Complex numbers

What result does zz̄ give?

Re(z)² + Im(z)²

What is the complex conjugate geometrically

Reflection in the real axis (horizontally)

What is the modulus of a complex number?

|z| = square root (zz̄)/

What is wrong with saying z < n

Can’t compare complex numbers like this unless it’s real

What is the triangle inequality?

What is the reverse triangle inequality?

How would you prove the triangle inequality

If z = x + iy, how would you define x and y in terms of the modulus and argument? (Think of diagram)

X = |z| cos (arg( z))

Y = |z| sin (arg( z))

What is arg (0)?

What is arg (z̄)

Doesn’t exist

-arg(z)

What is polar form

Z = r (cosx + isinx)

How is the principle argument denoted

Arg z

(arg(z) is general argument)

What is Eulers formula?

What are the properties of the exponential of an imaginary number?

Prove:

What is the exponential form of an imaginary number?

Using the exponential formula, what is a quick way to multiply AND divide complex numbers

Multiply the moduli, sum the arguments

Divide the moduli, subtract the arguments

Why is it generally not true that Arg(z) + Arg(w) = Arg(zw)?

As principle Argument is strictly in range -pi, pi

How would you show that e^ix differentiates to ie^ix?

What is Eulers identity and how is it obtained?

e^i(pi) + 1 = 0

Writing -1 in exponential form and rearrange

What is de Moivres theorem

How would you prove de Moivres theorem

What are the complex solutions of a quadratic equation

How do you obtain the reverse triangle inequality from the triangle inequality?

1. Express modulus of z with w

2. Rearrange

How do you convert exponential form to x + iy form?

EULERS FORMULA (re^ix = r( cosx + isinx) )