SECTION 8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS 439
8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS In Section 8.1, we mentioned that the complex zeros of a polynomial with real coefficients occur in conjugate pairs. For instance, we saw that the zeros ofp͑x͒ ϭ x2 Ϫ 6x ϩ 13 are 3 ϩ 2i and 3 Ϫ 2i. In this section, we examine some additional properties of complex conjugates. We begin with the definition of the conjugate of a complex number.
Definition of the The conjugate of the complex number z ϭ a ϩ bi, is denoted by z and is given by Conjugate of a z ϭ a Ϫ bi. Complex Number
From this definition, we can see that the conjugate of a complex number is found by changing the sign of the imaginary part of the number, as demonstrated in the following ex- ample.
EXAMPLE 1 Finding the Conjugate of a Complex Number
Complex Number Conjugate (a) z ϭϪ2 ϩ 3i z ϭϪ2 Ϫ 3i (b) z ϭ 4 Ϫ 5i z ϭ 4 ϩ 5i (c) z ϭϪ2i z ϭ 2i (d) z ϭ 5 z ϭ 5
REMARK: In part (d) of Example 1, note that 5 is its own complex conjugate. In gen- eral, it can be shown that a number is its own complex conjugate if and only if the number is real. (See Exercise 29.)
Geometrically, two points in the complex plane are conjugates if and only if they are reflections about the real (horizontal) axis, as shown in Figure 8.5. Figure 8.5 Imaginary Imaginary axis axis z = −2 + 3i z = 4 + 5i 3 5 4 2 3 2 1 Real Real axis − − axis −4 −3 −112 33572 2 6 −2 − − 3 2 −4 − −3 5 − z = −2 − 3i z = 4 5i Conjugate of a Complex Number 440 CHAPTER 8 COMPLEX VECTOR SPACES
Complex conjugates have many useful properties. Some of these are given in Theorem 8.1.
Theorem 8.1 For a complex number z ϭ a ϩ bi, the following properties are true. ϭ 2 ϩ 2 Properties of 1. zz a b 2. zz Ն 0 Complex Conjugates 3.zz ϭ 0 if and only if z ϭ 0. 4. (z) ϭ z
Proof To prove the first property, we letz ϭ a ϩ bi. Then z ϭ a Ϫ bi and zz ϭ ͑a ϩ bi͒͑a Ϫ bi͒ ϭ a2 ϩ abi Ϫ abi Ϫ b2i 2 ϭ a 2 ϩ b2. The second and third properties follow directly from the first. Finally, the fourth property follows the definition of complex conjugate. That is,
(z) ϭ (a ϩ bi) ϭ a Ϫ bi ϭ a ϩ bi ϭ z.
EXAMPLE 2 Finding the Product of Complex Conjugates Find the product ofz ϭ 1 Ϫ 2i and its complex conjugate.
Solution Since z ϭ 1 ϩ 2i we have zz ϭ ͑1 Ϫ 2i͒͑1 ϩ 2i͒ ϭ 12 ϩ 22 ϭ 1 ϩ 4 ϭ 5.
The Modulus of a Complex Number
Since a complex number can be represented by a vector in the complex plane, it makes sense to talk about the length of a complex number. We call this length the modulus of the complex number.
Definition of the The modulus of the complex number z ϭ a ϩ bi is denoted by͉z͉ and is given by Modulus of a ͉z͉ ϭ ͙a2 ϩ b2. Complex Number
REMARK: The modulus of a complex number is also called the absolute value of the number. In fact, when z ϭ a ϩ 0i is a real number, we have ͉z͉ ϭ ͙a2 ϩ 02 ϭ ͉a͉. SECTION 8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS 441
EXAMPLE 3 Finding the Modulus of a Complex Number
Forz ϭ 2 ϩ 3i andw ϭ 6 Ϫ i, determine the following. (a)͉z͉ (b)͉w͉ (c) ͉zw͉
Solution (a) ͉z͉ ϭ ͙22 ϩ 32 ϭ ͙13
(b) ͉w͉ ϭ ͙62 ϩ (Ϫ1͒2 ϭ ͙37 (c) Since zw ϭ ͑2 ϩ 3i͒͑6 Ϫ i͒ ϭ 15 ϩ 16i, we have
͉zw͉ ϭ ͙152 ϩ 162 ϭ ͙481.
