4.3
Complex Numbers

4.3.1 An Introductory Tale

The study of numbers generally begins with children and counting numbers 1, 2, 3, … then progressing to negative numbers, and then to fractions, and finally to the real numbers. To most students of mathematics, the complex numbers come last, if at all. Throughout history, every enlargement of the meaning of number had practical motivation. Negative numbers were required to solve x + 3 = 1, rational numbers were required to solve 3x = 5, and real numbers were a response to the equation x2 = 2. Finally, complex numbers came about when people wanted to solve equations like x2 + 1 = 0.

Square roots of negative numbers first appeared in Ars Magna (1545) by the Italian mathematician Gerolamo Cardano (1501–1576) in his solution of the simultaneous equations

equation

getting the solution

equation

Cardano did not give any interpretation of the square root of negative numbers, although he did say that they obeyed the usual rules of algebra and that solutions containing them could be verified. Cardano and other mathematicians at the time would go to great lengths to avoid both negative and complex numbers, referring to negative solutions as “false1” solutions and complex solutions as “useless.” Even the great mathematician Carl Gauss said, as late as 1825, “the true metaphysics of images is illusive.” He overcame his doubts, however, by 1831 when he applied complex numbers to number theory. His acceptance of complex numbers provided a great boost to the acceptance of complex numbers in the mathematical community.

Today, complex numbers are crucial in the study of many areas of mathematics, including harmonic analysis, ordinary and partial differential equations, analytic number theory, analytic function theory, as well as being applied in many areas of engineering and science, including theoretical physics, where analytic function theory constitutes much of the foundations of quantum mechanics.

Someone once argued that real numbers are more “natural” than complex numbers since real numbers measure things we can all see and feel, like a person's height or weight, whereas no one can physically “experience” complex numbers. The person who makes such a claim just does not know where to look. Every engineer, physicist, and student of differential equations knows it is complex numbers that allows for the description of oscillatory motion. The next time you hear an orchestra tuning their instruments to the standard A above Middle C, you are hearing the complex number 440 i, the complex number that describes 440 oscillations per second.

4.3.2 Complex Numbers

There are a number of ways to introduce complex numbers, each of which has its merits and demerits. We choose to define complex numbers formally as pairs of real numbers and then introduce a “more friendly” notation.

The question we ask is do these arithmetic operations result in a legitimate arithmetic system having properties similar to those of the real numbers? We have reason to believe this to be true since we can think of the complex numbers ℂ as an extension of the real numbers ℝ in the sense that complex numbers of the form (a, 0) behave exactly like the real numbers:

equation

That is, we can think of the real numbers as those complex numbers where the second coordinate is zero.

4.3.3 Complex Numbers as an Algebraic Field

The complex numbers along with the above binary operations of addition and multiplication forms an algebraic field similar to the real numbers, with (0, 0) being an additive identity and (1, 0) the multiplicative identity. We will not list all the field properties for the complex numbers, but a few of them you might recall are

  1. (a, b), (c, d) ∈ ℂ ⇒ (a, b) + (c, d) ∈ ℂ
  2. (a, b), (c, d) ∈ ℂ ⇒ (a, b) · (c, d) ∈ ℂ
  3. (a, b), (c, d) ∈ ℂ ⇒ (a, b) + (c, d) = (c, d) + (a, b)
  4. (a, b) ∈ ℂ ⇒ (a, b) + (0, 0) = (a, b)
  5. (a, b) ∈ ℂ ⇒ (a, b) + (−a, − b) = (0, 0)
  6. (a, b), (c, d) ∈ ℂ ⇒ (a, b) · (c, d) = (c, d) · (a, b)
  7. images
  8. (0, 1) · (0, 1) = (−1, 0)

The last identity (h) introduces the special complex number (0, 1), which we denote by the letter i, and since we identify the complex number (−1, 0) with the real number −1, we can interpret this statement as i · i = −1 or i2 =  − 1. For this reason, one often calls the complex number i as the square root of −1 and often written images. If we give the complex number (0, 1) the special name i, then we can write the general complex number (a, b) as

equation

which normally is expressed as a + bi. Using this notation to express complex numbers, we can express addition and multiplication of complex numbers as

equation

The real number a is called the real part of the complex number a + bi, and b is called the imaginary part, often denoted by

equation

4.3.4 Imaginary Numbers and Two Dimensions

A good way of thinking about complex numbers is to think of them as a two‐dimensional vector of real numbers, and when plotting these vectors we call the x‐axis the real axis and the y‐axis the imaginary axis.

