The Joy of X: A Guided Tour of Mathematics, from One to Infinity

by Steven H. Strogatz

Here “complex” doesn’t mean “complicated”; it means that two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2 + 3i. Complex numbers are magnificent, the pinnacle of number systems. They enjoy all the same properties as real numbers — you can add and subtract them, multiply and divide them — but they are better than real numbers because they always have roots. You can take the square root or cube root or any root of a complex number, and the result will still be a complex number.

“complex” doesn’t mean “complicated”; it means that two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2 + 3i. Complex numbers are magnificent, the pinnacle of number systems. They enjoy all the same properties as real numbers — you can add and subtract them, multiply and divide them — but they are better than real numbers because they always have roots. You can take the square root or cube root or any root of a complex number, and the result will still be a complex number.

intro to complex numbers

complex numbers visualised

Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating.

if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points. In this analogy, the function is like the elevation of each step compared to ground level. The rises of individual steps are like the derivative. Integrating the derivative is like summing the rises. And the two points are the top and the bottom.

integral fundamental

differential equations, which describe how interlinked variables change from moment to moment, depending on their current values.

differential equation broad purpose

The ball’s velocity relative to the court is the sum of two vectors: the ball’s velocity relative to you (a vector pointing down the line, as intended), and your velocity relative to the court (a vector pointing sideways, since that’s the direction you’re running).

tennis

divergence of a vector field measures: how fast the area of a small circle of corks grows.

divergence of a vector field

it indicates how strongly the field is swirling about a given point.

vector curl

(The natural logarithm behaves like the ordinary logarithm encountered in high school, except it’s based on the number e instead of 10.

natural logorithm