422 Chapter TeA
For questions 5 and 6, refer to the properties listed in section 10.2.
10.2 Exercises 1. Simplify the following (n
1J).
a) 1
d) i6
g) j4fl
b) i4 c) i5
e) —1
h)
f) i2
i) 42
4fl+l
2. Simplify the following.
a) (5—i)--(4+3i) b) (—1+i)—(1 —i)
c) 4(3+2i)—2(6+i) d) (2+z)2—(3—2i)2 e) (5+3i)(3 —i)+3(1 +i)(1 —1) —4(3+7i)i 3. Express in the form a + ib, where a 11 and
b€i. 1+4i a) 1.
—l
5+21 5 — 21
+
6—21
b) x+yi= 3+5i
1 —
1
41 1
h) 6+5i — (6+51)2 4. Givenz=cosO-i-isinOand w = cos
— 8. a) Prove thatz+ i—
andy. a) X+yi 4+i =4—i
1
e) 2+3i f) 3+4i g) 3 + 4t +
7. Provethatifzw=0,thenz=Oorw=0. (Hint:ifz=a+biandw=c+di,youmust provea=b=Oorc=d=0.)
9. The equation az2 + bz + c 0 is such that b2 — 4ac < 0, where a, b, and c are real. Prove that the roots of the equation are complex conjugates. 10. Solve the following for the real numbers x
7 — 31 .
1
6. Prove the properties of conjugates 1, 2, 4, 5, and 6.
1
1 + 41
d)
a) properties E, S and P. b) the properties of conjugates 1,2, and 3. c) Is the set Iii also a field?
b) Simplify3—2i+ 3—21
b) 2+i c)
5. By making the imaginary part zero, verify that the following properties in C also hold true in R.
0 — isin
0, prove the following.
(Use the formulas on page 542.) a) z + w = 2cos 0 b) z — w = 2isin 0
c) zw=1 d) z2 = cos 20 + isin 20 e) w2 = cos 20 — isin 20 1 = + --tan0 f)
1+w 2 2 2
11. Find the real and imaginary parts of
2+1 a) 1+51 + ____ 1—51
b) (1 +i) 12 Given z = 5z—4+31—4 , find the real and 1 — 2i i imaginary parts of z, and of z2.
13. Simplify (1 + i)4(4 — 31)2(1 — i)(4 + I
14. Find the number b such that I
2—
6
31)2.
3i/ = 2.
+ bi
15. Find z in terms of cos a and sin a, if
z2—2zcosa+1=0.