Elementary Linear Algebra: For University of Lethbridge Math 1410

We now move on to the study of the set of complex numbers. As you may recall, the complex numbers fill an algebraic gap left by the real numbers. There is no real number \(x\) with \(x^2 = -1\text{,}\) since for any real number \(x^2 \geq 0\text{.}\) However, we could formally extract square roots and write \(x = \pm \sqrt{-1}\text{.}\) We build the complex numbers by relabelling the quantity \(\sqrt{-1}\) as \(i\text{,}\) the unfortunately misnamed imaginary unit, \(i\text{.}\) The number \(i\text{,}\) while not a real number, is defined so that it plays along well with real numbers and acts very much like any other radical expression. For instance, \(3(2i) = 6i\text{,}\) \(7i-3i = 4i\text{,}\) \((2-7i) + (3 + 4i) = 5-3i\text{,}\) and so forth. The key properties which distinguish \(i\) from the real numbers are listed below.

Complex numbers include things you’d normally expect, like \(3+2i\) and \(\frac{2}{5} - i\sqrt{3}\text{.}\) However, don’t forget that \(a\) or \(b\) could be zero, which means numbers like \(3i\) and \(6\) are also complex numbers. In other words, don’t forget that the complex numbers include the real numbers, so \(0\) and \(\pi - \sqrt{21}\) are both considered complex numbers. We want to study the arithmetic of complex numbers, but before we can do so, we first need to make sure we understand what it means for two complex numbers to be equal.

The arithmetic of complex numbers is as you would expect. The definitions of addition and multiplication are as follows.

Addition of complex numbers is defined by simply adding the corresponding parts. The definition of multiplication looks complicated, but it’s simply an application of the “F.O.I.L.” rule for multiplying binomials, where we have to account for the fact that \(i^2=-1\text{.}\)

As long as we remember the two properties in Definition 1.4.1, we can treat expressions involving \(i=\sqrt{-1}\) as we would with any other radical. Let’s work through an example to see how this works.

To find \(\overline{6}\text{,}\) we note that \(\overline{6} = \overline{6+0i}= 6 - 0i = 6\text{,}\) so \(\overline{6} = 6\text{.}\) Similarly, \(\overline{4i} = -4i\text{,}\) since \(\overline{4i} = \overline{0 + 4i} = 0 - 4i = -4i\text{.}\) The properties of the conjugate are summarized in the following theorem.

Essentially, Theorem 1.4.7 says that complex conjugation works well with addition, multiplication and powers. The proofs of these properties can best be achieved by writing out \(z = a+bi\) and \(w = c+di\) for real numbers \(a\text{,}\) \(b\text{,}\) \(c\) and \(d\text{.}\) Next, we compute the left and right sides of each equation and verify that they are the same.

The last property is a characterization of real numbers. If \(z\) is real, then \(z = a + 0i\text{,}\) so \(\overline{z} = a - 0i = a = z\text{.}\) On the other hand, if \(z=\overline{z}\text{,}\) then \(a+bi = a - bi\) which means \(b=-b\) so \(b=0\text{.}\) Hence, \(z = a +0i = a\) and is real.

It is worth noting that although the arithmetic of complex numbers seems, at first impression, to be very different and strange compared to the arithmetic of real numbers, it actually satisfies all the same properties, as outlined in the following theorem.

We leave the proof of Theorem 1.4.8 as a long (but straightforward) exercise. Working through the proof is a good way to confirm for yourself that you understand the corresponding rules for real number arithmetic from Section 1.2, and how the properties for complex arithmetic are inherited from their real counterparts.

It is no coincidence that the non-real solutions to the equations in Example 1.4.9 appear in complex conjugate pairs. Any time we use the Quadratic Formula to solve an equation with real coefficients, the answers will form a complex conjugate pair owing to the \(\pm\) in the Quadratic Formula. This is stated formally in the following theorem.

Theorem 1.4.10 tells us that if ever we obtain non-real zeros to a quadratic function with real coefficients, the zeros will be a complex conjugate pair. (Do you see why?) Next, we note that in Example 1.4.6, part 6, we found \((x-[1+2i])(x-[1-2i])=x^2-2x+5\text{.}\) This demonstrates that the factor theorem holds even for non-real zeros, i.e, \(x=1+2i\) is a zero of \(f(x)=x^2-2x+5\text{,}\) and, sure enough, \((x-[1+2i])\) is a factor of \(f(x)\text{.}\) It turns out that polynomial division works the same way for all complex numbers, real and non-real alike, so the Factor and Remainder Theorems hold as well. But how do we know if a general polynomial has any complex zeros at all? We have many examples of polynomials with no real zeros. Can there be polynomials with no zeros whatsoever? The answer to that last question is “No.” and the theorem which provides that answer is The Fundamental Theorem of Algebra.

The Fundamental Theorem of Algebra is an example of an “existence” theorem in mathematics. It guarantees the existence of at least one zero, but gives us no algorithm to use in finding it. It took mathematicians literally hundreds of years to prove the theorem in its full generality, and some of that history can be found by looking up the Fundamental Theorem on Wikipedia

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en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

. Note that the Fundamental Theorem of Algebra applies to not only polynomial functions with real coefficients, but to those with complex number coefficients as well.

The Fundamental Theorem of Algebra has since been proved many times, using many different methods, by many mathematicians. There are probably very few, if any, results in mathematics with the variety of proofs this result has. Unfortunately, none of the proofs can be understood within the realm of this text, but if the reader is sufficiently interested, a collection of proofs can be found at www.cut-the-knot.org/fta/analytic.shtml

Suppose \(f\) is a polynomial of degree \(n \geq 1\text{.}\) The Fundamental Theorem of Algebra guarantees us at least one complex zero, \(z_{1}\text{,}\) and as such, the Factor Theorem guarantees that \(f(x)\) factors as \(f(x) = \left(x - z_{1}\right) q_{1}(x)\) for a polynomial function \(q_{1}\text{,}\) of degree exactly \(n-1\text{.}\) If \(n-1 \geq 1\text{,}\) then the Fundamental Theorem of Algebra guarantees a complex zero of \(q_{1}\) as well, say \(z_{2}\text{,}\) so then the Factor Theorem gives us \(q_{1}(x) = \left(x - z_{2}\right) q_{2}(x)\text{,}\) and hence \(f(x) = \left(x - z_{1}\right) \left(x - z_{2}\right) q_{2}(x)\text{.}\) We can continue this process exactly \(n\) times, at which point our quotient polynomial \(q_{n}\) has degree \(0\) so it’s a constant. This argument gives us the following factorization theorem.