Chapter 1 told how the signs of calculus were born: Newton’s dots, Leibniz’s $d$, the $\int$, Cauchy’s $\lim$. This chapter asks how those signs are read and what they do when one calculates.
Calculus deals with three things that are hard to write: what changes, what accumulates and what never ends. For each of them, mathematics invented more than one way of writing, and each way shows one part and hides another. The $\frac{dy}{dx}$ behaves like a fraction almost always, and misleads precisely when it does not. The $D$ of operators lets you factorise a differential equation as if it were a polynomial. The $\infty$ looks like a number and is not. And a sum with infinitely many terms can change its value just because the order of the terms has been changed.
Texts in this chapter
- Four ways to write the derivative
- From Δ to d: the difference that shrinks
- Is dy/dx a fraction?
- The D that factorises: the derivative as an operator
- The integral read sign by sign
- The fundamental theorem: when d and ∫ cancel
- Wallis’s ∞: a sign that is not a number
- Sums that never end: the = of series
- How to read calculus: a guide and a glossary
- References and credits