Complex Numbers. April 10, 2015

Transcription

1 Complex Numbers April 10, 2015 In preparation for the topic of systems of differential equations, we need to first discuss a particularly unusual topic in mathematics: complex numbers. The starting point of our discussion is the quadratic equation x = 0. We know that any number, whether positive or negative, when squared, will be positive. It is then pretty clear that there is no way that one could square any number x, add 1 to the result, and somehow have the sum be equal to zero. Thus it seems that the equation has no solution. As we will see later, it is convenient to create a number that will satisfy this equation. We give this number the (unfortunate) name imaginary number, and we denote it with the letter i. Since it solves the equation above, it is very convenient to think of it as i = 1, so that i = = 0. Another thing that is very useful for mathematics (and for science in general) is to consider combinations of real numbers and the imaginary number. This is described in the following definition. Definition 1. A complex number z is a number of the form z = a + bi, where a and b are real numbers, and i = 1. We call a the real part of z, and b is the imaginary part of z. Note that the imaginary part of a complex number z is a real number. You can think of it as the quantity of i s that you have. Also, we often reverse the order of i and its imaginary part. Thus a + bi = a + ib. 1

2 Example 1. The following are all examples of complex numbers i 1 + 2i 1 25i 3 + πi Just as with real numbers, we would like to be able to combine complex numbers through addition, subtraction, multiplication, and division. However, we have to explain what we mean by these operations, as we don t want funny things to happen. So we state explicitly what we mean by these operations in the context of complex numbers. Definition 2. If z = a + bi and w = c + di are complex numbers, we define: Addition: z + w = (a + c) + (b + d)i Subtraction: z w = (a c) + (b d)i Multiplication: z w = (a + bi)(c + di) = (ac bd) + (ad + bc)i Before we can define division of complex numbers, we must first introduce a concept that is unique to complex numbers. Definition 3. Let z = a + bi be a complex number. Then we define its complex conjugate, denoted z, as z = a bi. From this definition, it is easy to see that, for any complex number z, the product z z is a real number, given by z z = (a + bi)(a bi) = a 2 + b 2. With this definition, we can define division of complex numbers. Definition 4. If z = a + bi and w = c + di are complex numbers, then we define the quotient z/w as Division: z w = z w w w 2

3 Note that in the quotient above, the number at the bottom of the fraction is the real number w w. Thus, we define division of complex numbers in a way that doesn t actually involve dividing by any complex numbers. Mathematicians are devious that way. With these definitions in mind, it is worth noting that they were not arbitrarily chosen to be that way. They were chosen because we can think of any real number a as a complex number with imaginary part 0: a = a + 0i. Thus, all the operations above, when applied to real numbers, have to give you the same result as they would for real numbers. Fortunately for us, the following theorem tells us that this is indeed the case. Theorem 1. If z, w, and v are complex numbers, then the following properties hold: z + w = w + z z w = w z z + 0 = z 1 z = z z + (w + v) = (z + w) + v z (w v) = (z w) v z (w + v) = z w + z v 0 z = 0. The Complex Plane Just as we can represent numbers on a number line to represent them visually, we can do something similar for complex numbers. Given a complex number z = x + iy, we can associate to it a vector in R 2, namely the vector (x, y). Thus, we can think of the x-axis as representing the real part of z, and the y-axis as representing the imaginary part, as in the following diagram: 3

4 This diagram suggests another way of representing complex numbers. Suppose we form a right triangle where the hypotenuse is the vector (complex number) z, with the remaining sides being the components in the x and y axes, as shown below: We see that we form a triangle, with the angle formed by the hypotenuse and the x-axis denoted θ. In particular, the length of the hypotenuse, which we denote by r, is given by r = x 2 + y 2 = z z. In this context, we see that r is really just the length of the vector corresponding to z. Because of this, we use the notation z = zz. In addition, the notation and diagram above suggest another way of representing complex numbers. Given z = x + iy, we can also represent z by z = r cos θ + ir sin θ = r(cos θ + i sin θ), 4

5 where r = z and θ = tan 1 ( y ). This form of complex numbers is known as x polar form. To complete our discussion of complex numbers, we state the following important fact: Theorem 2. For any θ, the following equation holds: e iθ = cos θ + i sin θ. Thus, we may write a complex number in polar form as z = re iθ, with r and θ as before. 5