Note that in Example 3,͉zw͉ ϭ ͉z͉ ͉w͉. In Exercise 30, you are asked to show that this multiplicative property of the modulus always holds. The modulus of a complex number is related to its conjugate in the following way.
Theorem 8.2 For a complex number z, The Modulus of a ͉z͉2 ϭ zz. Complex Number
Proof Let z ϭ a ϩ bi, thenz ϭ a Ϫ bi and we have zz ϭ ͑a ϩ bi͒͑a Ϫ bi͒ ϭ a2 ϩ b2 ϭ ͉z͉2.
Division of Complex Numbers One of the most important uses of the conjugate of a complex number is in performing di- vision in the complex number system. To define division of complex numbers, let us con- sider z ϭ a ϩ bi and w ϭ c ϩ di and assume that c and d are not both 0. If the quotient z ϭ x ϩ yi w is to make sense, it would have to be true that z ϭ w͑x ϩ yi͒ ϭ ͑c ϩ di͒͑x ϩ yi͒ ϭ ͑cx Ϫ dy͒ ϩ ͑dx ϩ cy͒i. But, sincez ϭ a ϩ bi, we can form the following linear system. cx Ϫ dy ϭ a dx ϩ cy ϭ b 442 CHAPTER 8 COMPLEX VECTOR SPACES
Solving this system of linear equations for x and y we get ac ϩ bd bc Ϫ ad x ϭ and y ϭ . ww ww Now, sincezw ϭ ͑a ϩ bi͒͑c Ϫ di͒ ϭ ͑ac ϩ bd͒ ϩ ͑bc Ϫ ad͒i, we obtain the following definition.
Definition of The quotient of the complex numbers z ϭ a ϩ bi and w ϭ c ϩ di is defined to be Division of z a ϩ bi ac ϩ bd bc Ϫ ad 1 ϭ ϭ ϩ i ϭ ͑zw͒ Complex Numbers w c ϩ di c2 ϩ d 2 c2 ϩ d 2 ͉w͉2 provided c2 ϩ d 2 Þ 0.
REMARK: Ifc2 ϩ d 2 ϭ 0, thenc ϭ d ϭ 0, and therefore w ϭ 0. In other words, just as is the case with real numbers, division of complex numbers by zero is not defined.
In practice, the quotient of two complex numbers can be found by multiplying the nu- merator and the denominator by the conjugate of the denominator, as follows.
a ϩ bi a ϩ bi c Ϫ di (a ϩ bi)(c Ϫ di) ϭ ϭ c ϩ di c ϩ di c Ϫ di (c ϩ di)(c Ϫ di)
(ac ϩ bd) ϩ (bc Ϫ ad)i ϭ c2 ϩ d 2 ac ϩ bd bc Ϫ ad ϭ ϩ i c2 ϩ d 2 c2 ϩ d 2
EXAMPLE 4 Division of Complex Numbers 1 1 1 Ϫ i 1 Ϫ i 1 Ϫ i 1 1 (a) ϭ ϭ ϭ ϭ Ϫ i 1 ϩ i 1 ϩ i 1 Ϫ i 12 Ϫ i 2 2 2 2 2 Ϫ i 2 Ϫ i 3 Ϫ 4i 2 Ϫ 11i 2 11 (b) ϭ ϭ ϭ Ϫ i 3 ϩ 4i 3 ϩ 4i 3 Ϫ 4i 9 ϩ 16 25 25
Now that we are able to divide complex numbers, we can find the (multiplicative) in- verse of a complex matrix, as demonstrated in Example 5.
