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11.2a Geometric Representation of a Complex Number Write complex numbers in polar form. GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS A Complex Number is in the form: z = a+bi We can graph complex numbers on the axis shown below: 4 2 -5 5 -2 -4 Imaginary Axis Real axis ABSOLUTE VALUE OF A COMPLEX NUMBER z 3 4i 4 2 -5 5 -2 •An arrow is drawn from the origin to represent the complex number. -4 •The length of the arrow is the absolute value of the complex number. REPRESENTING COMPLEX NUMBERS USING RECTANGULAR VS. POLAR COORDINATES a r cos 8 (a,b)=(r,) 6 b r sin z a bi 4 b So, z r cos (r sin )i z r(cos i sin ) 2 a 5 We abbreviate this as “cis” z rcis Complex Numbers Rectangular Form: Polar Form: z z a bi rcis Convert to Rectangular • Formulas: a r cos b r sin Convert to Polar Form • Absolute Value of a Complex Nubmer ( Magnitude) • Same as a radius!!!!! z 3 2i 2 z 4 cis 3 Homework • 11.2 P. 406 1-12 all 11.2b Complex Numbers Expressing Products and Quotient’s of Complex Numbers in Polar and Rectangular Form. Operations with complex numbers • Do you want to go thru that every time? rcis tcis r t cis Multiply: 4cis 25 6cis35 How do you think we can work with division? 3cis165 4cis 45 SUMMARY To convert a+bi to polar: Formulas: r a2 b2 b tan a To convert rcis to rectangular: Formulas: a r cos b r sin Homework • Pg. 406, 17-21odd, 23, 24