Complex Numbers

Question

Explain the difference between rectangular form and polar form of a complex number.

Explain why there are two possible distinct results when simplifying (a+bi)^1/2

Answer 1

rectangular coordinates are the more well known graphical way to describe a point

(x,y) gives the x = horizontal distance from the origin to the point and y= vertical distance to the point

with x axis measuring the real value of a complex number and y measuring the imaginary value

for example, 1+ i is the point (1,1)

in polar coordinates 1+i = (sqr2, 45 degrees) or (sqr2, pi/4)

sqr2 =sqr(x^2 +y^2) = sqr(1+1) = radial coordinate, distance a crow flies from the origin to the given point

pi/4 = 45 degrees is the angular coordinate, the angle formed with the positive x axis and the line through the origin and the point

rectangular coordinates are for the cartesian coordinate system, named after a major philosopher who said "I think therefore I am" Polar coordinates are not named after polar bears. although the angles are often described in bearings.

sqr(a+bi) can be the positive or negative square root, so two possible values

= +/-[sqr(|z|+a)/2) + (b/|b|)sqr(|z|-a)/2)] where z=a+bi and b does not= 0

sqr(1+i) =+/-[sqr|1+i| +1)/2 + (1/1)sqr(|1+i|/2]