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How the square root of 2 became a number

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

Then, in the mid-1800s, Richard Dedekind, among others, realized that calculus — which had been developed 200 years earlier by Isaac Newton and Gottfried Leibniz — stood on a shaky foundation. Dedekind and others used his definition to prove major theorems in calculus for the first time — which allowed them not just to strengthen the edifice that Leibniz and Newton had built, but to add to it. Emmy Noether, a prolific mathematician who helped shape the field of abstract algebra in the early 20th century, is said to have told her students that “everything is already in Dedekind.”

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