Complex Numbers
Complex numbers are used for various mathematical purposes. A complex number consists of a real part and an imaginary part, often written in the form of $ ( X + iY ) $ Complex numbers can be manipulated using algebra just like any other number.
Addition
to add complex numbers we simply add the real parts and the imaginary parts together
EX: $ ( 3 + 2i) + ( 7 + 3i) = ( 10 + 5i) $
Subtraction
to subtract complex numbers we simply subtract the real parts and the imaginary parts
EX: $ ( 3 + 5i) - ( 2 - 3i ) = ( 1 + 8i) $
Multiplication
to multiply complex numbers we simply multiply each part of the complex number by each part of the other complex number as we would do when we multiply any polynomial by another polynomial.
EX: $ ( 2 + 3i ) * ( 4 + 2i ) = ( 8 + 4i + 12i + 6i^2 ) = ( 2 + 16i ) $
Division
to divide complex numbers we first get rid the complex portion of the denominator by multiply both top and bottom by it's complex conjugate and then we are left with a complex number over a real number.
EX: $ ( 1 + 3i ) / ( 2 + 2i) $ we multiply top and bottom by the complex conjugate of the denominator giving us $ ( 1 + 3i )* ( 2 - 2i) / (( 2 + 2i ) * ( 2 - 2i)) $
this simplifies to $ ( 8 + 4i ) / 8 $ or $ ( 1 + 0.5i ) $
