Robert Recorde invented $=$ in 1557 with a simple justification: no two things can be more equal than two parallel lines of the same length (chapter 1 tells this story, in the text on the signs of the operations). The sign caught on so well that today it does jobs Recorde never imagined. Look at these lines:

$$3+4=7,\qquad 2x+1=7,\qquad (a+b)^2=a^2+2ab+b^2,$$$$f(x)=x^2,\qquad \tfrac12=\tfrac24.$$

The sign is the same in all of them. What it says is not.

  • In $3+4=7$, the equals sign records a result. This is the reading the calculator teaches.
  • In $2x+1=7$, the equals sign asks a question: for which number is this true? It is an equation.
  • In $(a+b)^2=a^2+2ab+b^2$, the equals sign states a law, valid for all numbers. It is an identity.
  • In $f(x)=x^2$, the equals sign gives a name: from now on, $f(x)$ means $x^2$. It is a definition.
  • In $\frac12=\frac24$, the equals sign says that two different pieces of writing point to the same number.

The following texts in this chapter deal with each of these jobs. First, it is worth asking what they have in common.

The three laws of equality

Around 300 BC, Euclid opened the Elements with a few “common notions”, truths he thought too obvious to need proof. The first is this one:

Things which are equal to the same thing are also equal to one another.

— Euclid, Elements, book I, common notion 1

Today this sentence is called transitivity: if $A=B$ and $B=C$, then $A=C$. Together with two others, it forms the portrait of any equality:

Property In symbols In words
Reflexive $A=A$ everything is equal to itself
Symmetric if $A=B$, then $B=A$ the balance has no preferred side
Transitive if $A=B$ and $B=C$, then $A=C$ things equal to the same thing are equal to one another

These three laws are what allow calculations to be chained, $A=B=C=D$, so that the first term can be said to equal the last. All of school algebra depends on them, and they are almost never said out loud.

The relatives of the equals sign

Whenever one of the jobs of the equals sign became important enough, mathematics gave it a sign of its own. The result is a family of similar signs, each with its own reading:

Sign Reading Example Is it symmetric?
$=$ is equal to $3+4=7$ yes
$\equiv$ is identical to; is congruent to $\sin^2x+\cos^2x\equiv1$; $17\equiv2\pmod5$ yes
$:=$ is defined as $f(x):=x^2$ no: the new name always stays on the left
$\approx$ is approximately $\pi\approx3.14$ yes
$\cong$ is congruent to (figures) $\triangle ABC\cong\triangle DEF$ yes
$\sim$ is similar to $\triangle ABC\sim\triangle DEF$ yes
$\Leftrightarrow$ is equivalent to (sentences) $2x=6\Leftrightarrow x=3$ yes

Look at the last column. The $:=$ is not symmetric: you can write $x^2=:f(x)$, but the sign changes shape to keep the new name next to the colon. And $\approx$, which looks so innocent, hides a trap that the first exercise shows: it is not transitive.

Equals signs that break the rules

There is an equals sign that every computer science student knows and that breaks symmetry on purpose. When one writes

$$n^2+n=O(n^2),$$

the sign means “grows at most like”. But it makes no sense to write $O(n^2)=n^2+n$, and from $n=O(n^2)$ and $n^2=O(n^2)$ nobody concludes that $n=n^2$. The Dutch mathematician Nicolaas de Bruijn called this use a one-way equality. The right sign would be $\in$, “belongs to”: $n^2+n\in O(n^2)$. The $=$ stayed out of tradition.

Exercise: ≈ is not transitive

Say that two numbers are “approximately equal” when they differ by less than $0.05$. Then $1\approx1.04$ and $1.04\approx1.08$. But $1$ and $1.08$ differ by $0.08$, and therefore $1\not\approx1.08$. With small steps, a chain of $\approx$ can go from any number to any other. This is why, in a calculation with approximations, the error has to be tracked step by step: each $\approx$ adds a little.

Exercise: what job is the equals sign doing?

(a) $x^2-1=(x-1)(x+1)$. (b) $x^2-1=8$. (c) $g(x)=x^2-1$. (d) $9-1=8$.

(a) It states a law: it holds for every $x$, it is an identity. (b) It asks a question: only $x=3$ and $x=-3$ work, it is an equation. (c) It gives a name: it defines the function $g$. (d) It records the result of a calculation. Notice that (b) and (d) are linked: (d) is what you get by substituting $x=3$ in (b), and that is how a solution is checked.

Chapter index · ← Chapter 3: references and credits · Frege and the morning star: what a = b says →