10. Complex numbers

# 10. Complex numbers - Complex Numbers Origin of complex...

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©2009 by L. Lagerstrom Complex Numbers Origin of complex numbers The number i The rectangular representation Plotting a complex number The number j Complex number math The polar representation Complex numbers in Matlab The Euler equation and the exponential representation Multiplication and division as rotation in the complex plane Note: As a supplement to this presentation, see the associated video clip.

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©2009 by L. Lagerstrom The Origin of Complex Numbers You should be familiar with the basics of complex numbers, but we will provide a brief review. We also will introduce a way of writing complex numbers that is probably new to you (the so-called exponential representation). Consider the following simple and similar equations: The first is trivial to solve: x = +1 or -1. But the second, the form of which seems just as simple as the first, leads to a puzzle. We find: But what in the world does the square root of a negative number mean? 0 1 and 0 1 2 2 = + = - x x 1 - ± = x
©2009 by L. Lagerstrom The Number i The fact that you get square roots of negative numbers from simple (or complicated) equations was known to the earliest mathematicians over two thousand years ago. They didn't quite know what to do with them, so they carried them around in their equations and hoped that eventually two of them could be multiplied together, thus yielding a square root of a positive number. Over time, however, progress was made in understanding and using them. One simple improvement was the introduction of the notation The "i" stands for "imaginary", because early mathematicians didn't consider square roots of negative numbers to be quite real, i.e., they were strange and different from "regular" numbers (what became known as real numbers). 1 or , 1 2 - = - = i i

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©2009 by L. Lagerstrom The Rectangular Representation Another important advance came in the definition of a "complex number" z as where x is the "real part" and y is the "imaginary part" of the number (and i is the square root of -1). Another way to write this is x = Re(z) and y = Im(z). Also note that x and y are real numbers. This is known as the "rectangular representation" of a complex number (and is probably how you first learned about them). With this representation, we see that it is possible to represent a complex number in two dimensions in the so-called complex plane. The horizontal axis is the real axis, where the real part of a complex number is plotted. The vertical axis is the imaginary axis, where the imaginary part of the number is plotted. (Next slide.) yi x z + =
©2009 by L. Lagerstrom Plotting a Complex Number The complex number z = 4 + 3i is plotted. Note that its magnitude |z| is its distance from the origin (in this case, 5). The magnitude can be found using the Pytha- gorean theorem with the sides 3 and 4 of the triangle shown, or more generally: 4 3i |z| = 5 z = 4 + 3i Real axis (x) Imaginary axis (yi) 2 2 )) (Im( )) (Re( z z z + =

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