We can give thanks to nonmathematicians Casper Wessel and Jean Robert Argand for their insight in representing complex numbers as points in the plane (see Figure 4.10).

Complex plane representation with solid circle markers labeled –1 + 4i, 3i, 2+3i, 5+I, –3, –1, 3, –3–2i, 2–2i, 4–2i, and –1–4i.

Figure 4.10 Complex plane representation of complex numbers.

The absolute value (or modulus) of a complex number z = x + iy is defined as the nonnegative real number

equation

which is the length of the line segment from the origin to z in the complex plane (see Figure 4.11). The conjugate of a complex number z = x + iy is defined to be images, which geometrically is the reflection of z through the real axis. The absolute value of a complex number can be written in terms of its conjugate by images.

Plane of magnitude and complex conjugate of a complex number displaying an increasing line to a solid circle marker for z =x+iy (top) and decreasing line to a solid circle marker for z =x – iy (bottom).

Figure 4.11 Magnitude and complex conjugate of a complex number.

The complex numbers do not form an ordered field like the real numbers, but do form what is called an algebraically closed field,3 which means that all polynomial equations with complex coefficients have complex roots (real roots being special cases of complex roots). This property contrasts with the real numbers where polynomial equations like z2 + 1 = 0 with real coefficients, but roots outside the real number system, i.e. x =  ± i. Algebraists would say that the complex numbers are an algebraically closed field, whereas the real numbers are not algebraically closed.

4.3.5 Polar Coordinates

Recall that a point (x, y) in the Cartesian plane can be written in terms of polar coordinates (r, θ), where the relationship between them is given by x = r cos θ, y = r sin θ. Hence, any complex number can be written in terms of polar coordinates as

equation

where images is the absolute value of z, and θ is the argument of z, written θ = arg(z), which measures the angle between the positive real axis and the line segment from 0 to z (see Figure 4.12).

Plane in a polar form of a complex number displaying dotdash lines labeled r and y with a vertex labeled x+iy.

Figure 4.12 Polar form of a complex number.

Since the argument θ can wrap around the origin several times, either clockwise or counterclockwise, the principle argument of a complex number is the unique argument that lies in the interval (−π, π]. Thus, the complex number i has argument π/2, and −1 has argument π, and −i has argument −π/2 (although sometimes we say 3π/2).

4.3.6 Complex Exponential and Euler's Theorem

Any student of calculus knows that the exponential function ex bears little relationship to the trigonometric functions sin x and cos x. The exponential function ex grows without bound as x gets large, while the trigonometric functions oscillate between plus and minus 1. In one of the most important discoveries in mathematics, Swiss mathematician Leonhard Euler showed in 1748 that although real exponential functions may be unrelated to trigonometric functions, complex exponentials and trigonometric functions have an intimate relationship. Euler does this by replacing the θ in the Taylor series expansion of eθ with the complex number , thus defining a new function e, called the complex exponential. Euler shows that this exponential had an interesting relationship with the trigonometric functions.

Euler's theorem allows one to work with exponentials and all their wonderful manipulative properties like

  • images
  • images

Euler's equation also allows us to write the important identity

equation

4.3.7 Complex Variables in Polar Form

Euler's equation allows us to write complex numbers either in Cartesian or polar form by equation:

equation

Note that

equation

We can visualize the complex exponential e as a point in the complex plane5 with argument θ lying on the unit circle. As θ moves from 0 to 2π, the exponential e moves counterclockwise around the unit circle (see Figure 4.13).6 For example7

equation
Plane of complex exponential eiθ displaying solid circle markers for eπi/2=I (top), e2πi=1 (right), e3πi/2=-I (bottom), and eπi=-1 (left).

