TERMINOLOGY AND NOTATION

4. TERMINOLOGY AND NOTATION

Both i and j are used to represent −1, j usually in any problem dealing with electricity since i is needed there for current. A physicist should be able to work with ease using either symbol. We shall for consistency use i throughout this book.

We often label a point with a single letter (for example, P in Figure 3.2 and A in Figure 3.3) even though it requires two coordinates to locate the point. If you have studied vectors, you will recall that a vector is represented by a single letter, say v, although it has (in two dimensions) two components. It is customary to use

a single letter for a complex number even though we realize that it is actually a pair of real numbers. Thus we write

(4.1) z = x + iy = r(cos θ + i sin θ) = re iθ .

Here z is a complex number; x is the real part of the complex number z, and y is the imaginary part of z. The quantity r is called the modulus or absolute value of z, and θ is called the angle of z (or the phase, or the argument, or the amplitude of z). In symbols:

2 +y 2 , (4.2) Im z = y (not iy),

Re z = x,

|z| = mod z = r =

angle of z = θ.

50 Complex Numbers Chapter 2

The values of θ should be found from a diagram rather than a formula, although we do sometimes write θ = arc tan(y/x). An example shows this clearly.

√ Example. Write z = −1−i in polar form. Here we have x = −1, y = −1, r =

2 (Figure 4.1). There are an infinite number of values of θ,

where n is any integer, positive or negative. The value θ = 5π/4 is sometimes called the principal angle of the complex number z = −1 − i. Notice carefully, however, that this is not the same as the principal value π/4 of arc tan 1 as defined in

Figure 4.1 calculus. The angle of a complex number must be in the same quadrant as the point representing the number. For our present work, any one of the values in (4.3) will do; here we would probably use either 5π/4 or −3π/4. Then we have in our example

= 2e 5iπ/4 .

[We could also write z = ◦ 2 (cos 225 + i sin 225 ).] The complex number x − iy, obtained by changing the sign of i in z = x + iy,

is called the complex conjugate or simply the conjugate of z. We usually write the conjugate of z = x + iy as ¯ ∗ z = x − iy. Sometimes we use z instead of ¯ z (in fields

such as statistics or quantum mechanics where the bar may be used to mean an average value). Notice carefully that the conjugate of 7i − 5 is −7i − 5; that is, it is the i term whose sign is changed.

Complex numbers come in conjugate pairs; for example, the conjugate of 2 + 3i is 2 − 3i and the conjugate of 2−3i is 2+3i. Such a pair of points in the complex plane are mirror images of each other with the x axis as the mirror (Figure 4.2). Then in polar form, z and ¯ z have the same r value, but their θ values are negatives of each other. If we write z = r(cos θ + i sin θ), then

Figure 4.2

(4.4) − ¯ iθ z = r[cos(−θ) + i sin(−θ)] = r(cos θ − i sin θ) = re .

PROBLEMS, SECTION 4

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x, y, r, θ in your head for these simple problems. Then

Section 5 Complex Algebra

plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

16. 5(cos 0 + i sin 0)

17. 2e −iπ/4

18. 3e iπ/2 19. 5(cos 20 ◦ + i sin 20 ◦ ) 20. 7(cos 110 ◦

− i sin 110 0 )