baric Posted June 17, 2011 Posted June 17, 2011 Begin with Euler's Identity: [math]e^{i\pi} + 1 = 0 [/math] [math]e^{i\pi} = -1 [/math] [math]e^{i 2\pi } = 1[/math] [math]e^{i 2\pi} = e^0[/math] [math]i 2\pi = 0[/math] [math]i = 0[/math] [math]1 = 0[/math] Euler says so!
DrRocket Posted June 17, 2011 Posted June 17, 2011 Begin with Euler's Identity: [math]e^{i\pi} + 1 = 0 [/math] [math]e^{i\pi} = -1 [/math] [math]e^{i 2\pi } = 1[/math] [math]e^{i 2\pi} = e^0[/math] [math]i 2\pi = 0[/math] [math]i = 0[/math] [math]1 = 0[/math] Euler says so! A good illustration of why in complex variables one must "choose a branch of the logarithm".
baric Posted June 17, 2011 Author Posted June 17, 2011 A good illustration of why in complex variables one must "choose a branch of the logarithm". I posted that "proof" on the whiteboard of a coworker and asked him to find the errant step.. without luck!
insane_alien Posted June 17, 2011 Posted June 17, 2011 basically what you are saying there is 2 pi radians equals zero radians which is true just as 360 degrees is the same angle as 0 degrees. i thinks its time for you to turn 360 degrees and walk away.
baric Posted June 17, 2011 Author Posted June 17, 2011 basically what you are saying there is 2 pi radians equals zero radians which is true just as 360 degrees is the same angle as 0 degrees. i thinks its time for you to turn 360 degrees and walk away. The last time I did that I tripped over my XBox and broke it. No thanks!
rktpro Posted June 17, 2011 Posted June 17, 2011 The last time I did that I tripped over my XBox and broke it. No thanks! Then you agree that 360 degree is not equal to zero.
Marat Posted June 18, 2011 Posted June 18, 2011 Or another puzzle: Let 0.999... = x Subtracting equals from equals: 9.999... = 10x - .999... = -1x 9.000... = 9x Dividing through by 9 1.000... = x But we started with x = 0.999...?! No wonder limits seemed so troubling to Leibniz and Newton in the 17th century.
Shadow Posted June 18, 2011 Posted June 18, 2011 Or another puzzle: Let 0.999... = x Subtracting equals from equals: 9.999... = 10x - .999... = -1x 9.000... = 9x Dividing through by 9 1.000... = x But we started with x = 0.999...?! No wonder limits seemed so troubling to Leibniz and Newton in the 17th century. http://lmgtfy.com/?q=0.999+%3D+1
Marqq Posted June 18, 2011 Posted June 18, 2011 (edited) x = (1/9) 10x = 10(1/9) = 10/9 10x - x = 10/9 - 1/9 9x = 9/9 = 1 x = 1/9 I deny your reality! I get it, but wouldn't the aforementioned proof simply show that special rules must be applied to repeating decimals? I'd like to see how this would transfer to a base 9 system, where .1 = (in base 10) 1/9 ... It seems to me that any mathematical proof should be able to transfer to different number systems as such. Without such transferability, I'm going to have to say that 1/9 is one ninth of 1, and not one ninth of .999..., and that .999... does not equal 1. Also, x = 1/9 = .111... x2 = .0111... x2/x = .1 x = .1 ??? x = 1/9 == .1 9x = 1 == .9 I call fallacy! But I've got to ask, is there any real use for .999... being equal to 1 outside of estimations? Edited June 18, 2011 by Marqq
Bignose Posted June 18, 2011 Posted June 18, 2011 x = 1/9 = .111... x2 = .0111... ... I call fallacy! But I've got to ask, is there any real use for .999... being equal to 1 outside of estimations? perhaps you should check this first. 0.111...2 = 0.0123456... 1
DrRocket Posted June 18, 2011 Posted June 18, 2011 perhaps you should check this first. 0.111...2 = 0.0123456... Rubbish [math] 0.111.... = \frac {1}{9}[/math] as any decent high school algebra student can show in 3 lines. So, [math]0.111... ^2 = \frac {1}{81} = 0.011111.....[/math] On the other hand [math]0.012345601234560123456..... = \frac {123456}{9999999}[/math] and [math]0.1234567890123456789..... = \frac {123456789}{999999999}[/math] So, no matter what reasonable interpretation is placed on "0.123456....." you are incorrect.
