Chapter 29: Complex Numbers

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Chapter Summary
Complex numbers are an extension of the ordinary "real" numbers. The extended numbers allow us to solve all polynomial equations. These solutions are useful in calculus where complex exponentials give all solutions to linear differential equations.

Complex numbers first arise in solving polynomial equations. For example, the quadratic equation

has real solutions


When b2-4ac<0, the square root in the quadratic formula above is not a real number. In particular, if a=1, b=0, and c=1, the solution

makes no sense in the "real" numbers.

Complex numbers begin by defining a new number with the property that

This gives the two solutions to x2=-1: and , because and .

The quadratic roots above in the case where b2-4ac<0 can be written

with and . Notice that the previous computation uses ordinary rules of algebra such as with .

These computations lead us to the basic question, "What are the properties of the complex numbers?" This question is just another way to ask, "How do we compute with complex numbers?"

29.1 Algebra of Complex Numbers

Section Summary
We define complex numbers and show that they have the "algebraic" properties of ordinary numbers. (They do not have the "order properties.")

We begin with a useful formal definition of the "expression" .

Definition: Complex Numbers Complex numbers are ordered pairs of real numbers (a,b) for any pair of real numbers a and b. The first component a is called the "real part" of (a,b) and the second component b is called the "imaginary part" of (a,b). Addition is defined componentwise,

and complex multiplication is defined by

Example CD-29.1 Specific Addition and Multiplication

We usually will not use ordered pair notation when computing with complex numbers because the expression is more useful once we understand its computational properties. The ordered pair definition is very clear and specific. It allows us to prove Theroem CD-29.2 and justify the procedure we first illustrate with some examples as follows.

Example CD-29.2 Complex Numbers as "Expressions in the Imaginary Unit"

It is helpful in doing algebra to use the "imaginary" symbol as the second place holder, so we also write complex numbers (a,b) as expressions of the form . Consider the previous examples as formal high school algebra computations just treating as a variable in an algebraic expression.

The formal sum computation first uses the associative law that says we can group addition parentheses any way we wish. Next, the commutative law that says we can interchange the order of summation. Then the distributive law .

Formal multiplication begins with the distributive law for ,

Next, the computation uses comutativity and associativity of multiplication, , etc. The evaluation is an easy substitute for the ordered pair definition of complex multiplication above. In general a complex product is computed with high school algebra as follows.

We equate the expressions associated with the pair (0,b) and write simply . If and , then z1=z2 if and only if x1=x2 and y1=y2.

Example CD-29.3 Real Numbers as Complex Numbers with Zero Imaginary Part

Real numbers with their usual operations are included in the complex numbers as the ordered pairs . In other words, the complex operations give the same results,

As a result of this extension, we write a complex number of the form simply as r and treat a complex number with zero imaginary part as a real number. In particular, and .

A complex number is zero only when both of its parts are zero, if , then z=0 if and only if both x=0 and y=0.

The properties that extend from real to complex algebra that we used in the formal computations with above are as follows.

Theorem CD-29.1 Algebraic Properties of Complex Numbers Complex numbers , and satisfy:

The commutative laws of addition and multiplication,

The associative laws of addition and multiplication,

The distributive law of multiplication over addition,

A formal proof of this theorem amounts to writing out both sides of the identities above as ordered pairs and using properties of the real components to show they agree. The importance of the result is that the procedure of computing with expressions treating like an algebraic variable and simplifying with makes the computations correct.

Theorem CD-29.2 Inverses The complex numbers form a "field:"

Each complex number has an additive inverse, -z, with z+(-z)=0.

Each nonzero complex number has a multiplicative inverse, , with .

Complex division has the properties that if and , then


For an additive inverse let then .

For a multiplicative inverse, multiply the numerator and denominator by the algebraic conjugate, ,

Exercise CD-29.1.2 asks you to compute the product

proving that this is the reciprocal.

Exercise CD-29.1.5 asks you to show the properties above for the complex reciprocal.

Example CD-29.4 A Specific Reciprocal

We use the complex conjugate to make division real. For example,

Exercise set CD-29.1

  1. Verify the following by computation:
    • 1.
    • 2.
    • 3.
    • 4.
    • 5. (HINT: See the next part of the exercise.)
    • 6.
    • 7.
    • 8.
    • 9.
    • 10. Check your work with the program CmplxNrs.

    • Compute the product , showing that is the reciprocal of . Where do you require that ?

    • Show that and both satisfy z2+z+1=0 and z3=1.

    • For any complex number , show that

    • Show that complex division has the properties that if and , then

29.2 Geometry of Complex Numbers

Section Summary
The order properties of real numbers mean they represent points on an idealized line. Complex numbers do not have the "order" properties of real numbers, but rather are two-dimensional vectors with a special "vector multiplication." This section extends the vector geometry of Chapter 15 to complex multiplication.

The real case of complex multiplication extends the scalar multiplication of Chapter 15 since


This means that complex numbers are two-dimensional position vectors that have the additional structure of complex multiplication.

The vector norm or complex absolute value of is given by

This gives the length of the vector or, in other words, the distance from the origin (0,0) to the point with coordinates (x,y).

Example CD-29.5 A Specific Length

Compute .


Example CD-29.6 Distance Between Vectors

Find the distance between the tip of the vector and .


