can anyone explain the mathematical sense behind using imaginary numbers?

Question

the imaginary number in question being i, where:

i = (-1)^(1/2)

and

(i)^(2) = -1

Answer 1

Without complex numbers you would not be able to solve many of the equations that engineers, scientists, and mathematicians face every day, like the following equation:

x^2 - 2x + 4 = 0

the two values of x that makes this equation work are
1 + i*sqrt(3)
1 - i*sqrt(3)

without complex numbers this equation has no solution.

Let me put it this way: electrical engineering and electronics would not have progressed beyond what it was around 1890 without the use of complex numbers. You would have a light bulb hanging from your ceiling, but it would stop there. No computers, no radios, no cell phones or any kind of phone. Forget about TV's. Just a lousy light bulb. Pretty primitive huh!

Answer 2

imaginary numbers is just a name given to the class of numbers

if you need to refer to or use the quantity (-1)^1/2 you could just call it (-1)^1/2 all the time, but it is easier to refer to it as "i"

there are many mathmatics applications where the quantity "i" is useful and some of those are applied to real world problems

for instance, calculating many aspects of alternating electrical current is done by describing the "phase angle" with a complex number (that is a term that is the sum of a rational and an imaginary number, like "4+3i")

math is just symbols we use to figure out things and they don't always relate exactly to real physical things

some people I have tutored have found negative numbers to be ridiculous, what is negative five apples anyway? and some people are pretty comfortable with abstraction

keep at it
math is cool
it gives you power

Answer 3

It's been a while since I took any math, but I remember that there are mathematical rules that pertain to square roots, etc. that "work" whether the number is real or imaginary. The square root of a negative doesn't exist, but in some situations you can just pretend it does by sticking in an "i" and your formulas will still work. Pretty weird.

Sorry I can't come up with a more concrete example. Again, it's been a while.

Answer 4

Here are at least two values to trying to understand and thinking about abstract things like imaginary numbers.
1. There are many many things in science that we can never see. We must be able to visualize how things work and what's going on without actually being able to see them. An atom is a good example. Molecules of chemicals we call medicines, and how they interact with cells and other things are all things we must be able to visualize without actually seeing.
2. They teach fortitude. The ability to stick with things that are complex or complicated. Everybody needs to learn fortitude so we can stick with a job where the boss can be difficult, stick with a marriage when communication might break down, stick with your children when they don't always make you happy, etc.

We need the abstract to stretch our brains a bit.

Answer 5

That question has been bothering me during my high school days. But as I study Engineering in the university I saw different applications of imaginary numbers. Imaginalry numbers are used to solve differential equations using Laplace. it is used to solve non-linear circuits(w/c capacitors and inductors) using Phasor Analysis). It can also be used in Signals analysis such as Fourier Transforms. How? A basic signal which is composed of a sinusoids can be modeled as cos(wt)+isin(wt) which is equal to e^(iwt) by Eulers Identity.

Answer 6

If you perceive the real number set as a straight line, 0 at the center, then you might want to perceive complex number set as a 2D plane, the real numbers line bisecting the plane and 0 + 0i in the middle of the 2D plane.

Hmmm... it might be interesting to study the existence of a suitable concept that could be perceived as a 3D space, with the complex number 2D plane bisecting it :)

Answer 7

A phasor is a rotation in the complex plane, an operation that corresponds to raising a complex number to a power.

I'm just in here to supplement. Not after the prize.

Answer 8

I totally could never understand it, and flunked math 9 times in college. I can balance my checkbook, do fractions, percents, add, subtract, divide -- but throw an x and little numbers in there nd I'm screwed.