Complex Numbers

The complex numbers, denoted , have the wonderful property that every polynomial with complex coefficients has a (complex) root. This fact is known as the Fundamental Theorem of Algebra.

One pleasant aspect of the complex numbers is that, whereas describing the real numbers in terms of the rationals is a rather complicated business, the complex numbers are quite easy to describe in terms of real numbers. Every complex number has the form

where and are real numbers, and is a root of the polynomial . Here and are called the real part and the imaginary part of the complex number, respectively. The real numbers are now regarded as special complex numbers of the form , with zero imaginary part. The complex numbers of the form with zero real part are called pure imaginary numbers. The complex number itself is called the imaginary unit and is distinguished by the fact that As the terms complex and imaginary suggest, these numbers met with some resistance when they were first used. This has changed; now they are essential in science and engineering as well as mathematics, and they are used extensively. The names persist, however, and continue to be a bit misleading: These numbers are no more “complex” than the real numbers, and the number is no more “imaginary” than .

Much as for polynomials, two complex numbers are declared to be equal if and only if they have the same real parts and the same imaginary parts. In symbols,

The addition and subtraction of complex numbers is accomplished by adding and subtracting real and imaginary parts: This is analogous to these operations for linear polynomials and , and the multiplication of complex numbers is also analogous with one difference: . The definition is With these definitions of equality, addition, and multiplication, the complex numbers satisfy all the basic arithmetical axioms adhered to by the real numbers (the verifications are omitted). One consequence of this is that they can be manipulated in the obvious fashion, except that is replaced by wherever it occurs, and the rule for equality must be observed.

As for real numbers, it is possible to divide by every nonzero complex number . That is, there exists a complex number such that . As in the real case, this number is called the inverse of and is denoted by or . Moreover, if , the fact that means that or . Hence , and an explicit formula for the inverse is

In actual calculations, the work is facilitated by two useful notions: the conjugate and the absolute value of a complex number. The next example illustrates the technique.

The key to this technique is that the product in the denominator turned out to be a real number. The situation in general leads to the following notation: If is a complex number, the conjugate of is the complex number, denoted , given by

Hence is obtained from by negating the imaginary part. Thus and . If we multiply by , we obtain

The real number is always nonnegative, so we can state the following definition: The absolute value or modulus of a complex number , denoted by , is the positive square root ; that is,

For example, and .

Note that if a real number is viewed as the complex number , its absolute value (as a complex number) is , which agrees with its absolute value as a real number.

With these notions in hand, we can describe the technique applied in Example ex:033897 as follows: When converting a quotient of complex numbers to the form , multiply top and bottom by the conjugate of the denominator.

The following list contains the most important properties of conjugates and absolute values. Throughout, and denote complex numbers.

All these properties (except property C12) can (and should) be verified by the reader for arbitrary complex numbers and . They are not independent; for example, property C10 follows from properties C2 and C6.

The triangle inequality, as its name suggests, comes from a geometric representation of the complex numbers analogous to identification of the real numbers with the points of a line. The representation is achieved as follows:

Introduce a rectangular coordinate system in the plane, and identify the complex number with the point , as shown in the figure below.

When this is done, the plane is called the complex plane. Note that the point on the axis now represents the real number , and for this reason, the axis is called the real axis. Similarly, the axis is called the imaginary axis. The identification of the geometric point and the complex number will be used in what follows without comment. For example, the origin will be referred to as .

This representation of the complex numbers in the complex plane gives a useful way of describing the absolute value and conjugate of a complex number . The absolute value is just the distance from to the origin. This makes properties C8 and C9 quite obvious. The conjugate of is just the reflection of in the real axis ( axis), a fact that makes properties C4 and C5 clear.

Given two complex numbers and , the absolute value of their difference

is just the distance between them. This gives the complex distance formula:

This useful fact yields a simple verification of the triangle inequality, property C12. Suppose and are given complex numbers. Consider the triangle in the figure below whose vertices are , , and .

The three sides have lengths , , and by the complex distance formula, so the inequality

expresses the obvious geometric fact that the sum of the lengths of two sides of a triangle is at least as great as the length of the third side.

The representation of complex numbers as points in the complex plane has another very useful property: It enables us to give a geometric description of the sum and product of two complex numbers. To obtain the description for the sum, let

denote two complex numbers. We claim that the four points , , , and form the vertices of a parallelogram. In fact, in the figure below, the lines from to and from to have slopes

respectively, so these lines are parallel. (If it happens that , then both these lines are vertical.)

