

Imaginary Numbers
Name: Paul
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
In an equation like M = Square root of (L / S)...if 'L'
is a negative number can the equation still be solved? I'm mathematically
illiterate, so please be gentle.
Replies:
The equation can be solved if you allow the sqare root of 1 to exist.
Often a lower case "i" is used to represent the square root of 1.
Factoring this out allows the sqare root of what remains, a positive number,
to be calculated. There is no proof that such a thing actually exists in
anything real. It just makes a great deal of mathematics involving negative
numbers easier to work with. The square root of 4 equals the square root of
4 multiplied by the square root of 1. The square root of 4 is 2i.
Kenneth Mellendorf
Paul, you need not apologize.
First, let's write the equation in the form: M
= sqrt[L/S]
By doing this, we can specify that M, L, and S are all positive numbers and
the worrisome minus sign sticks out explicitly. By "squaring" both sides we
could write the exactly equivalent equation: M^2 *S /L = 1 How can this
be? Well don't feel too bad, it took several hundred years before some
pretty clever mathematicians finally unraveled the same problem bothering
you. As a consequence numbers that behaved like M,L, and S were called
"imaginary" or "complex" with all the emotional baggage those terms carry.
Concurrent with all this turmoil physicists/mathematicians developed the
algebra of ordered pairs of numbers, for example: (x,y) or specifically, (2,
5) or (3,8). These are the familiar ordered pairs that students from about
grade 3 or 4 make "x" "y" graphs for straight lines and so on. So,
specifically the point (2,5) says move 2 units to the right of the origin
(0,0) along the "x" axis, and 5 units up from the origin along the "y" axis.
The reason for calling the numbers (x,y) "ordered" is that reversing the
order of the "x" and the "y" do not give the same point using the rules
explained in the previous paragraph. That is, the point (2,5) is NOT the
same as (5,2).
Addition and subtraction of such pairs was straightforward:
(x1,y1) + (x2,y2) = (x1+ x2, y1+y2) and everything was cool. The "rub" came
with defining the operations of multiplication and division. How should
mathematicians define:
(x1,y1) * (x2,y2) = ????? Remember in math we can define it any old way we
choose. It is only the utility and logical consistency that counts.
What was discovered was this: Let's make a change in notation. Rewrite:
(x1,y1) = x1 + i(y1) and rewrite (x2,y2) = x2 + i(y2). Then:
(x1,y1) * (x2,y2) = [x1 + i(y1)] * [x2 + i(y2)]. The mathematicians noted
that this could be written:
([x1*x2y1*y2], i*[x2*y1 + y2*x1]) which looks kind of complicated, but the
curtain rises when they realized that this was notationally identical to
calling "i" whatever you wanted to but making the rule that i^2 = i*i = 1.
THAT IS "i" = sqrt (1).
So there is neither imaginary nor complex. It's just how we define the
multiplication of ordered pairs!!!
And Thus Came the Whole Fertile Field we now call Complex Variable!
Vince Calder
Click here to return to the Mathematics Archives
 
Update: June 2012

