Complex Numbers

What is a complex number? i is the square root of -1 What would the square root of -3 be? -24? Complex plane: x-axis = real numbers y-axis = imaginary numbers points are written as a+bi

Conjugates Useful for rationalizing Conjugate of a+bi is a-bi

Polynomials A polynomial of degree n has n solutions, real or imaginary What are the 4 solutions to x^4 = 1?

Exercises If (a+bi)(c+di) is a real number, what equations do we have in a, b, c, and d? What is the square root of i? (Hint: Use the fact that complex numbers are written as a+bi)

Polar coordinates The point a+bi can also be represented in polar form. How?

Operations Multiply the r’s, add the angles! What are the solutions to x^9 = 1 in polar form?

Roots of Unity What are the solutions of x^9 = 1? Let x = (r,Θ). Then x9 = (r9, 9Θ), and 1 = (1, 0 + 2πk). These two are equal if r9 = 1 or 9Θ = 2πk. r must be a positive integer by definition, so r = 1, and Θ = 2πk/9. Thus x = (1, 2πk/9), where k = 0, ±1, ±2, etc.