Reply To This Post         Return to Posts Index           VegSource Home

From:	Rod Everson (69-179-102-86.dyn.centurytel.net)
Subject:	Re: Can anyone help me understand imaginary numbers?
Date:	November 19, 2007 at 9:29 pm PST

In Reply to: Can anyone help me understand imaginary numbers? posted by Suzy in IL on November 15, 2007 at 11:40 am:

Why does this matter? & When are they used?

Basically, they matter because the concept of imaginary numbers comes into play in some advanced math work.

As for understanding the concept, it's best to relate them to what you already know, the concept of a square root.

If you draw a graph with an x and y axis, and take x from zero out to positive numbers, plotting the square root of x as the y value, you get a parabola open to the right side of the graph. (some of the values are x=0,y=0; x=1,y=1 and -1, x=4, y=2 and -2, x=9, y=3 and -3, etc.

Show this to your son and then point to the left side of the x axis (running from 0 to -infinity) and explain that since any number squared must be a positive number, then -1, -4, and -9 have no square roots. Thus there is no value on the graph associated with x=-1, -4, -9 or any other -x IN THE REAL NUMBER SYSTEM AS IT IS DEFINED.

However, if we add to the the set of all real numbers just one more entity (by defining it) we can have values for the square roots of negative numbers. That addition was the concept of i-squared =-1 (by definition, remember.) In that sense, we "imagine" (or define out of thin air) the concept of i.

And, since the square root of any number x is just that number which, when multiplied by itself, results in x, then by that definition, when i is multiplied by itself (squared, that is) it is defined as -1, so then the square root of -1 is i.

Puzzle that last bit through until it's clear. Then all that is left is to realize that any negative number can be expressed as -1 times the same positive number. So, -4 = -1 times 4, etc.

With this new concept of an imaginary number i, we can now calculate square roots of negative numbers, for the square root of, say -16 can be expressed as the square root of (-1 times 16) or as square root of -1 times the square root of 16, or as i times 4, usually expressed as 4i. By the same logic, -4i is also a solution. This enables one to "solve" problems which have no real number solution, and it turns out that this comes in handy at higher levels of math in certain situations.

The important thing for your son to "get" at this level is just that the imaginary number i is not a member of the set of real numbers, but creating it (by defining it) does enable us to talk about solutions that cannot be expressed in the real number system, such as the square roots of -16, for instance.

I don't know if this will be clear enough to help, but thought I'd give it a try.

Rod

Reply To This Post         Return to Posts Index           VegSource Home