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 i to represent the square root of a negative number. Thus √−1 = i , and √−25 = √25 × √−1 = 5 i .
Imaginary numbers were largely a stepchild in mathematics until the nineteenth century. Then, they were incorporated into another mathematical concept known as complex numbers. A complex number is a number that consists of a real part and an imaginary part. For example, the number 5 + 3 i is a complex number because it contains a real number (5) and an imaginary number (3 i ). One reason complex numbers are important is that they can be manipulated in ways so as to eliminate the imaginary part.
[ See also Complex numbers ]