EXAMPLE 5 Finding the Inverse of a Complex Matrix Find the inverse of the matrix 2 Ϫ i Ϫ5 ϩ 2i A ϭ ΄ ΅ 3 Ϫ i Ϫ6 ϩ 2i Ϫ1 ϭ and verify your solution by showing that AA I2. SECTION 8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS 443
Solution Using the formula for the inverse of a 2 ϫ 2 matrix given in Section 2.3, we have 1 Ϫ6 ϩ 2i 5 Ϫ 2i AϪ1 ϭ ΄ ΅. ͉A͉ Ϫ3 ϩ i 2 Ϫ i Furthermore, because ͉A͉ ϭ ͑2 Ϫ i͒͑Ϫ6 ϩ 2i͒ Ϫ ͑Ϫ5 ϩ 2i͒͑3 Ϫ i͒ ϭ ͑Ϫ12 ϩ 6i ϩ 4i ϩ 2͒ Ϫ ͑Ϫ15 ϩ 6i ϩ 5i ϩ 2͒ ϭ 3 Ϫ i we can write 1 Ϫ6 ϩ 2 i 5 Ϫ 2i A Ϫ1 ϭ ΄ ΅ 3 Ϫ i Ϫ3 ϩ i 2 Ϫ i 1 1 (Ϫ6 ϩ 2i) (3 ϩ i) (5 Ϫ 2i) (3 ϩ i) ϭ ΄ ΅ 3 Ϫ i 3 ϩ i (Ϫ3 ϩ i) (3 ϩ i) (2 Ϫ i) (3 ϩ i) 1 Ϫ20 17 Ϫ i ϭ ΄ ΅ . 10 Ϫ10 7 Ϫ i To verify our solution, we multiply A and AϪ1 as follows. 2 Ϫ i Ϫ5 ϩ 2i 1 Ϫ20 17 Ϫ i AA Ϫ1 ϭ ΄ ΅ ΄ ΅ 3 Ϫ i Ϫ6 ϩ 2i 10 Ϫ10 7 Ϫ i
1 10 0 ϭ ΄ ΅ 10 0 10 1 0 ϭ ΄ ΅ 0 1
TECHNOLOGY NOTE If your computer or graphing utility can perform operations with complex matrices, then you can verify the result of Example 5. For instance, on the HP 48G, you would enter the matrix A on the stack and then press the 1/x key. On the TI-85, if you have stored the matrix A, then you should evaluate AϪ1.
The last theorem in this section summarizes some useful properties of complex conjugates.
Theorem 8.3 For the complex numbers z and w, the following properties are true. 1. z ϩ w ϭ z ϩ w Properties of 2. z Ϫ w ϭ z Ϫ w Complex Conjugates 3. zw ϭ z w 4. z͞w ϭ z͞w 444 CHAPTER 8 COMPLEX VECTOR SPACES
Proof To prove the first property, let z ϭ a ϩ bi andw ϭ c ϩ di. Then z ϩ w ϭ (a ϩ c) ϩ (b ϩ d)i ϭ ͑a ϩ c͒ Ϫ ͑b ϩ d͒i ϭ ͑a Ϫ bi͒ ϩ ͑c Ϫ di͒ ϭ z ϩ w.
The proof of the second property is similar. The proofs of the other two properties are left to you.
SECTION 8.2 ❑ EXERCISES
In Exercises 1–4, find the complex conjugatez and graphically rep- In Exercises 21–26, determine whether the complex matrix A has an Ϫ resent both z and z. inverse. If A is invertible, find its inverse and verify that AA 1 ϭ I. 6 3i Ϫ Ϫ 1.z ϭ 6 Ϫ 3i 2. z ϭ 2 ϩ 5i ϭ ϭ 2i 2 i 21.A ΄ Ϫ ΅ 22. A ΄ ΅ 3.z ϭϪ8i 4. z ϭ 4 2 i i 3 3i
In Exercises 5–10, find the indicated modulus, where z ϭ 2 ϩ i, 1 Ϫ i 2 1 Ϫ i 2 ϭϪ ϩ ϭϪ 23.A ϭ ΄ ΅ 24. A ϭ ΄ ΅ w 3 2i, and v 5i. 1 1 ϩ i 0 1 ϩ i 5.͉z͉ 6.͉z 2͉ 7. ͉zw͉ 1 0 0 i 0 0 ͉ ͉ ͉ ͉ ͉ 2͉ 8.wz 9.v 10. zv 25.A ϭ ΄0 1 Ϫ i 0 ΅ 26. A ϭ ΄0 i 0΅ 11. Verify that͉wz͉ ϭ ͉w͉͉z͉ ϭ ͉zw͉, wherez ϭ 1 ϩ i and 0 0 1 ϩ i 0 0 i w ϭϪ1 ϩ 2i. 12. Verify that͉zv 2͉ ϭ ͉z͉͉v 2͉ ϭ ͉z͉͉v͉2, wherez ϭ 1 ϩ 2i and In Exercises 27 and 28, determine all values of the complex number v ϭϪ2 Ϫ 3i. z for which A is singular. (Hint: Set det(A) ϭ 0 and solve for z.) In Exercises 13–18, perform the indicated operations. 