Re: Which one result in maths has surprised you the most?
On Tue, Jul 9, 2013 Jason Resch jasonre...@gmail.com wrote: If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 There is no disputing matters of taste but I think the original equation is more beautiful because it shows a relationship between 5 of the most important numbers in all of mathematics. Your new equation only has 4 important numbers, it doesn't include zero, it has the multiplicative identity but not the additive identity. If you want to see all the constants at once there is an easy correction: e^(t*i) - 1 = 0 Then it has the additive identity but not the multiplicative identity and I still prefer Euler's original. John K Clark -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Thu, Jul 11, 2013 at 10:59 AM, John Clark johnkcl...@gmail.com wrote: On Tue, Jul 9, 2013 Jason Resch jasonre...@gmail.com wrote: If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 There is no disputing matters of taste but I think the original equation is more beautiful because it shows a relationship between 5 of the most important numbers in all of mathematics. Your new equation only has 4 important numbers, it doesn't include zero, it has the multiplicative identity but not the additive identity. If you want to see all the constants at once there is an easy correction: e^(t*i) - 1 = 0 Then it has the additive identity but not the multiplicative identity and I still prefer Euler's original. What is the mutliplicative identity in the original that is missing from this one? Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Thu, Jul 11, 2013 at 1:44 PM, Jason Resch jasonre...@gmail.com wrote: If you want to see all the constants at once there is an easy correction: e^(t*i) - 1 = 0 Then it has the additive identity but not the multiplicative identity and I still prefer Euler's original. What is the mutliplicative identity in the original that is missing from this one? 1. John K Clark -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
1 is in the modified version I provided: e^(t*i) - 1 = 0 Unless you were reading that as e^(t*i) + (-1) = 0 Also, if the more important numbers that can be included, the more beautiful you find the equation, we can also throw in 2, arguably the next most important number: e^(2*t*i) - 1 = 0, but I don't think trying to include as many important numbers into one equation as possible is what makes for an elegant equation. What makes for an elegant equation is showing an important connection between two concepts. e^(t*i) = e^(0) = 1, but t*i != 0. This is much more surprising than if you try the same with Pi, as you will find ln(e^(Pi*i)) = Pi*i, but ln(e^(t*i)) = 0. Jason On Thu, Jul 11, 2013 at 1:46 PM, John Clark johnkcl...@gmail.com wrote: On Thu, Jul 11, 2013 at 1:44 PM, Jason Resch jasonre...@gmail.com wrote: If you want to see all the constants at once there is an easy correction: e^(t*i) - 1 = 0 Then it has the additive identity but not the multiplicative identity and I still prefer Euler's original. What is the mutliplicative identity in the original that is missing from this one? 1. John K Clark -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Thu, Jul 11, 2013 at 3:31 PM, Jason Resch jasonre...@gmail.com wrote: 1 is in the modified version I provided: e^(t*i) - 1 = 0 I only see a -1. 1* X is always equal to X but -1*X is never equal to X unless X=0. John K Clark -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Thu, Jul 11, 2013 at 3:23 PM, John Clark johnkcl...@gmail.com wrote: On Thu, Jul 11, 2013 at 3:31 PM, Jason Resch jasonre...@gmail.com wrote: 1 is in the modified version I provided: e^(t*i) - 1 = 0 I only see a -1. 1* X is always equal to X but -1*X is never equal to X unless X=0. Perhaps this suits you better then: e^(t*i) = 1 + 0 Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On 08 Jul 2013, at 23:22, Johnathan Corgan wrote: On 07/08/2013 02:16 PM, Jason Resch wrote: This one is very interesting, but the fact that Pi was a poor choice for the constant makes the equation considerably more ugly than it should be. There is a growing movement to usurp the number Pi with the much more important constant 2*Pi (see: http://www.math.utah.edu/~palais/pi.html ). If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 I think part of the appeal of the original formulation is realizing that the result of an exponentiation of a positive number can be a negative number. While this is unremarkable with complex exponents, many people are only used to seeing real (or even just integer) exponents. I like and often give the following exercise: compute i^i. Is it real or imaginary? Since sometimes my most amazing result in math is the Turing universality of the diophantine polynomials. My favorite simple result is the irrationality of sqr(2). A good exercise, using the fundamental theorem of arithmetic (existence and uniqueness of decomposition of numbers in prime factors) generalizes this for sqr(n) for any n not being a square. A result often attributed to Theaetetus. Of course the existence of universal numbers is also an amazing, stunning results, especially if you know how weak are the pretense of universality for mathematical notions. This needs Church thesis, which I consider as the most amazing thesis in cognitive science. Bruno Johnathan -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Mon, Jul 8, 2013 at 5:16 PM, Jason Resch jasonre...@gmail.com wrote: I think the fact that e^i*PI +1 = 0 surprises almost everyone when they first hear of it. This one is very interesting, but the fact that Pi was a poor choice for the constant makes the equation considerably more ugly than it should be. There is a growing movement to usurp the number Pi with the much more important constant 2*Pi (see: http://www.math.utah.edu/~palais/pi.html ). If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 There is no disputing matters of taste but I think the original equation is more beautiful because it shows a relationship between 5 of the most important numbers in all of mathematics. Your new equation only has 4 important numbers, it doesn't include zero, it has the multiplicative identity but not the additive identity. John K Clark -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Tue, Jul 9, 2013 at 2:20 PM, John Clark johnkcl...@gmail.com wrote: On Mon, Jul 8, 2013 at 5:16 PM, Jason Resch jasonre...@gmail.com wrote: I think the fact that e^i*PI +1 = 0 surprises almost everyone when they first hear of it. This one is very interesting, but the fact that Pi was a poor choice for the constant makes the equation considerably more ugly than it should be. There is a growing movement to usurp the number Pi with the much more important constant 2*Pi (see: http://www.math.utah.edu/~palais/pi.html ). If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 There is no disputing matters of taste but I think the original equation is more beautiful because it shows a relationship between 5 of the most important numbers in all of mathematics. Your new equation only has 4 important numbers, it doesn't include zero, it has the multiplicative identity but not the additive identity. If you want to see all the constants at once there is an easy correction: e^(t*i) - 1 = 0 Circles are defined by their radius, not their diameter. The mistake of using Pi leads to circles being 2*Pi radians, rather than tau radians. The area formula for circles obscures the fact that an integration took place (1/2) tau r^2 makes it clearer that there was an integration. The period of sin and cosine are tau, cos(t) = 1 rather than -1, etc. Pi is simply a less elegant circle constant than tau. Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
