On Wed, 24 Mar 2010 21:04:19 -0500, the infamous snipped-for-privacy@host122.r-bonomi.com (Robert Bonomi) scrawled the following:
Lay her down, roll her over, and do it again?
I like it! I wonder how many people actually tried that.
-- If we attend continually and promptly to the little that we can do, we shall ere long be surprised to find how little remains that we cannot do. -- Samuel Butler
I must be doing something wrong: ( sqrt(2)/2 - sqrt(2)/2*i ) **2
=(sqrt(2)/2)**2 -*((sqrt(2)/2*sqrt(2)/2*i) - (sqrt(2)/2*i)**2 = 2/4 -2*(2/4*i) - 2/4*(i*i) = 2/4 - 4/4*i + 2/4 = 1-i
I'm tempted to make a facitous remark about an 'off by one' error. :)
Admittedly, I haven't played with this stuff for 30+ years, but I've got a vague recollection of 'j' (the 'hyper-imaginary'??) as sqrt(-i).
Of course, on my first exposure to the imaginary exponential, I immediately and rather vehemently questioned "exp(2*pi*i) = 1". As follows:
exp(0) = 1 by definition exp(2*pi*i) = exp(0) 2 thins equal same thing, equal to each other (2*pi*i) = 0 if bases (non-zero, and non-multiplicative- identity) equal, exponents equal.
This rather upset the 8th grade math teacher. He *knew* my reasoning was incorrect, but pointing out _where_ the flaw was was not obvious.
Simple reasoning on complex issues can lead one to into trouble.
Once it was clarified that '2*pi' was angular measure, things clarified It is true that 2*pi == 0 (plus 1 revolution), even though neither of the multiplicands is zero.
Robert, Nice Try! The line above should be 2/4 -2*(2/4*i) +2/4*(i*i)
= 2/4 -4/4i -2/4 = -i, as advertised. You had me worried for a moment.
Yes, you assumed that the function e^x is 1-to-1. And it is definitely not. It wraps the imaginary axis around the unit circle infinitely often. Indeed, it maps every horizontal strip of height 2PI in the complex plane onto the whole complex plane minus the point 0. I use this property of e^x as a building block to help construct other mappings which are infinity-to-one. Of course, there is nothing special abought e^x in this context, any exponential function a^x (a>0, a!=1) may be written in the form e^(kx) for some real number k so similar properties hold for a^x. What is special about e^x is that d/dx (e^x) = e^x, and only constant multiples of it have this property (of being equal to their derivative on a suitable domain).
..Not obvious to an 8th grade math teacher. They have their hands full trying to convince students that (a+b)^2 is not a^2 + b^2!!! Tough job!
Actually, my 8th grade math teacher was one of my favorite teachers, K-12.
It is of course Not true that 2*pi =0. What is true is that e^(2pi*i) = e^0=1.
What is also true is that, for real values x, e^(ix) = cos(x) + i sin (x), which I see is what you were referring to by "plus 1 revolution" above.
Best, Bill
ARGH!!! like I said, "I must be doing something wrong'
I inverted the sign on the last term _and_ left the i^2 in. one or the other, but not both. Takes more than 2 i's to find it. :)
Well, It _is_, for exponents without an imaginary component. The concept of multiplying something by itself an imaginary number of times is utterly lacking in any intuitive foundation.
His own fault! _He_ was the one that sprung it on me, cold, and wanted me to agree it was true. I'd stuck my head in the classroom, after school was out, to ask him something, and he threw that at me, from the middle of a discussion he was having with somebody else -- "e to the two pi eye equals one, right?" I thought for about half a second and said, firmly, "no". He asked "why not?" And I wrote the above 'disproof' on the black- board. Finding -where- to kick a hole in it is difficult. Every line _does_ follow validly from the prior one. It's merely that the 'meaning' of the numbers changed.
I think my 5th or 6th grade class spent -maybe- one afternoon on that. Re-write the '^2' as an explicit multiplication, and apply the distributive property (a total of 3 times) to eliminate the parens, then consolidate the like terms.
The error is also 'painfully obvious' if you simply work through a couple of 'concrete' examples.
Admittedly, I was learning this stuff _just_before_ the "new math" teaching started. (My younger brother, 3 years behind me, got a *very* different math education!)
Depends on what you mean by '='.
They're both the same "direction". "equal" in that sense.
Anything derived from the point-value (ignoring the 'path' that got you there) is the same for both.
Quoting a line from an old comic strip -- where they got it absolutely wrong (unintentionally!) in context: "How do you tell a pilot he's 360 degrees off-course?"
Yes, think of expressions like 2^i as exp^(i *ln 2), then you can use your intuition a little more once you are adequately aquainted with the function e^x.
What you REALLY mean is that: 2*pi radians ~ 0 radians!
0 and 2*Pi are complex numbers, different ones, which are over 6 units apart in the complex plane. We save "=" for when the numbers coincide.Again, The terminal side of angle of 0 degrees is the same of one of 360 degrees. But, turning a gear or the steering wheel of your car 360 degrees or -360 degree or 0 degree are 3 different things, right? I agree it probably doesn't matter how a pilot turns west, if west is the direction he is supposed to go! : )
Best, Bill
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