I'm not sure I would say this is the best evidence I've seen of _decadence_, per se, Gobble. After all, this is a little mind bending. I do wish people would accept the proofs faster, though.

Another nice proof of the e^(ix) thing is using differential equations. Say f(x) = e^(ix). Then f '(x) = if(x), and f''(x) = -f(x). But there are only two linearly independent solutions to f''(x) = -f(x), and they're sin(x) and cos(x). So f(x) = A cos(x) + B sin(x).

Now f(0) = 1, and f '(0) = i, which gives A = 1, B = i, and you're done.

(The sentence "But there are only two linearly independent solutions..." would be more clearly accurate if it said, "There are only two linearly independent solutions to f''(x) = -f(x), and two such solutions are sin(x) and cos(x)").

semck, is it possible to prove that there are only 2 linearly independent solutions to f"(x) = -f(x), which are sin(x) and cos (x)?

everyone look here

1 = x
100 = 100x
100x = 100
50x = 50
25x = 50-25x
5x = 10-5x
3x = 10-7x
3x-2 = 8-7x
-7x+8 = -2+3x
-7x = -2-8+3x
-3.5x+-3.5x = -10+3x
-3.5x = -10+6.5x
-10+6.5x = -3.5x
6.5x-10 = -3.5x
10x-10 = 0
x-1 = 0
-1 = -x
1 = x

YOU CANT EXPLAIN THAT

@Gobbledydook,

I'm ashamed to say I'd have to look up the proof, and in any case I'm not going to reproduce it here, but f'' = -f is a second-order linear differential equation with constant coefficients, and it's a theorem that any such equation has k linearly independent solutions, where k is the order of the highest derivative in the equation.

That means that since sine and cosine ARE solutions, and are linearly independent, any other solution can be expressed as a linear combination of them.

that circle on the chalkboard is not only that it's not a perfect circle, but that it's merely a representation of a circle, not the circle itself. but it's good to see this argument is continuing.

yes Abgemacht, I meant the imaginary number that is the square root of minus one,
we use the symbol i in cursive form in Australia to represent it.

Ghug, you are wrong in saying that this imaginary number is useless.
Modern air traffic control systems could not function without the number.

without i you cannot produce an answer to the square root of any negative number
eg the square root of positive 16 is plus or minus 4
ie minus 4 times minus 4 gives positive 16, and
positive 4 times positive 4 also equals 16.

to find the square root of minus 16 you convert minus 16 to
positive 16 times minus one
you geta solution to the square root of minus 16 as
plus or minus 4 (times ) i

For modern radar systems to provide a "speed" to or away from the radra location
you apparently need to use an algorithm, or equation that analyses the frquency
difference between the frequency of the emmitted pulse and returned pulse that
results from the Doppler effect.
the algorithm uses a square root of a negative value.
so you need this abstract concept, the imaginary number to solve this algorithm.

Maths is an abstract construction of human intellectual capacity that we use to
describe, understand and change our natural environment.

Perhaps, along with the human ability to create tools ( and there are some animals
that are basic tool creators & users, but humans have made the greatest use from
the creation of tools & machines of any species on this planet ) this ability of
human intellect to create and use maths is the reason we have been able to construct
the societies and civilisations in which we now live.

So don't knock the old maths caper Ghug,
you seriously underestimate the power of maths.

as to the does 0.9999 recurring being the same as 1 question
I liked the answer that said, they are not the same numbers, but they are equal
(in value)

Abge and I were joking, MM.

@MM, if two numbers are equal, then they are the same number. That's what it means to be equal.

Can we all agree that they are not the same numeral then? :-D

And that one is the loneliest number?

Here's a better question: Does MM = MajorMitchell?

Some insist it does. I do not believe it does.

I was actually confused when I first saw "MM" in this thread because I couldn't find a post by you, Marx.

Ouch, sorry MadMarx. I wasn't thinking.

Check and mate, if a mod says it, it is so.

semck83, please refer to me as MM.

:-)

Agreed 100%. Imaginary numbers are an invaluable tool that allow our daily lives to be as sophisticated as they are.

Ah k semck. So since there are only 2 L.I. solutions to f" = -f, and sin and cos are 2 of them, sin and cos are therefore the solutions.

Actually, the existence of a second MM on the forums has been bugging me for a while now. Apologies, Marx.

Exactly, GD.

@MM, OK, will do. But to avoid confusion, how about we compromise on MadM? When in a hurry, we can shorten to Ma'M. ;-)

semck83, be careful, I know where you live!

ghug, apology accepted!

Well, that's really creepy.

Oh wait, I know more than enough to figure out where he lives. Now I feel creepy too.

ghug: Re-shun.

He being semck, not Ma'M. I only have a general idea of where you live.

i have a related question for you guys, since you say 0.99... exists, does that mean -0 exists and if so does -0 = 0?