The use of the radius instead of diameter is historic and constructive: the circumference was make by turning a rope or a compass a full turn instead of turning a rigid stick half a turn around his center. The former is easier. 2013/7/9 Jason Resch jasonre...@gmail.com On Tue, Jul 9, 2013 at 2:20 PM, John Clark johnkcl...@gmail.com wrote: On Mon, Jul 8, 2013 at 5:16 PM, Jason Resch jasonre...@gmail.com wrote: I think the fact that e^i*PI +1 = 0 surprises almost everyone when they first hear of it. This one is very interesting, but the fact that Pi was a poor choice for the constant makes the equation considerably more ugly than it should be. There is a growing movement to usurp the number Pi with the much more important constant 2*Pi (see: http://www.math.utah.edu/~palais/pi.html ). If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 There is no disputing matters of taste but I think the original equation is more beautiful because it shows a relationship between 5 of the most important numbers in all of mathematics. Your new equation only has 4 important numbers, it doesn't include zero, it has the multiplicative identity but not the additive identity. If you want to see all the constants at once there is an easy correction: e^(t*i) - 1 = 0 Circles are defined by their radius, not their diameter. The mistake of using Pi leads to circles being 2*Pi radians, rather than tau radians. The area formula for circles obscures the fact that an integration took place (1/2) tau r^2 makes it clearer that there was an integration. The period of sin and cosine are tau, cos(t) = 1 rather than -1, etc. Pi is simply a less elegant circle constant than tau. Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
I think the fact that e^i*PI +1 = 0 surprises almost everyone when they first hear of it. I was surprised to learn that infinity times infinity is just the same old infinity but 2 to the power of infinity yields a larger infinity, and I was surprised to learn that there is a proof that some things are true but a proof that they are true will never be found because it does not exist. Probability gave me a lot of surprises, in grade school I was surprised to learn that in a group of just 23 people the chance that two of them have the same birthday is 50% and with 57 people it's 99%; and it took me an embarrassingly long time to understand the Monty Hall puzzle, the idea that if you change your guess when Monty gives you the opportunity your chance of winning the car behind the door increases from 1 chance in 3 to 2 chances in 3. John K Clark -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On Mon, Jul 8, 2013 at 12:07 PM, John Clark johnkcl...@gmail.com wrote: I think the fact that e^i*PI +1 = 0 surprises almost everyone when they first hear of it. This one is very interesting, but the fact that Pi was a poor choice for the constant makes the equation considerably more ugly than it should be. There is a growing movement to usurp the number Pi with the much more important constant 2*Pi (see: http://www.math.utah.edu/~palais/pi.html ). If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 Jason -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
On 07/08/2013 02:16 PM, Jason Resch wrote: This one is very interesting, but the fact that Pi was a poor choice for the constant makes the equation considerably more ugly than it should be. There is a growing movement to usurp the number Pi with the much more important constant 2*Pi (see: http://www.math.utah.edu/~palais/pi.html ). If we call that new number tau (t). Then Euler's identity becomes: e^(t * i) = 1 I think part of the appeal of the original formulation is realizing that the result of an exponentiation of a positive number can be a negative number. While this is unremarkable with complex exponents, many people are only used to seeing real (or even just integer) exponents. Johnathan -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
Now for me the most surprising thing is Homotophy type theory that unifies spaces, proofs, computations and category theory in a different foundation for mathematics. Redefine a proof as the existence of paths that connect objects in a space with homological properties, but not distances. It is constructive and it is free from the Russell paradox and the Gödel paradox, since type theory where made with this purpose (and set theory is a particular case). http://existentialtype.wordpress.com/2013/06/22/whats-the-big-deal-with-hott/ 2013/7/6 Telmo Menezes te...@telmomenezes.com http://math.stackexchange.com/questions/2949/which-one-result-in-maths-has-surprised-you-the-most -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Which one result in maths has surprised you the most?
That 1+2+3+4+5+..to infinity equals minus 1/12 On Sun, Jul 7, 2013 at 4:40 AM, Alberto G. Corona agocor...@gmail.comwrote: Now for me the most surprising thing is Homotophy type theory that unifies spaces, proofs, computations and category theory in a different foundation for mathematics. Redefine a proof as the existence of paths that connect objects in a space with homological properties, but not distances. It is constructive and it is free from the Russell paradox and the Gödel paradox, since type theory where made with this purpose (and set theory is a particular case). http://existentialtype.wordpress.com/2013/06/22/whats-the-big-deal-with-hott/ 2013/7/6 Telmo Menezes te...@telmomenezes.com http://math.stackexchange.com/questions/2949/which-one-result-in-maths-has-surprised-you-the-most -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Which one result in maths has surprised you the most?
http://math.stackexchange.com/questions/2949/which-one-result-in-maths-has-surprised-you-the-most -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
