To elaborate on each of these formulas in detail would make this review too long, but they range from $1+1=2$, which is about the very origin of counting and the different number systems that have been used by different cultures, up to Whitehead and Russell's proof of 1 + 1 = 2 in the $\textit{Principia Mathematica}$. The last chapter is an explanation of $\aleph_0+\aleph_0=\aleph_0$ introducing countability, cardinal numbers, and the continuum hypothesis.

The geometry of Greek antiquity is covered in the next chapter, starting with stories on the sum of the angles of a triangle being 180 degrees or $\pi$ radians and an excursion about the parallel hypothesis and non-Euclidean geometries. There is of course discussion of the Pythagorean theorem in two and more dimensions and its analogue in Fermat's last theorem regarding $a^n+b^n=c^n$, as well as the formulas for the circumference and area of a circle. The latter two involve $\pi$, that most remarkable number in mathematics which turns up in so many unexpected places.

The equation $\phi^2=\phi+1$ introduces the golden ratio, Fibonacci numbers and their generalizations (the Lucas numbers) and their fascinating relation to Pascal's triangle, including an excursion on tessellations and Penrose tilings. A similar equation $x^2=x$ is used to justify the introduction of Boole algebra (19th century), set theory and switching circuits.

There are also many stories for the previous 17th-18th centuries which I quickly mention: prime numbers (those named after Mersenne, Fermat, and Euler), combinatorics (as related to probability), and Euler's formula for polyhedra $V-E+F=2$ (where the letters represent the number of vertices, edges and faces).

On the calculus side Briggs and Napier proposed the logarithmic function, and taking the square root of a negative number became acceptable after the introduction imaginary and complex numbers. Hamilton then generalized the idea to quaternions. Also, complex numbers have a convenient geometric interpretation as a point in the 2-D plane. The relation $e^{ix}=\cos(x)+i\sin(x)$ easily implies de Moivre's formulas, and $e^{i\pi}=-1$ is considered the most beautiful formula in mathematics.

Another story describes Euler's solution to be the Basel problem, showing that $\sum_{n\geq 1} \frac{1}{n^2}=\frac{\pi^2}{6}$. This is directly connected to Riemann's zeta function $\zeta(s)=\sum_{n\geq 1} \frac{1}{n^s}$. The location of all its poles (formulated in the Riemann hypothesis) is still an open problem with deep consequences for the distribution of prime numbers.