Complex Numbers: A Review

• Students in a linear systems course often have difficulty comprehending the concept of a complex number which leads, among other things, to difficulties manipulating complex numbers in calculations. In an effort to address this problem, I offer you this brief review. It is by no means mathematically rigorous, but hopefully it is accessible and useful to the target audience. First, let's review some hopefully already familiar concepts:

The Positive Integers

These are probably the first type of numbers you encountered. How many fingers am I holding up? Addition: 1+1 = 2, 1 + 1 + 1 = 3, 2 + 3 = (1 + 1 )+ (1 + 1 + 1) = 1 +1+1+1+1 = 5, etc. Multiplication: First, 1 times (X) anything gives you the same thing back: 1 X n = n X 1 = n Now, multiplication can be defined in terms of the distributive law: 2 X 3 = (1+1) X 3 = 1 X 3 + 1 X 3 = 3 + 3 = 6. , etc. Zero: n + 0 = 0+ n = n for any number n Subtraction: The inverse of addition: m "minus" m = m - n is defined as the number k such that n + k = m This is well defined, so far, as long as m is greater than or equal to n. What if m is less than n? Well, you probably learned that in this case, m-n = - (n-m), but that involves extending the set of numbers we deal with to the set of all integers, or simply the set of integers. More extensions of this sort will lead us naturally to the set of complex numbers.

The Integers

0 minus m, or simply -m, is defined in terms of subtraction above: m + (-m) = m - m = 0 The number line is illustrated above. It gives us a graphical way of thinking about numbers, which is quite useful, e.g. : 0 times anything equals 0, i.e. 0 X n = n X 0 = 0 for any n. Why?: 0 X n = (1 - 1) X n = 1 X n + (-1) X n = n - n = 0 Minus one times anything "changes the direction" of the number. i.e. (-1) X n = -n. Why? (-1) X n = (0 - 1) X n = (0 X n) - (1 X n) = 0 - n = -n The change in direction is 180 degrees, i.e. the new direction is opposite. This concept will be the key intuition leading to the extension to the set of complex numbers! Division: The inverse of multiplication. If q is not zero, p divided by q, written p/q , is defined as the number r such that r X q = p. Often, there will be no such integer r so satisfying this condition, so we invent such a number r This causes us again to extend the set of numbers we deal with to the set of rational numbers.

The Rational and Real Numbers



The set of rational numbers is the set of numbers of the form p/q , where p and q are integers. Many times, it is convenient to work with numbers that cannot be expressed in the form of p/q. For example, consider the concept of the square root of a number, say the square root of 2. We may consider the set of rational numbers r, whose square, r X r, is less than or equal to 2. The square root of 2 is formally defined as the least upper bound to this set of numbers. It can be shown that this least upper bound is not a rational number -- it is thus called an irrational number. Another example of an irrational number is pi, defined as 1/2 of the ratio of the circumference of a circle to the diameter of the circle. Thus, the set of rational numbers is not complete, in some sense, -- there are conceptually numbers "in between." The set of rational numbers together their limits are called the set of real numbers . An irrational number is a real number that is not rational. The set of real numbers is probably the set of numbers you are most used to dealing with and thinking about. Contrary to what you might have thought, the set of real numbers is still not complete, in a sense. We extended the set of rational numbers to the real numbers so that we had a well defined square root of 2, for example. You may have learned in high school that there is "no such thing" as the square root of a negative number. It is true that there are no real numbers whose square gives you a negative number. However, if we are willing to extend our concept of real numbers a bit, we can define square roots of negative numbers in a way consistent with the mathematical operations we already have defined. This leads us to extend the set of real numbers to the set of complex numbers.

The Complex Numbers


Conceptually, we may think of complex numbers as points in a two dimensional plane (called the complex plane and illustrated above), whereby a complex number z is an ordered pair of real numbers (x,y), i.e. z = (x,y). This the so called rectangular form of a complex number. There is some utility to this way of thinking, but this is not the entire story. For example, if we were two add to complex numbers z_1 and z_2 , we define the sum z_1 + z_2 as if we were adding two vectors, i.e. addition is component-wise. But how do we define multiplication between two complex numbers? In physics, you may have heard of the dot (scalar) product between two vectors, or the so called "cross product" between two vectors, but it turns out we want to define multiplication differently, so that we maintain consistency with our familiar mathematical operations we are already used to. To define multiplication, it is convenient to represent complex numbers in polar form , i.e. instead of representing a number of z by the ordered pair (x,y), we can represent z by the ordered pair (r,A), where r is called the magnitude of z, and A is called the angle (or argument) of z. The magnitude r of z is the distance from z to the origin in the complex plane. It is a generalization of the concept of the "absolute value" of a number that you are probably already familiar with, thus r is sometimes written as r = |z|. (For example, -2 is a complex number, and its distance to the origin is 2, so we write |-2| = 2). The angle of z, sometimes written A = arg(z), is the angle (in radians or degrees) that z makes with the positive real axis. For example, for the number z illustrated above A is approximately 45degree. Clearly, r and A completely specify the number z. The product between two numbers z_1 = (r_1 , A_1 ) and z_2 = (r_2 , A_2 ) is defined as z_1 X z_2 = (r_1 X r_2 , A_1 + A_2 ). That is to multiply two numbers, we multiply their magnitudes and add their angles. It is instructive to consider some examples.