Similarly, the lines from to and from to are also parallel, so the figure with vertices , , , and is indeed a parallelogram. Hence, the complex number can be obtained geometrically from and by completing the parallelogram. This is sometimes called the parallelogram law of complex addition. Readers who have studied mechanics will recall that velocities and accelerations add in the same way; in fact, these are all special cases of vector addition.

In this exploration we return to the complex numbers of Example ex:033865, and we analyze them from a geometric point of view. Let and .
(a)
Plot , , and in the on the same plane to see an example of the parallelogram law of complex addition.
(b)
What happens geometrically when you multiply a complex number by ? What if you multiply by a second time? Verify that this is the same as multiplying by .

Polar Form

The geometric description of what happens when two complex numbers are multiplied is at least as elegant as the parallelogram law of addition, but it requires that the complex numbers be represented in polar form. Before discussing this, we pause to recall the general definition of the trigonometric functions sine and cosine. An angle in the complex plane is in standard position if it is measured counterclockwise from the positive real axis as indicated in the figure below.

Rather than using degrees to measure angles, it is more natural to use radian measure. This is defined as follows: The circle with its centre at the origin and radius (called the unit circle) is drawn as in the above figure. It has circumference , and the radian measure of is the length of the arc on the unit circle counterclockwise from to the point on the unit circle determined by . Hence , , , and a full circle has the angle . Angles measured clockwise from are negative; for example, corresponds to (or to ).

Consider an angle in the range . If is plotted in standard position as in the above figure, it determines a unique point on the unit circle, and has coordinates (, ) by elementary trigonometry. However, any angle (acute or not) determines a unique point on the unit circle, so we define the cosine and sine of (written and ) to be the and coordinates of this point. For example, the points

plotted in the figure are determined by the angles , , , , respectively. Hence

Now we can describe the polar form of a complex number. Let be a complex number, and write the absolute value of as

If , the angle shown in the figure below is called an argument of and is denoted

This angle is not unique ( would do as well for any
). However, there is only one argument in the range , and this is sometimes called the principal argument of .

Referring to the figure below, we find that the real and imaginary parts and of are related to and by

Hence the complex number has the form

The combination is so important that a special notation is used: is called Euler’s formula after the great Swiss mathematician Leonhard Euler (1707–1783). With this notation, is written This is a polar form of the complex number . Of course it is not unique, because the argument can be changed by adding a multiple of .

In Euler’s formula , the number is the familiar constant from calculus. The reason for using will not be given here; the reason why is written as an exponential function of is that the law of exponents holds:

where and are any two angles. In fact, this is an immediate consequence of the addition identities for and : This is analogous to the rule , which holds for real numbers and , so it is not unnatural to use the exponential notation for the expression . In fact, a whole theory exists wherein functions such as , , and are studied, where is a complex variable. Many deep and beautiful theorems can be proved in this theory, one of which is the so-called fundamental theorem of algebra mentioned later (Theorem th:034196). We shall not pursue this here.

The geometric description of the multiplication of two complex numbers follows from the law of exponents.

In other words, to multiply two complex numbers, simply multiply the absolute values and add the arguments. This simplifies calculations considerably, particularly when we observe that it is valid for any arguments and .

Roots of Unity

If a complex number is given in polar form, the powers assume a particularly simple form. In fact, , , and so on. Continuing in this way, it follows by induction that the following theorem holds for any positive integer . The name honors Abraham De Moivre (1667–1754).

Proof
The case has been discussed, and the reader can verify the result for . To derive it for , first observe that In fact, by the multiplication rule. Now assume that is negative and write it as , . Then If , this is De Moivre’s theorem for negative .

De Moivre’s theorem can be used to find th roots of complex numbers where is positive. The next example illustrates this technique.

The same type of calculation gives all complex th roots of unity; that is, all complex numbers such that . As before, write and

in polar form. Then takes the form using De Moivre’s theorem. Comparing absolute values and arguments yields Hence , and the values of all lie in the range . As in Example ex:034107, every choice of yields a value of that differs from one of these by a multiple of , so these give the arguments of all the possible roots.

The th roots of unity can be found geometrically as the points on the unit circle that cut the circle into equal sectors, starting at . The case is shown in the figure below, where the five fifth roots of unity are plotted.

The method just used to find the th roots of unity works equally well to find the th roots of any complex number in polar form. We give one example.

An expression of the form , where the coefficients , , and are real numbers, is called a real quadratic. A complex number is called a root of the quadratic if . The roots are given by the famous quadratic formula:

The quantity is called the discriminant of the quadratic , and there is no real root if and only if . In this case the quadratic is said to be irreducible. Moreover, the fact that means that , so the two (complex) roots are conjugates of each other: The converse of this is true too: Given any nonreal complex number , then and are the roots of some real irreducible quadratic. Indeed, the quadratic has real coefficients ( and is twice the real part of ) and so is irreducible because its roots and are not real.

