Roots of a complex number To view this power point, right click on the screen and choose “Full screen”. Roots of a complex number Chapter 7 review # 31 & 35 & 33

#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.

#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

#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

#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

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)

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

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