1 A Review of Complex Numbers Ilya Pollak ECE 301 Signals and Systems Section 2, Fall 2010 Purdue University A complex number is represented in the form z = x + jy, where x and y are real numbers satisfying the usual rules of addition and multiplication, and the symbol j, called the imaginary unit, has the property j 2 = −1. The numbers x and y are called the real and imaginary part of z, respectively, and are denoted by x = ℜ(z), y = ℑ(z). We say that z is real if y = 0, while it is purely imaginary if x = 0. Example 0.1. The complex number z = 3 + 2j has real part 3 and imaginary part 2, while the real number 5 can be viewed as the complex number z = 5 + 0j whose real part is 5 and imaginary part is 0. Geometrically, complex numbers can be represented as vectors in the plane (Fig. 1). We will call the xy-plane, when viewed in this manner, the complex plane, with the x-axis designated as the real axis, and the y-axis as the imaginary axis. We designate the complex number zero as the origin. Thus, x + jy = 0 means x = y = 0. In addition, since two points in the plane are the same if and only if both their x- and y-coordinates agree, we can define equality of two complex numbers as follows: x1 + jy1 = x2 + jy2 means x1 = x2 and y1 = y2 . ℑ y " " " R " " " θ " " x z ℜ Figure 1. A complex number z can be represented in Cartesian coordinates (x, y) or polar coordinates (R, θ). 2 Thus, we see that a single statement about the equality of two complex quantities actually contains two real equations. Definition 0.1 (Complex Arithmetic). Let z1 = x1 + jy1 and z2 = x2 + jy2 . Then we define: (a) z1 ± z2 = (x1 ± x2 ) + j(y1 ± y2 ); (b) z1 z2 = (x1 x2 − y1 y2 ) + j(x1 y2 + x2 y1 ); (c) for z2 6= 0, w = z1 z2 is the complex number for which z1 = z2 w. Note that, instead of the Cartesian coordinates x and y, we could use polar coordinates to represent points in the plane. The polar coordinates are radial distance R and angle θ, as illustrated in Fig. 1. The relationship between the two sets of coordinates is: x = R cos θ, y = R sin θ, p x2 + y 2 = |z|, R = y θ = arctan . x Note that R is called the modulus, or the absolute value of z, and it alternatively denoted |z|. Thus, the polar representation is: z = |z| cos θ + j|z| sin θ = |z|(cos θ + j sin θ). Definition 0.2 (Complex Exponential Function). The complex exponential function, denoted by ez , or exp(z), is defined by ez = ex+jy = ex (cos y + j sin y). In particular, if x = 0, we have Euler’s equation: ejy = cos y + j sin y. Comparing this with the terms in the polar representation of a complex variable, we see that any complex variable can be written as: z = |z|ejθ . 3 z = z1 z2 |z| = |z1 | · |z2 | ℑ θ = θ1 + θ2 z y " 1 " θ2 " "" " θ1 " " ℜ x Figure 2. Multiplication of two complex numbers z1 = |z1 |ejθ1 and z2 = |z2 |ejθ2 (z2 is not shown). The result is z = |z|ejθ with |z| = |z1 | · |z2 | and θ = θ1 + θ2 . Properties of Complex Exponentials. 1 jθ (e + e−jθ ), 2 1 jθ sin θ = (e − e−jθ ), 2j |ejθ | = 1, cos θ = ez1 ez2 e−z ez+2πjn = ez1 +z2 , 1 = z, e = ex (cos(y + 2πn) + j sin(y + 2πn)) = ex (cos y + j sin y) = ez , for any integer n. DT complex exponential functions whose frequencies differ by 2π are thus identical: ej(ω+2π)n = ejωn+2πjn = ejωn . We have seen examples of this phenomenon before, when we discussed DT sinusoids. It follows from the multiplication rule that z1 z2 = |z1 |ejθ1 |z2 |ejθ2 = |z1 ||z2 |ej(θ1 +θ2 ) . Therefore, in order to multiply two complex numbers, • add the angles; • multiply the absolute values. Multiplication of two complex numbers is illustrated in Fig. 2. Definition 0.3 (Complex Conjugate). If z = x + jy, then the complex conjugate of z is z ∗ = x − jy (sometimes also denoted z̄). 4 ℑ 2 C 2 C z = 2j ℑ z = x + jy y " " " " " " " " b ℜ x b b b b b b b z ∗ = x − jy −y 1 z = 1 + j, |z| = √ 2 θ = π/4 ℜ 1 @ @ 1/z = 1/2 − j/2 @ @ @ @ z∗ = 1 − j −1 (a) (b) Figure 3. (a) Complex number z and its complex conjugate z ∗ . (b) Illustrations to Example 0.2. This definition is illustrated in Fig. 3(a). Note that, if z = |z|ejθ , then z ∗ = |z|e−jθ . Here are some other useful identities involving complex conjugates: ℜ(z) = ℑ(z) = |z| = 1 (z + z ∗ ), 2 1 (z − z ∗ ), 2j √ zz ∗ , = z, ∗ ∗ (z ) ∗ z = z ⇔ z is real, (z1 + z2 )∗ = z1∗ + z2∗ , (z1 z2 )∗ = z1∗ z2∗ . Example 0.2. Let us compute the various quantities defined above for z = 1 + j. 1. z ∗ = 1 − j. √ √ 2. |z| = √12 + 12p= 2. Alternatively, p √ |z| = zz ∗ = (1 + j)(1 − j) = 1 + j − j − j 2 = 2. 3. ℜ(z) = ℑ(z) = 1. 4. Polar representation: z = √ 2(cos π4 + j sin π4 ) = √ π 2ej 4 . 5 5. To compute z 2 , square the absolute value and double the angle: π z 2 = 2(cos π2 + j sin π2 ) = 2j = 2ej 2 . The same answer is obtained from the Cartesian representation: (1 + j)(1 + j) = 1 + 2j + j 2 = 1 + 2j − 1 = 2j. 6. To compute 1/z, multiply both the numerator and the denominator by z ∗ : z∗ 1−j 1−j 1 j 1 = ∗ = = = − . z zz (1 + j)(1 − j) 2 2 2 Alternatively, use the polar representation: 1 z π 1 1 = √ e−j 4 j π4 2 2e √ √ ! π 2 2 1 1 π = √ cos − + j sin − =√ −j 4 4 2 2 2 2 1 j − . = 2 2 2 We can check to make sure that (1/z) · z = 1: 21 − 2j (1 + j) = 12 − 2j + 2j − j2 = 1. = √ These computations are illustrated in Fig. 3(b).