Review of Complex numbers 1 Exponential Form:Rectangular Form: Real Imag x y  r=|z|

The real and imaginary parts of a complex number in rectangular form are real numbers: Real Imag x=Re(z) y=Im(z) Therefore, rectangular form can be equivalently written as: Real & Imaginary Parts of Rectangular Form

Real Imag x  r=|z | The real and imaginary components of exponential form can be found using trigonometry: Geometry Relating the Forms y Real Imag  r=|z |

Geometry Relating the Forms: Real & Imaginary Parts Real Imag r=|z| The real and imaginary parts of a complex number can be expressed as follows:

Geometry Relating the Forms: Quadrants Real Imag x y  r=|z| Real Imag

Real Imag x y r=|z| Use Pythagorean Theorem

Real Imag x y  r=|z| adj opp hyp Use trigonometry

Summary of Algebraic Relationships between Forms Real Imag x y  r=|z|

Euler’s Formula

Rectangular Form : Exponential Form : Consistency argument If these represent the same thing, then the assumed Euler relationship says:

11 Euler’s Formula Can be used with functions:

Addition and subtraction of complex numbers is easy in rectangular form 12 Addition & Subtraction of Complex Numbers Addition and subtraction are analogous to vector addition and subtraction Real Imag a b d c x y a b d c

Multiplication of Complex Numbers 13 Multiplication of complex numbers is easy in exponential form Real Imag

14 Division of Complex Numbers Division of complex numbers is easy in exponential form Division of complex numbers is sometimes easy in rectangular form Multiply by 1 using the complex conjugate of the denominator

Complex Conjugate Real Imag x y  r=|z| The complex conjugate is a reflection about the real axis

The product of a complex number and its complex conjugate is REAL. Common Operations with the Complex Conjugate Addition of the complex number and its complex conjugate results in a real number Real Imag x y  r=|z|