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

by Steven H. Strogatz

Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.

math always involves both invention and discovery: we invent the concepts but discover their consequences. As we’ll see in the coming chapters, in mathematics our freedom lies in the questions we ask—and in how we pursue them—but not in the answers awaiting us.

Fractions always yield decimals that terminate or eventually repeat periodically—that can be proven—and since this decimal does neither, it can’t be equal to the ratio of any whole numbers. It’s irrational.