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Complex Numbers

At this point you should begin to have the feeling that this process of generating supersets of the numbers we already have figured out will never end. You would be right, and some of the extensions (ones we will not cover here) are actually very useful in more advanced physics. However, we have a finite amount of time to review numbers here, and complex numbers are the most we will need in this course (or even ``most'' undergraduate physics courses even at a somewhat more advanced level). They are important enough that we'll spend a whole section discussing them below; for the moment we'll just define them.

We start with the unit imaginary number11 , $ i$ . You might be familiar with the naive definition of $ i$ as the square root of $ -1$ :

$\displaystyle i = +\sqrt{-1}$ (10)

This definition is common but slightly unfortunate. If we adopt it, we have to be careful using this definition in algebra or we will end up proving any of the many variants of the following:

$\displaystyle -1 = i \cdot i = \sqrt{-1}\cdot\sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1$ (11)


A better definition for $ i$ that it is just the algebraic number such that:

$\displaystyle i^2 = -1$ (12)

and to leave the square root bit out. Thus we have the following cycle:
$\displaystyle i^0$ $\displaystyle =$ $\displaystyle 1$  
$\displaystyle i^1$ $\displaystyle =$ $\displaystyle i$  
$\displaystyle i^2$ $\displaystyle =$ $\displaystyle -1$  
$\displaystyle i^3$ $\displaystyle =$ $\displaystyle (i^2)i = -1 \cdot i = -i$  
$\displaystyle i^4$ $\displaystyle =$ $\displaystyle (i^2)(i^2) = -1 \cdot -1 = 1$  
$\displaystyle i^5$ $\displaystyle =$ $\displaystyle (i^4)i = i$  
$\displaystyle ...$     (13)

where we can use these rules to do the following sort of simplification:

$\displaystyle +\sqrt{- \pi b} = +\sqrt{i^2 \pi b} = +i\sqrt{\pi b}$ (14)

but where we never actually write $ i = \sqrt{-1}$ .

We can make all the purely imaginary numbers by simply scaling $ i$ with a real number. For example, $ 14i$ is a purely imaginary number of magnitude $ 14$ . $ i \pi$ is a purely imaginary number of magnitude $ \pi$ . All the purely imaginary numbers therefore form an imaginary line that is basically the real line, times $ i$ .

With this definition, we can define an arbitrary complex number $ z$ as the sum of an arbitrary real number plus an arbitrary imaginary number:

$\displaystyle z = x + iy$ (15)

where $ x$ and $ y$ are both real numbers. It can be shown that the roots of any polynomial function can always be written as complex numbers, making complex numbers of great importance in physics. However, their real power in physics comes from their relation to exponential functions and trigonometric functions.

Complex numbers (like real numbers) form a division algebra12 - that is, they are closed under addition, subtraction, multiplication, and division. Division algebras permit the factorization of expressions, something that is obviously very important if you wish to algebraically solve for quantities.

Hmmmm, seems like we ought to look at this ``algebra'' thing. Just what is an algebra? How does algebra work?

next up previous contents
Next: Algebra Up: Numbers Previous: Real Numbers   Contents
Robert G. Brown 2011-04-19