1. Roots of a complex number
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Roots of a complex number
Chapter 7 review
# 31 & 35 & 33

2. #31) Find the square root of -1 + i
Change the number from a + bi form to r(cos q + isin q) form.
Find q.
Imaginary number axis
Graph the ordered pair (a,b).
(-1,1) q
Real number axis
tan q = 1 -1
q = arctan (-1)
q = -45o = 315o in quadrant IV or q = 135o in quadrant II.

3. #31) Find the square root of -1 + i
Change the number from a + bi form to r(cos q + isin q) form.
Since q is in quadrant II, we will use 135o .
r = (-1)2 + (1)2
-1 + i = (cos 135o + isin 135o) 2
Now use the rule for finding the roots of a complex number. n n
(cos a + isin a)
a + bi = r
q + 360o k . n
Where a =
and k = 0, 1, 2 … n – 1

4. #31) Find the square root of -1 + i
Change the number from a + bi form to r(cos q + isin q) form.
Since q is in quadrant II, we will use 135o .
r = (-1)2 + (1)2
-1 + i = (cos 135o + isin 135o) 2
135o + 360o k 2
a =
and k = 0 and 1
(cos 67.5o + isin 67.5o)
-1 + i = 4 2 or = 4
(cos 247.5o + isin 247.5o) 2

5. #31) Find the square root of -1 + i
The answer could also expressed in radians.
Change the number from a + bi form to r(cos q + isin q) form. 3p 4
Since q is in quadrant II, we will use .
r = (-1)2 + (1)2 3p 4 3p 4
-1 + i = (cos i sin ) 2 3p 4
+ 2p k
a =
and k = 0 and 1 2
(cos i sin ) 3p 8 4 3p 8
-1 + i = 2 or 4
(cos i sin )
11p 8 = 3p 8 2

6. We use the same method to solve an equation.
# 35) Find all complex solutions of the equation.
x4 – i = 0
x4 = i
Find the 4th root of i. (Use 0 + i).
The ordered pair is (0,1)
r = (0)2 + (1)2
0 + i = 1(cos 90o + i sin 90o)

7. Find the 4th root of i. cos 22.5o + i sin 22.5o 0 + i =
Change the number from a + bi form to r(cos q + isin q) form.
0 + i = (cos 90o + isin 90o)
90o + 360o k 4
a =
and k = 0, 1, 2, 3 4
0 + i =
cos 22.5o + i sin 22.5o
= cos 112.5o + i sin 112.5o or
= cos 202.5o + i sin 202.5o or
= cos 292.5o + i sin 292.5o or

8. Find the 4th root of the real number 81.
#33)
Change the number from a + bi form to r(cos q + isin q) form.
81 + 0i = 81(cos 0o + isin 0o)
0o + 360o k 4
a =
and k = 0, 1, 2, 3 4
81 =
3(cos 0o + i sin 0o) = 3
= 3(cos 90o + i sin 90o) = 3i or
= 3(cos 180o + i sin 180o) = –3 or
= 3(cos 270o + i sin 270o) = –3i or