Fundamental Theorem of Algebra

As we mentioned earlier, the complex numbers are the culmination of a long search by mathematicians to find a set of numbers large enough to contain a root of every polynomial. The fact that the complex numbers have this property was first proved by Gauss in 1797 when he was 20 years old. The proof is omitted.

If is a polynomial with complex coefficients, and if is a root, then the Factor Theorem (see Precalculus by Stitz-Zeager, for instance) asserts that

where is a polynomial with complex coefficients and with degree one less than the degree of . Suppose that is a root of , again by the fundamental theorem. Then , so This process continues until the last polynomial to appear is linear. Thus has been expressed as a product of linear factors. The last of these factors can be written in the form , where and are complex (verify this), so the fundamental theorem takes the following form.

This form of the fundamental theorem, when applied to a polynomial with real coefficients, can be used to deduce the following result.

In fact, suppose has the form

where the coefficients are real. If is a complex root of , then we claim first that is also a root. In fact, we have , so where for each because the coefficients are real. Thus if is a root of , so is its conjugate . Of course some of the roots of may be real (and so equal their conjugates), but the nonreal roots come in pairs, and . By Theorem thm:034221, we can thus write as a product: where is the coefficient of in ; are the real roots; and are the nonreal roots. But the product is a real irreducible quadratic for each (see the discussion preceding Example ex:034182). Hence (eq:complexproduct) shows that is a product of linear and irreducible quadratic factors, each with real coefficients. This is the conclusion in Theorem th:034221.

Practice Problems

Solve each of the following for the real number .
(a)
(b)
(c)
(d)
Convert each of the following to the form .
(a)
(challenge problem)
(b)
(challenge problem)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(challenge problem)
In each case, find the complex number .
(a)
(b)
(challenge problem)

(c)
(d)
(e)
(f)
(challenge problem)

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In each case, find the roots of the real quadratic equation.
(a)
(b)

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(c)
(d)

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Find all numbers in each case.
(a)
(b)

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(c)
(d)

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In each case, find a real quadratic with as a root, and find the other root.
(a)
(b)

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(c)
(d)

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Find the roots of , any angle.
Find a real polynomial of degree with and as roots.

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Let and denote, respectively, the real and imaginary parts of . Show that:
(a)
(b)
(c)
(d)
(e)
(challenge problem) , and if is real
(f)
(challenge problem) , and if is real
In each case, show that is a root of the quadratic equation, and find the other root.
(a)
(challenge problem) ;
(b)
(challenge problem) ;
(c)
(challenge problem) ;
(d)
(challenge problem) ;
Find the roots of each of the following complex quadratic equations.
(a)
(b)

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(c)
(challenge problem)
(d)
(challenge problem)

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In each case, describe the graph of the equation (where denotes a complex number).
(a)
(b)

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Circle, centre at , radius

(c)
(d)

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Imaginary axis

(e)
(f)
, a real number

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Line

a.
Verify directly for and .
b.
Deduce (a) from properties C2 and C6.
Prove that for all complex numbers and .
If is real and , show that for some real number .
If and , show that for some in with .
Show that is real for all , using property C5.
Express each of the following in polar form (use the principal argument).
(a)
(b)

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(c)
(d)

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(e)
(f)

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Express each of the following in the form .
(a)
(b)

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(c)
(d)

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(e)
(f)

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Express each of the following in the form .
(a)
(b)

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(c)
(d)

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(e)
(f)

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Use De Moivre’s theorem to show that:
a.
;
b.
;
a.
Find the fourth roots of unity.
b.
Find the sixth roots of unity.
Find all complex numbers such that:
(a)
(b)

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(c)
(d)

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If in polar form, show that:
(a)
(b)
if
Show that the sum of the th roots of unity is zero.
for any complex number .
(a)
Let , , , , and be equally spaced around the unit circle. Show that .
for any complex number .
(b)
Repeat (a) for any points equally spaced around the unit circle.
The argument in (a) applies using . Then .
(c)
If , show that the sum of the roots of is zero.
If is real, , show that is real.
If , show that is real or pure imaginary.
If and are rational numbers, let and denote numbers of the form . If , define and . Show that each of the following holds.
(a)
(challenge problem) only if and
(b)
(c)
(d)
(e)
(f)
(challenge problem) If is a polynomial with rational coefficients and is a root of , then is also a root of .

Text Source

This section was adapted from Appendix A of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 581–594.