One letter, $f$, and three sentences:

$$f(x)=x^2,\qquad f(x)=x\cdot x,\qquad f(x)=4.$$

The first gives a name: from here on, $f$ is the function that squares. The second states a law: for every $x$, the value of $f$ is $x$ times $x$. The third asks a question: for which $x$ is the function equal to $4$? The answer is $x=2$ or $x=-2$.

The sign is identical in all three. The reader’s task is not:

Kind Question it answers What you do with it Example
Equation for which values is it true? you solve it $x^2=4$
Identity is it true for all of them? you prove it $(x-2)(x+2)=x^2-4$
Definition what does this name mean? you accept it $n!=1\cdot2\cdots n$

A definition is neither true nor false: it is a christening. It makes no sense to “prove” that $n!=1\cdot2\cdots n$, just as it makes no sense to prove that someone’s name is Mary. An identity, by contrast, has to be proved, and an equation has to be solved. Whoever confuses the three jobs does the right thing to the wrong object.

Signs that try to warn

Over the centuries, several mathematicians tried to give each job its own sign.

  • $\equiv$ for identities. According to the historian Florian Cajori, Bernhard Riemann was already using it in 1857 to say “equal for all values”. It still appears in physics and engineering books: $\sin^2x+\cos^2x\equiv1$.
  • $:=$ for definitions. The colon goes on the side of the new name: $f(x):=x^2$ reads “$f(x)$ is, by definition, $x^2$”. The sign became popular with programming languages, where it marks assignment, and spread into mathematics books at the end of the twentieth century.
  • $\overset{\text{def}}{=}$ or $\stackrel{\triangle}{=}$. Variations that put the word “definition” on top of the equals sign. More explicit, less used.

None of them won for good. Most books still write $=$ for everything and rely on context: words such as “let”, “define”, “for all”, “solve” do the work the sign does not. It is a typical case of a notation that leaves to pragmatics, in Morris’s sense, what syntax does not say (see chapter 2).

The freshman’s dream read as an equation

One of the most famous mistakes in algebra is to write

$$(a+b)^2=a^2+b^2.$$

In English it even has a nickname: the freshman’s dream. As an identity, the sentence is false: with $a=b=1$, the left side is $4$ and the right side is $2$. But look at what happens when it is read as an equation, a question about $a$ and $b$:

$$\begin{aligned} (a+b)^2=a^2+b^2&\iff a^2+2ab+b^2=a^2+b^2\\ &\iff 2ab=0\\ &\iff a=0\ \text{ or }\ b=0. \end{aligned}$$

The equation has an answer, and the answer is precise: the freshman’s dream is only true when one of the two letters is zero. Reading the equals sign as a question turned a mistake into a discovery, and showed exactly how big the mistake is.

The opposite also happens. A student asked to prove the identity $(x+1)^2-(x-1)^2=4x$ sometimes starts “solving”: moves everything to one side, simplifies, arrives at $0=0$ and does not know what to do with it. The $0=0$ is the sign that the sentence holds for every $x$: the equation the student tried to solve has all numbers as its solution set. The identity was proved without the student noticing.

Exercise: equation, identity or definition?

(a) $(x+1)^2=x^2+2x+1$. (b) $(x+1)^2=9$. (c) $\binom{n}{k}=\frac{n!}{k!\,(n-k)!}$. (d) $x+1=x+2$.

(a) Identity: it holds for every $x$. (b) Equation: $x+1=3$ or $x+1=-3$, so $x=2$ or $x=-4$. (c) Definition of the binomial coefficient (although in some books $\binom nk$ is defined as “the number of subsets”, and the formula becomes a theorem). (d) An equation with no solution: subtracting $x+1$ leaves $0=1$.

Exercise: is √(x²) = x an identity?

No. With $x=-3$, the left side is $\sqrt9=3$ and the right side is $-3$. The correct identity is $\sqrt{x^2}=|x|$. Read as an equation, $\sqrt{x^2}=x$ asks when $|x|=x$, and the answer is: for every $x\geq0$. The solution set is neither everything nor nothing, but half of the line.

Chapter index · ← Frege and the morning star: what a = b says · Equivalence: fractions, clocks and Gauss’s ≡ →