1 Exponential Form Rectangular Form Real Imag x y f r z The real and imaginary parts of a complex number in rectangular form are real numbers Real Imag x Rez ID: 1030911
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1. Review of Complex numbers1Exponential Form: Rectangular Form:RealImagxyfr=|z|
2. The real and imaginary parts of a complex number in rectangular form are real numbers:RealImagx=Re(z)y=Im(z) Therefore, rectangular form can be equivalently written as:Real & Imaginary Parts of Rectangular Form
3. RealImagxfr=|z| The real and imaginary components of exponential form can be found using trigonometry: Geometry Relating the FormsyRealImagfr=|z|
4. Geometry Relating the Forms: Real & Imaginary PartsRealImag r=|z| The real and imaginary parts of a complex number can be expressed as follows:
5. Geometry Relating the Forms: QuadrantsIn 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. RealImagxyfr=|z| RealImag
6. RealImagxyr=|z| Use Pythagorean Theorem to find in terms of and : Geometry Relating the Forms: in terms of and
7. Geometry Relating the Forms: in terms of and RealImagxyfr=|z| adjopp hypUse trigonometryto find in terms of and
8. Summary of Algebraic Relationships between FormsRealImagxyfr=|z|
9. Euler’s Formula
10. ) Rectangular Form:Exponential Form:Consistency argumentIf these represent the same thing, then the assumed Euler relationship says:
11. 11 Euler’s Formula Can be used with functions:
12. Addition and subtraction of complexnumbers is easy in rectangular form12Addition & Subtraction of Complex Numbers Addition and subtraction are analogous to vector addition and subtractionRealImagabdc xyabdc
13. Multiplication of Complex Numbers13Multiplication of complex numbers is easy in exponential form Multiplication by a complex number, , can be thought of as scaling by and rotation by RealImag Magnitude scaled by Angle rotated counterclockwise by
14. 14Division of Complex NumbersDivision 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. Complex ConjugateAnother important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c.: change i -i RealImagxyfr=|z| The complex conjugate is a reflection about the real axis
16. The product of a complex number and its complex conjugate is REAL.Common Operations with the Complex ConjugateAddition of the complex number and its complex conjugate results in a real number RealImagxyfr=|z| x