Chapter 2 showed that the equals sign is a balance, not an arrow. This chapter asks what else it is.
The answer is surprising: in $2x+1=7$, the equals sign asks a question; in $(a+b)^2=a^2+2ab+b^2$, it states a law; in $f(x)=x^2$, it gives a name; in $\frac12=\frac24$, it says that two different pieces of writing point to the same number. To keep these jobs apart, mathematics invented relatives of the equals sign: $\equiv$, $:=$, $\approx$, $\Leftrightarrow$. And when equalities are chained one after another, each one justified, a proof is born, and a proof has its own way of writing: “let”, “suppose”, “hence”, $\blacksquare$.
Texts in this chapter
- One sign, many equals
- Frege and the morning star: what a = b says
- Equation, identity, definition: three jobs for the same sign
- Equivalence: fractions, clocks and Gauss’s ≡
- ⇒ and ⇔: what is gained and what is lost when an equation is transformed
- The chain of equalities: how an identity is proved
- The words of a proof: let, suppose, hence, ∎
- Examples do not prove: Euler, Pascal and induction
- How to read a proof: a guide and a glossary
- References and credits