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• Complex numbers

An imaginary number is the square root of a negative real number. (The square root of a number is a second number that, when multiplied by itself, equals the first number.) As an example, √−25 is an imaginary number.

The problem with imaginary numbers arises because the square (the result of a number multiplied by itself) of any real number is always a positive number. For example, the square of 5 is 25. But the square of −5 (−5 × −5) is also 25. What does it mean, then, to say that the square of some number is −25. In other words, what is the answer to the problem √−25 = ?

As early as the sixteenth century, mathematicians were puzzled by this question. Italian mathematician Girolamo Cardano (1501–1576) is generally regarded as the first person to have studied imaginary numbers. Eventually, a custom developed for using the lowercase letter...