Bignose Posted June 18, 2011 Posted June 18, 2011 (edited) So, [math]0.111... ^2 = \frac {1}{81} = 0.011111.....[/math] Seriously? How do you get 1/81 = 0.011111? [math] \frac{1}{81} = 0.\overline{012345679}[/math] 0.0111111 is 1/90. Edited June 18, 2011 by Bignose 2
Cap'n Refsmmat Posted June 18, 2011 Posted June 18, 2011 Rubbish [math] 0.111.... = \frac {1}{9}[/math] as any decent high school algebra student can show in 3 lines. So, [math]0.111... ^2 = \frac {1}{81} = 0.011111.....[/math] Er, no, if [imath]\frac{1}{9} = 0.111\ldots[/imath], then [imath]0.0111\ldots = \frac{1}{90}[/imath]. Quoth the Mathematica: And also: Muphry's law strikes again.
DrRocket Posted June 18, 2011 Posted June 18, 2011 Seriously? How do you get 1/81 = 0.011111? Simple, I screwed up a long division. Actually, 0.0111......... = 1/90 So, (0.011111...... )^2 = 1/8100 = 0.0001234567900123456790012345679...... The repearing numbers are 0012345679 .
Marqq Posted June 19, 2011 Posted June 19, 2011 ok...my goof then...a mathematician I am not... Still, that would've been cool, right? So how about this: x = 1/9 = 0.111... x/10 = 1/90 = 0.0111... x - x/10 = 1/9 - 1/90 = 0.111... - 0.0111... = .1 .1x = 9/90 = .1 ya...well, I tried, and I'll keep trying, because of the fact that it doesn't transfer to other number systems with the same value. I think I'll never trust repeating decimals again... I'll still believe firmly that each value possible (all infinity of them) has only one numerical representation, barring significant digits. I kind of enjoy the fact that this number-system disparity points out that every value is an infinitely repeating decimal, and even an irrational number, when you consider real values of time and distance... I wonder how much better all this would work out if we didn't limit ourselves to working with one number system?
DrRocket Posted June 19, 2011 Posted June 19, 2011 (edited) ok...my goof then...a mathematician I am not... Still, that would've been cool, right? So how about this: x = 1/9 = 0.111... x/10 = 1/90 = 0.0111... x - x/10 = 1/9 - 1/90 = 0.111... - 0.0111... = .1 .1x = 9/90 = .1 ya...well, I tried, and I'll keep trying, because of the fact that it doesn't transfer to other number systems with the same value. I think I'll never trust repeating decimals again... I'll still believe firmly that each value possible (all infinity of them) has only one numerical representation, barring significant digits. I kind of enjoy the fact that this number-system disparity points out that every value is an infinitely repeating decimal, and even an irrational number, when you consider real values of time and distance... I wonder how much better all this would work out if we didn't limit ourselves to working with one number system? It is not true that infinite decimals are unique. There are some exceptions. 1.0000.... = 0.999999.... and variations on that theme. Repeating decimals represent rational numbers and only rational numbers. It is a theorem that a number is rational if and only if the decimal representation is finite or repeating. Irrational numbers have decimal representations that are neither finite nor repeating. Very very little of mathematics is dependent on any number base, so using a different base would not have much impact, and would just be inconvenient. Usually the term "number system", refers to some algebraic structure such as the natural numbers, the integers, the rational numbers, the real numbers, the complex numbers, or even the quaternions, Cayley, numbers, etc. Theree are also some truly abstract systems such as the "non-standard real numbers" and hypercomplex numbers. All of these things have been studied. What might surprise you is that almost all of them are logical consequences of the natural numbers {0,1,2,3,4,...} (some people leave out 0 and add it in later), and basic set theory. This is usually in the form of the Peano Axioms, or in formal set theory the Zermelo-Fraenkel axioms (usually with the axiom of choice added in for more esoteric applications). (The non-standard reals and hypercomplex numbers use a construction that requires the axiom of choice as I recall). For most of mathematics the integers, rationals, reals, complex numbers and things constructed from them with ordinary algebraic constructions (quotients, polynomials, field extensions, etc.) are all that you need. Edited June 19, 2011 by DrRocket
baric Posted June 19, 2011 Author Posted June 19, 2011 Then you agree that 360 degree is not equal to zero. Sorry if you did not get the XBox 360 joke reference.
rktpro Posted June 20, 2011 Posted June 20, 2011 Sorry if you did not get the XBox 360 joke reference. I got it. My sentence was just irony
khaled Posted July 22, 2011 Posted July 22, 2011 first, isn't [math]i = \sqrt{-1}[/math] ? and speaking about rounds, in rounding close numbers are considered the same [math]X1 = 0.1999999999... ~ 0.1999 = X2[/math] [math]X1 >~ X2[/math] [math]X1 * 10^c >> X2 * 10^c[/math] where: [math]c > 0[/math] I think playing with rounding depends on the application decimal degree of power
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