Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne 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*x2-y1*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

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs

NEWTON AND ASK A SCIENTIST
Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory