Complex numbers are numbers that are comprised of a "real" and "imaginary" component. Symbollically, a complex number is expressed in the form "a+jb" or "a+ib," where "j" = "i" = the square root of -1 and "a" and "b" represent whole real numbers.
Complex numbers are easily interpreted into graphical representation by method of the traditional Cartesian coordinate (x,y) plane where the y-axis is the imaginary (j or i) axis and the x-axis is the real axis.
Some complex numbers appear to be... well, complex. Fortunately, simple algebraic manipulations can be done to simplify the expression of fractional complex numbers. Take, for example, a fraction with the complex number (a+jb) in the numerator and (a-jb) in the denominator. A method of simplifying this fraction is by multiplying both the numerator and denominator by the "complex conjugate" of the complex number in the denominator. The complex conjugate of a complex number is determined by changing the sign of the operator between the real and imaginary components of the number. This simple deduction is expressed in the example below.
Complex number = a+jb Complex conjugate = a-jb
Multiplying both the numerator and denominator by the complex conjugate of the denominator will eliminate any complex number terms from the denominator and make the number far more useful in lengthy calculations. This example is illustrated below.
(a+jb)/(a-jb) = ((a+jb)/(a-jb))*((a+jb)/(a+jb)) where (a+jb) is the complex conjugate of (a-jb)
((a+jb)/(a-jb))*((a+jb)/(a+jb)) = ((a+jb)(a+jb)/(a-jb)(a+jb)) = (a^2 + j2ab - b^2)/(a^2 + b^2)
If you or a loved one are considering using complex numbers in the analysis of various mathematical models, please refer to this page whenever necessary!