Figure 4.13 The complex exponential e.

4.3.8 Basic Arithmetic of Complex Numbers

Complex numbers are an algebraic field just like the real numbers. They can be added, subtracted, multiplied, and divided and have interesting geometric interpretations in the complex plane.

4.3.9 Roots and Powers of a Complex Number

One of the benefits of representing complex numbers in polar form is visualizing the roots of the equation xn = a. Most beginning students of mathematics know there are two square roots of a positive number like four, knowing them to be ±2. However, if you answered two as the cube root of eight, you would be one‐third correct, since there are two other two cube roots. And what about the cube roots of a negative number like minus one? Certainly, minus one is one cube root, but there are two more.

This leads to the following problem. Given a complex number a ≠ 0 and an integer n greater or equal to two, find the roots of zn = a? To solve this equation, we write z and a in polar form, getting

equation

where |a| = rn. Taking the absolute value of each side of this equation yields rn = |a| from which we get r = |a|1/n. Plugging this back into the equation, the equation reduces to einθ = e from which we find = ϕ or θ = ϕ/n. Hence, putting all this together, the nth root of zn = a is

equation

However, this is not the only root since

equation

also satisfies zn = a, which can be seen by direct computation

equation

These results can be summarized as follows.

Figure 4.19 shows the six roots of the equation z6 = 1 called the six roots of unity.

Plane of six roots of unity displaying a circle with solid circle markers with corresponding equations.

Figure 4.19 Six roots of unity.

Plane displaying a circle with solid circle markers with corresponding equations connected by lines forming angles measuring 120° to point of origin.

Figure 4.20 Three cube roots of i.

Problems

  1. Complex Numbers

    Let

    1. w = i, z = 1 + i
    2. w = 2, z =  − i
    3. w = i, z =  − i
    4. w = − 1 − i, z = 1 + i
    5. w = 1 + i, z = − 1 + i

    Find the following values.

    • |z|
    • θ = arg(z)
    • w + z
    • wz
    • images
  2. Convert to Polar Form

    Convert the following complex numbers to polar form.

    1. 2i Ans: re = 2ei(π/2)2
    2. −1 + i
    3. i Ans: re = ei(3π/2)
    4. 2 − 2i
  3. Convert to Cartesian Form

    Convert the following complex numbers to Cartesian form x + iy.

    1. e3πi Ans: e3πi = cos 3π + i sin 3π = − 1
    2. 2eiπ/4
    3. e2πi
    4. e5πi
    5. 5e3πi/4
  4. Evaluate (1 + i)100.
  5. Useful Identity I

    Show that the complex conjugate of the sum of two complex numbers is the sum of the conjugates; that is images.

  6. Useful Identity II

    Verify the identity images for z = 2 + 3i.

  7. Taking a Complex Function Apart

    Find the real and imaginary parts of the following.

    1. z3 Ans: Re(z3) = x3 − 3xy2, Im(z3) = 3x2y − y3
    2. 1/z
  8. Compute the following.
    1. images
    2. images
    3. images
    4. images
    5. images
  9. de Moivre's Formula

    Use Euler's theorem to prove de Moivre's formula

    equation

    for any positive integer n. Hint: Use induction.

  10. Primitive Roots of Unity

    The n roots the equation zn = 1 are called the n roots of unity. Find and plot the roots when n = 1, 2, 3, 4, and 8.

  11. Fractional Powers

    Find the following.

    1. i3/2
    2. (−1)3/4
    3. images
    4. ii
  12. Hmmmmmmmmm

    Show images

  13. Internet Research

    There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward applications of complex numbers, history of complex numbers, and roots of unity.

Notes

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