Steven Strogatz in the NYT’s Opinionator:
For more than 2,500 years, mathematicians have been obsessed with solving for x. The story of their struggle to find the “roots” — the solutions — of increasingly complicated equations is one of the great epics in the history of human thought.
And yet, through it all, there’s been an irritant, a nagging little thing that won’t go away: the solutions often involve square roots of negative numbers. Such solutions were long derided as “sophistic” or “fictitious” because they seemed nonsensical on their face.
Until the 1700s or so, mathematicians believed that square roots of negative numbers simply couldn’t exist.
They couldn’t be positive numbers, after all, since a positive times a positive is always positive, and we’re looking for numbers whose square is negative. Nor could negative numbers work, since a negative times a negative is, again, positive. There seemed to be no hope of finding numbers which, when multiplied by themselves, would give negative answers.
We’ve seen crises like this before. They occur whenever an existing operation is pushed too far, into a domain where it no longer seems sensible. Just as subtracting bigger numbers from smaller ones gave rise to negative numbers and division spawned fractions and decimals, the free-wheeling use of square roots eventually forced the universe of numbers to expand…again.
Historically, this step was the most painful of all.