Understanding Imaginary Numbers

Date: 09/18/2002 at 20:03:39
From: Anonymous
Subject: Imaginary number 

Dr. Math,

I have asked and looked many places trying to understand the 
imaginary number.  I don't have it in school yet, but I have to do a 
report on it. Why is the square root of i negative, and what about 
when it is cubed? Why is i times i equal to -i?  Also, could 
i be considered real? Not real as in real number, but real as in real 
life. If so, how can we apply it today? I just want to know why it has 
its properties. Please help me. Thanks

Date: 09/18/2002 at 22:04:51
From: Doctor Ian
Subject: Re: Imaginary number 

Hi,

The following item from our archive should get you started: 

   Complex Numbers
   http://mathforum.org/library/drmath/view/53877.html 

Complex numbers follow the same rules of exponents as real numbers:

  i^3 = i * i * i

      = (i * i) * i

      = -1 * i

      = -i

>Why is i times i equal to -i?

Recall the definition of i. We _define_ i such that

  i * i = -1

This is a little bit like defining the product of two negatives:

  -1 * -1 = 1

With negatives, you can pair up the signs, and each pair disappears, 
because it's equivalent to 1:

  -2 * -3 * -4 * -5 * -6 = -----(2 * 3 * 4 * 5 * 6)

                         = ---(2 * 3 * 4 * 5 * 6)

                         = -(2 * 3 * 4 * 5 * 6) 

With i, you get a similar effect, but you have to get _four_ i's if 
you want to cancel:

  i * i * i * i = (i * i) * (i * i)

                = -1 * -1

                = 1

so

  2i * 3i * 4i * 5i * 6i = iiiii(2 * 3 * 4 * 5 * 6)

                         = i(2 * 3 * 4 * 5 * 6)

You can think of it this way. Multiplying by -1 is like making a 
180-degree turn on the number line. But multiplying by i is like 
making a left turn on the coordinate plane. It takes only two 
180-degree turns to cancel out. But you need to make four left turns 
to get back to the direction you were going in originally. 

Does that make sense? 

>Also, could i be considered real?

When you get right down to it, no numbers are 'real'.  Numbers are a 
device that we make up in order to keep track of things. Counting 
numbers are good for keeping track of of sets of distinct objects 
(sheep, voters, pennies). Rational and irrational numbers are good for 
keeping track of quantities that can take values in continuous ranges, 
(lengths, intervals of time, areas, volumes).  

There's an old Calvin and Hobbes cartoon that I really like. The 
family drives over a bridge with a sign that says the bridge has a 
weight limit of, say, 20 tons. Calvin asks his father how they know 
what the limit is. His father says that they build the bridge, and 
then they drive heavier and heavier trucks over it until it collapses.  
Then they build another one just like it, and put up the sign with the 
weight of the last truck that crossed safely.

Now obviously, this isn't the way it's done! But it illustrates why we 
find it so convenient to work with numbers. We can use numbers to 
model the behavior of real things, as a way of predicting how those 
things will interact in the real world. The things are real, but the 
models aren't, and the numbers that make up the models aren't. 

What are complex numbers good for? One thing that complex numbers are 
really good for is keeping track of things that behave like waves.  
The most important kinds of waves that we deal with are light, sound, 
and electric and magnetic fields.  

(Complex numbers might also be used in the design of a bridge, if the 
bridge is long enough; since one thing that can make long bridges fall 
is that they can start vibrating in resonance with winds, so when you 
build a bridge, you need to know what its preferred frequencies of 
vibration are. Vibrations are modeled with waves, and waves are 
modeled with complex numbers.) 

It's important to realize that what each new kind of number gives you 
is a notation that's easier to work with. We could get along with 
nothing but counting numbers, but it would be very cumbersome. Using 
negative numbers gives us a very compact notation for talking about 
quantities that have opposites (east vs. west, up vs. down, credit vs. 
debt), but that's all it does. Using rational numbers allows us to 
break larger units into partial units, so we can measure things in 
feet and fractions of feet instead of some smallest unit (like the 
radius of an electron).  

Similarly, there is nothing that we can do with complex numbers that 
we couldn't do without them. It would just be more cumbersome to do 
calculations, that's all. It doesn't make them 'real'.  But it sure 
makes them useful!

Some of the stuff I'm telling you I've only figured out within the 
last few weeks, while trying to explain to someone else why the 
product of two negatives has to be positive. In case you've ever 
wondered what the volunteer math doctors get out of doing this, this 
is a big part of it: We're forced to re-examine things that we already 
think we 'know', and find out that maybe we don't know them as well as 
we think we did. Believe it or not, that's fun. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

See also these answers from the Dr. Math archives:

Imaginary Numbers in Real Life
http://mathforum.org/library/drmath/sets/shortcuts/dm_imaginary.html

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/17/2003 at 16:28:48
From: Anonymous
Subject: Imaginary number 

Well, it's been a while since I asked all this about imaginary 
numbers, and I don't think that I ever thanked you for replying 
so wonderfully.  This really helped me understand it and I'm so 
grateful.  I know where to come to if I ever need more help.  

Thanks!