# Complex Numbers

## Motivation

Complex numbers are a way to combine the idea of number (addition, multiplication, distribution, etc.) with the idea of vectors in the plane, producing a powerful tool.

The power is provided mathematically by the fundamental theorem of algebra. Recall that this theorem says that any algebraic polynomial equation of the nth degree in the variable z-- something of the form

anzn + an-1zn-1 + ... a1z + a0 = 0

always has exactly n complex roots; that is, it can be factored into the form

(z - z1)(z - z2)(...)(z - zn = 0

where the zi are (possibly complex) constants, the solutions of the equation. We will see this give us tremendous power for figuring out solutions of ordinary linear (i.e., the unknown comes in as the first power in every term) differential equations with constant coefficients.

The power is provided physically whenever we have a physical system in which there are two things (numbers or functions) that are related in the way complex numbers are. One such case is for oscillations of objects governed by Newton's laws. Since these are second order differential equations, sometimes linear, sometimes with constant coefficients as is our prototype oscillator example, the SHO) the two solutions look like sines and cosines. These are related in the way that the parts of a complex function are.

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