Review of Complex numbers Rectangular Form Imag rz
Review of Complex numbers Rectangular Form: Imag r=|z| Exponential Form: y f x Real 1
Real & Imaginary Parts of Rectangular Form The real and imaginary parts of a complex number in rectangular form are real numbers: Imag Therefore, rectangular form can be equivalently written as: y=Im(z) x=Re(z) Real
Geometry Relating the Forms Imag The real and imaginary components of exponential form can be found using trigonometry: r=|z| f x Imag Real r=|z| y f Real
Geometry Relating the Forms: Real & Imaginary Parts Imag The real and imaginary parts of a complex number can be expressed as follows: r=|z| Real
Geometry Relating the Forms: Quadrants Imag r=|z| y Imag Real x f Real
Imag Use Pythagorean Theorem r=|z| y x Real
Use trigonometry hyp opp adj Imag r=|z| y f x Real
Summary of Algebraic Relationships between Forms Imag r=|z| y f x Real
Euler’s Formula
Consistency argument If these represent the same thing, then the assumed Euler relationship says: Rectangular Form: Exponential Form:
Euler’s Formula Can be used with functions: 11
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 y Imag b a Real c d b c a x d 12
Multiplication of Complex Numbers Multiplication of complex numbers is easy in exponential form Imag Real 13
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 14
Complex Conjugate Imag r=|z| y f Real x The complex conjugate is a reflection about the real axis
Common Operations with the Complex Conjugate Addition of the complex number and its complex conjugate results in a real number Imag r=|z| y The product of a complex number and its complex conjugate is REAL. f x Real
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