2 2i 1 ϩ i 5 z 27.A ϭ ΄ ΅ 28. A ϭ 1 Ϫ i Ϫ1 ϩ i z 2 ϩ i 1 3i 2 Ϫ i ΄ ΅ 13. 14. 1 0 0 i 6 ϩ 3i 29. Prove that z ϭ z if and only if z is real. 3 Ϫ ͙2i 5 ϩ i 15. 16. 30. Prove that for any two complex numbers z and w, the follow- 3 ϩ ͙2i 4 ϩ i ing are true. (2 ϩ i)(3 Ϫ i) 3 Ϫ i 17. 18. (a) ͉zw͉ ϭ ͉z͉ ͉w͉ 4 Ϫ 2i (2 Ϫ i)(5 ϩ 2i) (b) If w Þ 0, then ͉z͞w͉ ϭ ͉z͉͉͞w͉. In Exercises 19 and 20, find the following powers of the complex 31. Describe the set of points in the complex plane that satisfy the number z. following. Ϫ Ϫ (a)z 2 (b)z 3 (c)z 1 (d) z 2 (a)͉z͉ ϭ 3 (b) ͉z Ϫ 1 ϩ i͉ ϭ 5 19.z ϭ 2 Ϫ i 20. z ϭ 1 ϩ i (c)͉z Ϫ i͉Յ 2 (d) 2 Յ ͉z͉Յ 5 SECTION 8.3 POLAR FORM AND DEMOIVRE’S THEOREM 445
32. Describe the set of points in the complex plane that satisfy the 1 ϩ i 2 34. (a) Verify that ϭ i. following. ͙2 (a) ͉z͉ ϭ 4 (b) ͉z Ϫ i͉ ϭ 2 (b) Find the two square roots of i. 4 ϩ (c)͉z ϩ 1͉Յ 1 (d) ͉z͉Ͼ 3 (c) Find all zeros of the polynomial x 1. 33. (a) Evaluate ͑1͞i͒n for n ϭ 1, 2, 3, 4, and 5. (b) Calculate͑1͞i͒57 and ͑1͞i͒1995. (c) Find a general formula for ͑1͞i͒n for any positive integer n.
8.3 POLAR FORM AND DEMOIVRE’S THEOREM
Figure 8.6 At this point we can add, subtract, multiply, and divide complex numbers. However, there Imaginary axis is still one basic procedure that is missing from our algebra of complex numbers. To see this, consider the problem of finding the square root of a complex number such as i. When (a, b) we use the four basic operations (addition, subtraction, multiplication, and division), there seems to be no reason to guess that
r 1 ϩ i 1 ϩ i 2 ͙i ϭ . That is, ϭ i. b ͙2 ͙2 θ Real To work effectively with powers and roots of complex numbers, it is helpful to use a polar a 0 axis representation for complex numbers, as shown in Figure 8.6. Specifically, if a ϩ bi is a nonzero complex number, then we let be the angle from the positive x-axis to the radial ϩ Complex Number: a + bi line passing through the point (a, b) and we let r be the modulus of a bi. Thus, Rectangular Form: (a, b) a ϭ r cos , b ϭ r sin , and r ϭ ͙a2 ϩ b2 Polar Form: (r, θ ) and we have a ϩ bi ϭ ͑r cos ͒ ϩ ͑r sin ͒i from which we obtain the following polar form of a complex number.
Definition of Polar Form The polar form of the nonzero complex number z ϭ a ϩ bi is given by of a Complex Number z ϭ r ͑cos ϩ i sin ͒ where a ϭ r cos , b ϭ r sin , r ϭ ͙a2 ϩ b2, and tan ϭ b͞a. The number r is the modulus of z and is called the argument of z.