Roots of Complex Numbers Sec 6 6 c

Roots of Complex Numbers Sec. 6. 6 c

FROM LAST CLASS: The complex number is a third root of – 8 is an eighth root of 1

DEFINITION A complex number v = a + bi is an nth root of z if n v =z If z = 1, then v is an nth root of unity.

FINDING NTH ROOTS OF A COMPLEX NUMBER If , then the n distinct complex numbers where k = 0, 1, 2, …, n – 1, are the nth roots of the complex number z.

LET’S SEE THIS IN PRACTICE: Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 0:

LET’S SEE THIS IN PRACTICE: Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 1:

LET’S SEE THIS IN PRACTICE: Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 2:

LET’S SEE THIS IN PRACTICE: Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 3: How would we verify these algebraically? ? ?

LET’S SEE THIS IN PRACTICE: Find the cube roots of – 1 and plot them. First, rewrite the complex number in trig. form: Use the new formula, with r = 1, n = 3, k = 0 – 2,

LET’S SEE THIS IN PRACTICE: Find the cube roots of – 1 and plot them. First, rewrite the complex number in trig. form: Use the new formula, with r = 1, n = 3, k = 0 – 2,

LET’S SEE THIS IN PRACTICE: Find the cube roots of – 1 and plot them. First, rewrite the complex number in trig. form: Use the new formula, with r = 1, n = 3, k = 0 – 2,

LET’S SEE THIS IN PRACTICE: Find the cube roots of – 1 and plot them. First, rewrite the complex number in trig. form: Use the new formula, with r = 1, n = 3, k = 0 – 2, The cube roots of – 1 Now, how do we sketch the graph? ? ?