The operation of taking the square root (or any other roots) of a real number has an interesting history which we will not review here. Two aspects of number theory that have grown directly out of exploring square roots are, however, irrational numbers (since the square root of most integers can be shown to be irrational) and imaginary numbers. The former will not interest us as we already work over at least the real numbers which include all rationals and irrationals, positive and negative. Imaginary numbers, however, are a true extension of the reals.
Since the product of any two non-negative numbers is non-negative, and the product of any two negative numbers is similarly non-negative, we cannot find any real number that, when squared, is a negative number. This permits us to ``imagine'' a field of numbers where the square root of a nonzero negative number exists. Such a field cannot be identical to the reals already discussed above. It must contain the real numbers, though, in order to be closed under multiplication (as the square of an ``imaginary'' number is a negative real number, and the square of that real number is a positive real number).
If we define the unit imaginary number to be: i = +-1 such that ±i^2 = -1 we can then form the rest of the field by scaling this imaginary unit through multiplication by a real number (to form the imaginary axis) and then generating the field of complex numbers by summing all possible combinations of real and imaginary numbers. Note that the imaginary axis alone does not form a field or even a multiplicative group as the product of any two imaginary numbers is always real, just as is the product of any two real numbers. However, the product of any real number and an imaginary number is always imaginary, and closure, identity, inverse and associativity can easily be demonstrated.
The easiest way to visualize complex numbers is by orienting the real axis at right angles to the imaginary axis and summing real and imaginary ``components'' to form all of the complex numbers. There is a one-to-one mapping between complex numbers and a Euclidean two dimensional plane as a consequence that is very useful to us as we seek to understand how this ``imaginary'' generalization works.
We can write an arbitrary complex number as for real numbers and . As you can easily see, this number appears to be a point in a (complex) plane. Addition and subtraction of complex numbers are trivial - add or subtract the real and imaginary components separately (in a manner directly analogous to vector addition).
Multiplication, however, is a bit odd. Given two complex numbers
, we have:
z = z_1 ·z_2 = x_1 x_2 + i (x_1 y_2 + y_1 x_2) - y_1 y_2
so that the real and imaginary parts are
&real#Re;z & = & x_1 x_2 - y_1 y_2
&image#Im;z & = & x_1 y_2 + y_1 x_2
This is quite different from any of the rules we might use to form
the product of two vectors. It also permits us to form the so-called
complex conjugate of any imaginary number, the number that one can
multiply it by to obtain a purely real number that appears to be
the square of the Euclidean length of the real and imaginary components
z & = & x + iy
z^* & = & x - iy
|z|^2 = z^* z = z z^* & = & x^2 + y^2
A quite profound insight into the importance of complex numbers can be
gained by representing a complex number in terms of the plane polar
coordinates of the underlying Euclidian coordinate frame. We can use
the product of a number
and its complex conjugate
that is the polar distance of the
complex number from the complex origin. The usual polar angle
can then be swept out from the positive real axis to identify the
complex number on the circle of radius
. This representation can
then be expressed in trigonometric forms as:
z & = & x + i y = |z| (&thetas#theta;) + i|z|(&thetas#theta;)
& = & |z| ( (&thetas#theta;) + i(&thetas#theta;) )
& = & |z| e^i&thetas#theta; where the final result can be observed any number of ways, for example by writing out the power series of for complex and matching the real and imaginary subseries with those for the cosine and sine respectively. In this expression &thetas#theta;= ^-1 yx determines the angle in terms of the original ``cartesian'' complex coordinates.
Trigonometric functions are thus seen to be quite naturally expressible in terms of the exponentials of imaginary numbers. There is a price to pay for this, however. The representation is no longer single valued in . In fact, it is quite clear that: z = |z| e^i&thetas#theta;±2 n &pi#pi; for any integer value of . We usually avoid this problem initially by requiring (the ``first leaf'') but as we shall see, this leads to problems when considering products and roots.
It is quite easy to multiply two complex numbers in this representation:
z_1 & = & |z_1| e^i&thetas#theta;_1
z_2 & = & |z_2| e^i&thetas#theta;_2
z = z_1 z_2 & = & |z_1||z_2| e^i(&thetas#theta;_1 + &thetas#theta;_2) or the amplitude of the result is the product of the amplitudes and the phase of the result is the sum of the two phases. Since may well be larger than even if the two angles individually are not, to stick to our resolution to keep the resultant phase in the range we will have to form a suitable modulus to put it back in range.
Division can easily be represented as multiplication by the inverse of a complex number: z^-1 = 1|z| e^-i&thetas#theta; and it is easy to see that complex numbers are a multiplicative group and division algebra and we can also see that its multiplication is commutative.
One last operation of some importance in this text is the formation of roots of a complex number. It is easy to see that the square root of a complex number can be written as: z = ±|z|e^i&thetas#theta;/2 = |z| e^i(&thetas#theta;/2 ± n&pi#pi;) for any integer . We usually insist on finding roots only within the first ``branch cut'', and return an answer only with a final phase in the range .
There is a connection here between the branches, leaves, and topology - there is really only one actual point in the complex plane that corresponds to ; the rest of the ways to reach that point are associated with a winding number that tells one how many times one must circle the origin (and in which direction) to reach it from the positive real axis.
Thus there are two unique points on the complex plane (on the principle branch) that are square roots (plus multiple copies with different winding numbers on other branches). In problems where the choice doesn't matter we often choose the first one reached traversing the circle in a counterclockwise direction (so that it has a positive amplitude). In physics choice often matters for a specific problem - we will often choose the root based on e.g. the direction we wish a solution to propagate as it evolves in time.
Pursuing this general idea it is easy to see that where is an integer are the points |z|^1n e^i(&thetas#theta;/n ±2m&pi#pi;/n) where as before. Now we will generally have roots in the principle branch of and will have to perform a cut to select the one desired while accepting that all of them can work equally well.