More Of A Math Question

Could some one please explain how this works? π=pi=3.14159... e=2.718 i= the square root of -1 e^(πi) = e^(π/i) = e^(-πi)=-1 , but πi is not equal to π/i (one is a positive value, the other is a negative value), but π/i = -πi i figured all this during a bout of insomnia thanks

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honestly, from the transitive property, it appears that πi = -πi ...

that is of course assuming your equality statement is correct... that is...

e^(πi) = e^(-πi)

or e^x = e^x, where x = {πi,-πi} --- however, my work with non-reals is essentially non-real....

I don't know, though, unless this was done computationally and non-reals were in some way approximated, or somehow the result of raising e to a non-real gives some sort of approximate answer involving merely a distance from 0...

good luck... but I will attempt to check your equality statement...

by any chance could you explain to what this pertained?

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I'm afraid that you are mixing two ways of representing complex numbers, and that's why what you say is disturbing.
You are talking about "complex" numbers, which have properties that "real" numbers don't have. It's what is teached in the so-called "modern" mathematics. Integer numbers were the first numbers discovered. However, if you divide an integer by another integer, you don't obtain an integer number, so were invented the rational numbers, which are a ratio of two integer numbers. The integer numbers were a subset of the rational numbers.
Now, historically, it was a shame that the square of a rational is a rational, but the square root of a rational is not a rational. Then was invented the the notion of "real" numbers. The rational numbers are a subset of the real numbers. And the square root of a real number is a real number.
then came another philosophical problem : there is no square root for a negative number.
This meant that we needed a new extension to the old space of real numbers. We decided to think in terms of a two-dimensional surface, instead of thinking of one-dimensional line. Exactly like you think about the road in front of your house : you can think in terms of kilometers starting from your house, townwards (positive kilometers) or landwards (negative kilometers). But you can go out of this street by means of going through your garden till your neighbour's house. This situation is slightly more complex for describing where you are : in front of my house (kilometer zero accross the road) but two houses away perpendicularly to the road.
Here come the so-called "complex" numbers. The "real" numbers are along the horizontal line of the complex plan. The other axis is perpendicular to the real axis. And what is exactly "not the real axis" is "the imaginary axis".
The unit on the real axis is 1.
The unit on the imaginary axis is named "i", like "imaginary".
Have a look, draw the map : you have the real axis, unit 1. If you turn leftwards once, you arrive to the imaginary plan, on the "i" point.
If you turn once again, you arrive back to the "real" axis, but oppositely to the "1" point, you are at the "-1" coordinate of the real axis. That's why if you multiply i by i you have "-1".
Now, you see that a complex number is a point on the plan, represented by it's two coordinates on the real and on the imaginary axis. That's notated c = a + ib
If another complex number is named d = e + if,
if you multiply d*c you obtain (a+ib)*(e+if)=(ae + ifa + ibe + iibf)
and because i*i=-1, d*c=(ae)bf + i(af+be).

Another error is mixing both ways of representing a complex number. You cannot say that both ways are equal, you have to say that they are the coordinates of the same dot on the plan.
a+ib is one way of representing this number, by his coodinates on each axis.
r*exp(itheta) is the way of representing the same dot of the plan, where r is the distance to the zero of the axes, and theta is the angle with the horizontal axis.
You see that you cannot say a + ib = rexpt(itheta), it's hard to see where a is and where b is.
Except if you know that exp(itheta)=cos(theta) + isin(theta), you see that then rcos(theta)=a and rsin(theta)=b.
The third error is that the "square root" relationship is not symmetric.
The square of 2 is 4, "squareing" gives a single value.
But "the number whose square is 4" is not necessary 2, it could also be "-2". So, this is typically an equation that has two values.
So, if you are in your bed thinking with complex numbers (which were the nightmare of a lot of mathematicians) and if you mix two representions you fall into the case where simplifying notions start beint complacing things.
Hope this helped.
Yordan

 

 

 

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alright, thanks. I think i got it

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My hats off to you yordon! This whole discussion brings back nightmares of MATH 431 – Complex Variables. I failed it the first time around and barely got through it the second time. I will certainly say that you retained much of your higher education where I was glad to forget it the first chance I got.

The biggest thing that you should remember that “i" is not simply the square root of -1. It is better represented as a+ib. The prior is what is commonly taught in high school math.

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Nicely said, tansqrx. a + ib is part of a large region of mathematics and physics. "i" is simply the particular case where a=0 and b=1.
And remember, in light transmission an nuclear physics collision, "ib" is the "lost" part, for instance light intensity decreasing because a glass is not fully transparent, and material lost whan neutrons cross a wall.
And we can come back to the other topic about time reversal if you are looking about emotions, all using the complex numbers!

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