The displacement vector from z2 to z1 is parallel to z1-z2 (see vector difference in Chapter 15), so the distance is

Example CD-29.7 Another Length

Compute .


Notice that this shows algebraically that the complex number lies on the unit circle centered at the origin.

Figure CD-29.1:

Theorem CD-29.3 The Triangle Inequalities


Draw a triangle with one side along the position vector z1 and the other side along the displacement arrow z2 translated so its tail starts at the tip of z1. The position vector z1+z2 forms the third side of a triangle (see the vector sum in Chapter 15). The inequality on the left says the length of the third side is no more than the sum of the lengths of the other two sides.

Draw a triangle with sides on the position vectors z1 and z2 and a third side connecting the tips of the two vectors. The displacement vector from the tip of z2 to the tip of z1 is parallel to z1-z2 (see vector difference in Chapter 15.) The inequality on the right asserts that the length of the third side is at least as great as the difference between the lengths of the other two sides.

Theorem CD-29.4 The Length of the Product is the Product of the Lengths


Let for k=1,2. The first two expressions are

Expanding both we have

These agree, so .

Exercise CD-29.2.7 asks you to prove the quotient property.

Example CD-29.8 The Polar Form of Complex Numbers

The unit length vector u in direction is . (See Chapters 28 and 16 and the program CmplxNrs.) If the vector (x,y) points in a direction that is an angle radians from the positive x-axis, then

because the length of z is |z| and the direction of z is the same as . The expression is called the polar form of z. It is the real length of z times the unit length complex number in the same direction as z.

Theorem CD-29.5 Multiplication Adds Angles The direction of the product is the sum of the directions of z and w.


Suppose we have two unit length complex numbers, and . The angle of the product is the sum of the angles of the numbers because of the following computation using the addition formulas for sine and cosine (see Chapter 28).

In general,

One simple consequence of this Theorem is that we can find nth roots of complex numbers by finding the positive real nth root of the length and dividing the angle by n. Before we illustrate this we want to give the exponential version of the polar form of a complex number.

Example CD-29.9 Euler's Formula

In Chapter 23, we prove Euler's Formula

This can be used with the polar form of a complex number to give the number by the real length times a complex exponential,

The exponential identity

applied to this situation also tells us that when complex numbers are multiplied, lengths multiply and angles add, if and ,

Figure CD-29.2: Geometric complex product

In the following examples we will write the polar forms of complex numbers using both trig functions and complex exponentials. They are equal by Euler's formula, but the use of the exponential identity is sometimes clearer.

Example CD-29.10 Square Roots by Geometry

Suppose we want to find the square roots of . These components lie on a triangle with sides 1, and hypotenuse 2, a "30-60-90" triangle. This makes the direction angle from the x-axis 60 degrees or . The length of the vector is 2.

We seek the square root and its length squared must be 2, so the length of the square root is . The angle of the square root is half the angle of the original number because we multiply the square root times itself, adding its angle to itself, and must arrive at .

Notice that division of the angle by 2 agrees with the laws of exponents when the polar form is written with Euler's Formula.

There is another square root. We can find it formally by representing the original number with another angle.

Example CD-29.11 Cube Roots by Geometry

Suppose we want to find the cube roots of . In this case the length is and the angle is 45 degrees, so

but the sine and cosine do not have simple values.

Two more cube roots could be obtained by dividing the angles in the polar representations


All the cube roots can also be obtained from one of them and the 3 cube roots of unity. We describe this approach next.

Example CD-29.12 Roots of Unity

Let n be a positive integer. We seek all the solutions to zn=1 or . The real solution z=1 is one possibility, but there are n-1 others equally distributed around the unit circle. We can represent . Dividing this angle by n gives us

For example, if n=3,

the point one third of the way around the unit circle.

The numbers

are equally spaced around the unit circle and satisfy (unk)n=1 because n times the angles and .

Finally, if is one nth root of z,

Then the numbers

also satisfy

Exercise set CD-29.2

  1. (Computer Exercise) Use the program CmplxNrs to draw the following complex numbers as position vectors. Compute their length (or absolute value) and direction angle from the positive x-axis.
    • 1.
    • 2.
    • 3.
    • 4.
    • 5.

    • Sketch the vectors and find the distance between:
      • 1. and
      • 2. and
      • 3. and

      • Complex Multiplication
        • 1. Plot the unit vectors and .
          • (a) Compute the angle each makes with the horizontal x-axis.
          • (b) Show that the product . ()
          • 2. Show that the product . ()
          • 3. Plot the unit vectors and .
            • (a) Compute the angle each makes with the horizontal x-axis.
            • (b) Compute the complex product
            • (c) Use your calculator or the computer to plot the unit vector and show that it makes an angle of with the horizontal. ()

            • Use the polar (either trig or complex exponential) form to show the following:
              • 1.
              • 2.
              • 3.

              • Use Euler's Formula to prove

                • 1. Show that 1, , , , -1, , , and are eight distinct eighth roots of unity. HINT: Draw them on the unit circle and show that they are equally spaced.
                • 2. Find 12 distinct 12th roots of unity and sketch them on the unit circle.
                • 3. Find 3 distinct cube roots of and sketch them on the unit circle.
                • 4. Find 6 distinct 6th roots of and sketch them on the unit circle.
                • 5. Find 3 distinct cube roots of and sketch them on the unit circle.

                • Show that if , then .

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