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Unformatted text preview: Section 8.3
Complex Plane; De Moivr'e Theorem Complex Plane
A complex number z = x + yi and be interpreted as the point (x,y). Each point in the xyplane corresponds to a complex number, so when we are talking about complex numbers we refer to the collection of complex points as the complex plane. The xaxis is called the real axis and the yaxis is called the imaginary axis. Modulus and Conjugate
The magnitude or modulus of z, denoted  z, is defined as the distance from the origin to the point (x,y). z = (x2 + y2) If z = x + yi then its conjugate is zbar= x yi Since zzbar = x2 + y2 z = (zzbar) Polar Form of a Complex Number
When a complex number is written in standard form z = x + yi, it is said to be in rectangular, or Cartesian, form. If r 0 and 0 2, then the complex number z = x + yi may be written in polar form as z = x + yi =(r cos)+(r sin)i =r (cos+i sin) In this form is called the argument of z. Since we require r to be positive we have z = r Theorem
Let z1 = r1(cos1+ i sin1) and z2 = r2(cos2+ i sin2) be two complex numbers. Then z1z2 = r1r2[cos(1 + 2) i sin(1 + 2)] If z2 0 then z1/z2 = r1/r2[cos(1 2) i sin(1 2)] Partial Proof
z1z2 = r1(cos1 + i sin1)*r2(cos2 + i sin2) =r1r2 (cos1 + i sin1)(cos2 + i sin2) =r1r2 (cos1cos2sin1sin2) + i(sin1cos2+cos1sin2) =z1z2 = r1r2[cos(1 + 2) i sin(1 + 2)] De Moivre's Theorem
If z = r (cos+i sin) is a complex number, then z = rn [cos(n)+i sin(n)] Where n is a positive integer. Theorem Finding Complex Roots
Let = r(cos0 + i sin0) be a complex number and let n 2 be an integer. If 0, there are n distinct complex roots of , given by the formula zk = r1/n[cos(0/n + 2k/n) + i sin(0/n + 2k/n)] Where k = 0, 1, 2, ..., n1. ...
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 Fall '08
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 Trigonometry, Complex Numbers

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