sentence about what someone knows or wrote, not always.
  5. Do the three laws hold? If the relation is not symmetric ($:=$, the $=$ of big $O$) or not transitive ($\approx$), it is not a true equals sign, and chains $A=B=C$ stop working.

Five questions for a proof

  1. What exactly is being claimed? Find the statement and its letters: which are arbitrary, which are given.
  2. Where is the “let”? That is where the general becomes particular. Check that nothing was assumed about the particular case beyond what the statement gives.
  3. What kind of arrow links the lines? A chain of $=$ and $\Leftrightarrow$ can be read both ways. A chain of $\Rightarrow$ only works from beginning to end.
  4. Where does each hypothesis open and close? Every time “suppose” appears, look for where it ends. Letters introduced inside do not exist outside.
  5. What was left out? Every “it is easy to see”, every ellipsis, every “similarly” is a step the reader should be able to fill in.

A complete reading of a short proof

Claim. For every integer $n$, the number $n^3-n$ is divisible by $6$.

Proof. Let $n$ be an integer. Then $n^3-n=(n-1)\,n\,(n+1)$, the product of three consecutive integers. Among three consecutive integers, one is a multiple of $3$ and at least one is even. Hence the product is a multiple of $2$ and of $3$, and therefore of $6$. $\blacksquare$

  1. The statement speaks of every integer: $n$ is a generalised number.
  2. “Let $n$ be an integer” is Euclid’s setting-out. Nothing is assumed about $n$ beyond its being an integer.
  3. $n^3-n=(n-1)n(n+1)$ is an identity, proved by the chain $(n-1)n(n+1)=n(n^2-1)=n^3-n$. The equals sign here replaces one piece of writing by another with a more useful sense: the factorised form shows the three consecutive numbers, the original does not.
  4. “Among three consecutive integers, one is a multiple of 3” is a fact invoked without proof. Behind it lies an equivalence: modulo $3$, the three consecutive numbers fall into the drawers $0$, $1$ and $2$, one in each.
  5. “Hence… therefore” marks two $\Rightarrow$ steps. The second uses the fact that $2$ and $3$ have no common divisor, another fact left out.
  6. The $\blacksquare$ closes the proof and returns the reader to the general statement.

Glossary

  • Equality. A relation between two expressions that point to the same object. Reflexive, symmetric and transitive.
  • Sense and reference. The reference of an expression is the object it designates; the sense is the way it presents that object. $2^{10}$ and $1024$ have the same reference and different senses.
  • Equation. An equality with letters read as a question: for which values is it true? Its solution set is the answer.
  • Identity. An equality that is true for all values of the letters. Sometimes written with $\equiv$.
  • Definition. An equality that gives a name to an expression. Sometimes written with $:=$.
  • Equivalence relation. A relation that is reflexive, symmetric and transitive. It divides objects into equivalence classes.
  • Congruence. $a\equiv b\pmod m$ when $m$ divides $a-b$. The equivalence relation of remainders, introduced by Gauss in 1801.
  • Implication. $P\Rightarrow Q$: whenever $P$ is true, $Q$ is true as well.
  • Logical equivalence. $P\Leftrightarrow Q$: $P$ and $Q$ are true in exactly the same cases. Read “if and only if”, abbreviated iff.
  • Extraneous root. A solution that appears at the end of a solving process but does not satisfy the original equation, introduced by a $\Rightarrow$ step that is not $\Leftrightarrow$.
  • Calculational proof. A chain of equalities in which each link carries its justification.
  • Proof by contradiction. A proof that assumes the opposite of what is wanted and reaches a contradiction.
  • Mathematical induction. A proof in two pieces, base case and step, that a claim holds for all natural numbers.
  • Counterexample. A single case in which a general claim fails. One is enough to refute it.
  • Q.E.D. and $\blacksquare$. End-of-proof marks: quod erat demonstrandum and Halmos’s tombstone.
Exercise: apply the guide for the equals sign

Read $\displaystyle\cos2x=\cos^2x-\sin^2x$. It makes a statement for every $x$: it is an identity. The two sides have the same reference for each $x$, but different senses: the left one speaks of the doubled angle, the right one of the single angle. Read as an equation, the question “for which $x$ does it hold?” would have the answer “all of them”. If someone writes $\cos2x=1$, then there is a question, with the answer $x=k\pi$, for integer $k$.

Exercise: apply the guide for proofs

Find the step that was left out: “If $n^2$ is even, then $n$ is even. Proof: if $n$ were odd, $n=2k+1$, and $n^2=4k^2+4k+1$ would be odd. $\blacksquare$”

The proof opens a hypothesis (“if $n$ were odd”), shows that it leads to $n^2$ being odd, and stops there. The missing step is the closing: “this contradicts $n^2$ being even; hence $n$ is not odd, and therefore it is even”. The proof also uses, without saying so, that every integer is either even or odd.

Chapter index · ← Examples do not prove: Euler, Pascal and induction · References and credits →