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Arithmetic of Complex Numbers Multiplying by the conjugate to rationalize the denominator Converting vectors between rectangular form and polar form Objectives Multiply and divide complex numbers in polar form Raise a complex number to a power Find the roots of a complex number University of Minnesota Multiplying Complex Numbers/DeMoivre’s
5 3 Complex Multiplication 7 a Find z and arg(z) Then plot the complex number z b Calculate z 2 and z 3 c Determine the absolute value and argument of z 2 and z 3 Then plot these complex numbers and draw the modulus 10 Consider the two complex numbers ????????= ?2 2 + ?2 2 a Compare and contrast the two complex numbers b
Numbers of the form (8) are called complex numbers The set of all complex numbers will be denoted by C = fz= x+ iy: x;y2Rg: (9) As we shall see hereafter, C is an algebraic eld, i e , is possible to de ne in C addition and multiplication operations satisfying the same axioms of we have seen in Lecture 1 for R ( eld axioms) Page 1
Proposition 1 4 3 In polar coordinates, the multiplication rule for complex numbers becomes: (r;?)·(s;?) = (rs;?+?) which is a wonderfully simple geometric description In English: Multiplication Rule: To multiply two complex numbers in polar coordinates, add their angles and multiply their distances from 0
Multiplication by a real scalar : z 1 = a 1 + b 1i: Multiplication between complex numbers: z 1z 2 = (a 1 + b 1i)(a 2 + b 2i) = a 1a 2 + a 1b 2i+ a 2b 1i+ b 1b 2i 2 = (a 1a 2 b 1b 2) + (a 1b 2 + a 2b 1)i: All rules are identical to those for multiplication between real numbers, just remember that i2 = 1 Length/magnitude of a complex number z