Section 8.3 - Section 8.3 Complex Plane; De Moivr'e Theorem...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

Ask a homework question - tutors are online