How can I calculate imaginary numbers whether in polar form or normal form

For calculating using the polar form, this site is useful:  http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Forms.aspx

If you have specific questions, you can post them here.

The imaginary unit "i" is the square root of -1:

 

i = √-1

 

An imaginary number (#) is a product of a real # and the imaginary unit.

 

h = b*i     where h is an imaginary # and b is a real # [a # you can find on the (real) # line]

 

 

 

A complex # is a sum of a real # and an imaginary #:

 

z = a +ib      Where z is a complex #, and a and b are real #'s.

 

 

When you add together two complex #'s, you add real parts w/ real parts, and imaginary parts w/ imaginary parts, ex:

 

z₁ = 5 +3i and z₂ = 7 - 2i

 

z₁ + z₂ = 5 +3i + 7 - 2i = 12 + i

 

When you multiply them, you do the same as you would do w/ real #'s, except that you have to remember that i² = -1:

 

z₁ * z₂ = (5 +3i)(7 - 2i) = 35 - 10i + 21i -6i² = 35 +11i - (-6) = 41 +11i 

When you divide two complex #'s, you have to multiply both the numerator and the denominator w/ the conjugate of the denominator. The conjugate of a complex number defined by "a + ib" is "a - ib". 

Remember that:

(x + y)(x - y) = x² - y²

 

Thus we get: 

 

(5+3i)/(7-i) = (5+3i)(7+i)/(7-i)(7+i) = (35 + 5i + 21i + 3i²)/(49-i²) =

(35 + 26i -3)/[49-(-1)] = (32 + 26i)/50 = 16/25 + 13i/25