Roots of Complex Numbers Sec. 6.6c HW: p. 558 39-59 odd.

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Presentation transcript:

Roots of Complex Numbers Sec. 6.6c HW: p odd

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

Definition A complex number v = a + bi is an nth root of z if v = z n 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 now do an example… Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 0:

fourth root continued… Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 1:

fourth root continued… Find the fourth roots of Use the new formula, with r = 5, n = 4, k = 0 – 3, k = 2:

fourth root done! 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???

A new example… 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,

third root continued… 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,

third root continued… 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,

third root done! Find the cube roots of –1 and plot them. Now, how do we sketch the graph??? 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