By definition the square root of a number a is a number r with the property that the square of r is equal to a. Thus 10 is a square root of 100, and –11 is a square root of 121. Every positive real number has exactly two square roots, one of which is positive and the other negative. Therefore we may agree to call the positive of the two roots the square root. In this sense, 1.414213… is the square root of 2, and 7 is the square root of 49.
Since the square of every real number is positive or 0, no negative real number can have a real square root. However, it is possible to extend the domain of real numbers to the even larger domain of complex numbers, in which all the real numbers do have square roots.
If you imagine the real numbers as points on a line, say the x-axis in a coordinate system, then the complex numbers may be viewed as the set of all the points in the plane. We may then define for complex numbers an addition (which functions like vector addition) and a multiplication (which takes some getting used to) so that all the usual properties of numbers are satisfied: the commutative, associative, and distributive laws. We note in particular that there is a complex number whose square corresponds precisely to the point –1 on the x-axis. The number that is associated with the coordinates (0, 1) has this property. It is called the imaginary unit and is denoted by the letter i.
Thus i is a square root of –1. Furthermore, –i (this number corresponds to the point (0, –1) is also a square root of –1. In contrast to the case of the real numbers, here there is no obvious preference for one of the roots over the other. (Therefore, mathematicians are not entirely happy about saying ‘i is the square root of –1’; just as in everyday speech we don’t say that we ran into the brother of Irene if she in fact has two brothers.)
Complex numbers are a fertile domain for the existence of square roots: the equation x2 – a = 0 always has a solution for any complex number a.
Remarkably, every equation of the form
a0+a1x+a2x2+ ••• +anxn = 0
with arbitrary (real or complex) coefficients can be solved in the domain of complex numbers.
Because of their importance in solving such algebraic equations, these numbers were employed intuitively centuries ago. But for a long time, mathematicians had no secure understanding of how to operate in this domain. One spoke of ‘false roots’ and so on. It was only in the nineteenth century that methods were proposed for eliminating such difficulties.
Today, professional mathematicians and those who use mathematics in applied fields consider complex numbers to be just as valid as the natural numbers 1, 2, 3, ….
(Ehrhard Behrends, Freie Universität Berlin)