Review of Complex numbers

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

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

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

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

5. Imag Imag r=|z| y f Real Real x Geometry Relating the Forms: Quadrants In exponential form, the positive angle, , is always defined from the positive real axis. If the complex number is not in the first quadrant, then the “triangle” has lengths which are negative numbers.

6. Imag y Real x Geometry Relating the Forms: in terms of and Use Pythagorean Theorem to find in terms of and : r=|z|

7. Imag r=|z| hyp y opp f Real x adj Geometry Relating the Forms: in terms of and Use trigonometry to find in terms of and

8. Imag r=|z| y f Real x Summary of Algebraic Relationships between Forms

9. Euler’s Formula

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

11. Euler’s Formula Can be used with functions:

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

13. Imag Real Multiplication of Complex Numbers Multiplication of complex numbers is easy in exponential form Multiplication by a complex number, , can be thought of as scaling by and rotation by Angle rotated counterclockwise by Magnitude scaled by

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

15. Imag r=|z| y f Real x Complex Conjugate Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c.: change i -i The complex conjugate is a reflection about the real axis

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