Where are complex numbers used?

Answer 1

Complex numbers are necessary in order for the fundamental theorem of algebra to be complete.

The fundamental theorem of algebra states that for every polynomial function of degree n (the highest exponent on the variable) we will get n number of solutions.

Take this quadratic for example:

f(x) = x2 - 4

The fundamental theorem of algebra states that we will get two values of x for every value of f(x) we pick. In most cases we are concerned with the "roots" (typically x-intercepts) where the function crosses the line f(x) = 0.

So for the quadratic above when f(x) is set to 0 we get x = 2 or x = -2.

Now take a look at this quadratic:

f(x) = x2 + 1

A quick look at the graph of this function will show that it never crosses the line f(x) = 0. But according to the fundamental theorem of algebra we should get two values for x at f(x) = 0.

This is where the complex numbers come in.

When we solve this equation for f(x) = 0 we get the following conundrum:

-1 = x2

Now we know that we cannot take the square root of a negative number and get a real solution. There is no real number asuch that a2 will equal a negative number. So we move into the complex numbers.

The following solutions work for the above problem, x = i, x = -i.

The complex numbers make it to where the fundamental theorem of algebra holds for any polynomial function.

Complex numbers are also used in physics and electrical engineering. I do not know the specifics of those applications though so feel free to add to this.

Through the use of Euler's formula, derived by in the 1700's Leonhard Euler, we have: ei*ÆŸ = cos(ÆŸ) + i*sin(ÆŸ), where 'e' is the base of natural logarithms (approx 2.718281828), i is the imaginary unit number sqrt(-1), and ÆŸ is an angle in radians.

Some examples of where this is useful is: analysis of certain electric circuits, signal processing, control systems, electric power transmission lines. The angle ÆŸ usually represents a phase shift between two signals. Graphing in the complex plane can reveal behavior of the system, that is not readily evident with other methods.