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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 xy-plane 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 x-axis 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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