Roots of Complex Numbers

1. Roots of Complex Numbers Sec. 6.6c

2. From last class: The complex number The complex number is a third root of –8 is an eighth root of 1

3. 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.

4. 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.

5. 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:

6. 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:

7. 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:

8. 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???

9. 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,

10. 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,

11. 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,

12. 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???