Complex number

A complex number is an ordered pair of real numbers. (A real number may take any value from -infinity to +infinity. Real numbers are commonly represented as points on the "real number line", i.e., a straight line of infinite length.)

The two components of a complex number (a,b) are the real part (a) and the imaginary part (b). Complex numbers may be represented as points on an infinite two-dimensional plane surface, with the real part as the "X" coordinate and the imaginary part as the "Y" coordinate.

The operations of addition and multiplication are defined for complex numbers:

(a,b) + (c,d) = (a+c, b+d), and

(a,b) x (c,d) = (ac-bd, ad+bc)

Complex numbers may also be represented using "i" (or "j" in engineering contexts). The symbol "i" refers the the complex number (0,1). If "i" is interpreted as the square root of -1, we can write complex numbers in the form

(a,b) = a + ib

The addition and multiplication operators work out in a simple way, if we remember to collect real and imaginary terms and remember that i x i = -1. Thus,

(a+ib) x (c+id) = ac+aid+ibc+iibd = ac+i(ad+bc)+(-1)bd = ac-bd + i(ad+bc)

Complex numbers are often used in scientific and engineering applications to describe systems where amplitude and phase of a narrow band signal are important. If V = (re, im) is a complex value (say voltage), the amplitude and phase of V are

$$Amp = \sqrt{re^2 + im^2}\,$$

and

$$Phase = \arctan \big(\frac{im}{re}\big)$$

A sinusoidal voltage with frequency $$\omega = 2 \pi F$$ may be considered to be the real part of a complex voltage

$$V(t) = V_0\, \exp(j \omega t+ j\phi) = V_0\, ( \cos(\omega t+\phi) + j \sin(\omega t+\phi)\,)$$

with amplitude $$V_0$$ and  phase $$\phi$$.