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Things I learned recently...

91Joe95

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Aug 15, 2003
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Why bicycles are as stable as they are - scientists don't actually know.



Volatile singer Johnny Rotten (John Lydon), lead singer of the Sex Pistols, is actually a homebody, married since 1979. I would not have guessed that one.

 
Why bicycles are as stable as they are - scientists don't actually know.



Volatile singer Johnny Rotten (John Lydon), lead singer of the Sex Pistols, is actually a homebody, married since 1979. I would not have guessed that one.


You can change your BWI username every month with little effort using the ‘Account Details’ option under Settings. So, don’t even need a new email address.
 
I'm assuming i isn't the square root of -1. I threw that equation into my phone calculator using both base 10 and natural log with i=5 and a few other numbers and got back undefined.
Yes, i=sqrt(-1). It’s a complex logarithm. I just now took down a complex analysis book and I’ll get to the bottom of it.
 
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Learning is hard. Fortunately, the interwebs makes it easier.
Unfortunately, while making overall learning easier, the interwebs make learning accurate information much more difficult. ;)
 
Why bicycles are as stable as they are - scientists don't actually know.



Volatile singer Johnny Rotten (John Lydon), lead singer of the Sex Pistols, is actually a homebody, married since 1979. I would not have guessed that one.

Is learning that you didn't learn something actually learning?
 
Something most people will learn eventually....the longer you live, the more loved ones, friends and acquaintances you will lose. Instead most humans you interact with will be strangers. You will wake up in the morning and realize that the world you once knew is largely gone forever.
Just when you thought you would cross the finish line ,and qualify for a medal, all the rules have been changed.
 
Is that derivable from Euler's formula? Or totally independent?

Yes, it is derivable from Euler's formula. I fooled around with this a bit this afternoon and the derivation is as follows:



244165458_10209149304847047_612029841383158570_n.jpg
 
Yes, it is derivable from Euler's formula. I fooled around with this a bit this afternoon and the derivation is as follows:



244165458_10209149304847047_612029841383158570_n.jpg
That must have been fun. It would have been helpful for me to have thought to use the fact that C (the set of complex numbers) is a field, and so it’s easy to show that (1-i)/(1+i)=-i. (Just multiply by 1= (1-i)/(1-i).) Once you do this, it’s easy to show that e^(-i*pi/2)=-i (via Euler’s formula) and, bang, you’re done. I’m out of practice.
 
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Why bicycles are as stable as they are - scientists don't actually know.



Volatile singer Johnny Rotten (John Lydon), lead singer of the Sex Pistols, is actually a homebody, married since 1979. I would not have guessed that one.

Racecar spelled backwards is Racecar….
 
That must have been fun. It would have been helpful for me to have thought to use the fact that C (the set of complex numbers) is a field, and so it’s easy to show that (1-i)/(1+i)=-i. (Just multiply by 1= (1-i)/(1-i).) Once you do this, it’s easy to show that e^(-i*pi/2)=-i (via Euler’s formula) and, bang, you’re done. I’m out of practice.
So Euler published his revolutionary and brilliant formula ( exp(ix) = cos(x) + i sin (x) ) in 1714. Giulio Fagnano "discovered" his formula in 1750. But a jabroni like me derived Fagnano's formula from Euler's equation in about 45 minutes using high school algebra. I would conclude that Fagnano is a fraud and should not be given any credit for his "discovery". I bet if Fagnano was around today he would likely be a pitt football fan. Trying to make something special out of nothing. Fagnano's "accomplishment" is like winning games in the ACC Coastal -- essentially useless and of no value.
 
The closest state capital to Hamilton, Bermuda is Providence, RI
Parts of Southwest Virginia are closer to 9 other state capitals than it is to Richmond.

Lee County is 350 miles from Richmond. But only 129 miles from Frankfort, Kentucky; 164 miles from Charleston, West Virginia; 177 miles from Nashville, Tennessee; 201 miles from Atlanta, Georgia; 233 miles from Columbia, South Carolina; 234 miles from Columbus, Ohio; 257 miles from Indianapolis, Indiana; 286 miles from Raleigh, North Carolina and 329 miles from Montgomery, Alabama.
 
Yeah, once I gave this a little thought, no, they’re not equivalent. (Meaning, you can’t get Euler’s formula from this identity.) One of the stupider things I’ve ever put my name to. Oh well, math teaches you humility.
Fagnano's equation is simply an instance of Euler's universal formula. You get Fagnano's equation by simply setting x = Pi/2 in the Euler Equation and doing a bit of high school algebra. Fagnano's equation is useless. It is like somebody publishing an equation using E = m x c^2 while setting m = 5 kg and then saying they discovered something new. Fagnano's equation should be dismissed and ignored.
 
Fagnano's equation is simply an instance of Euler's universal formula. You get Fagnano's equation by simply setting x = Pi/2 in the Euler Equation and doing a bit of high school algebra. Fagnano's equation is useless. It is like somebody publishing an equation using E = m x c^2 while setting m = 5 kg and then saying they discovered something new. Fagnano's equation should be dismissed and ignored.
I suspect there's more to this than what meets the eye.

Fagnano's equation is no more or less "useless" than

IaVh3s-Aq_qZrPTIPb23ulgZ9CI0zW3a6f9R0uB-ax5NU359aZ5Sn8RCxlbd_7-8-KCitCE3ShWbiwqOVEVal2dwso3ltso1f7DMff9txuKA5iBfUppDsFCisjeYGM0

or (due to Ramanujan)

Ramanujan.jpg


or (due to Wallis)
maxresdefault.jpg


Formulas like this aren't used to calculate the digits of pi; for example, pi had been calculated to 112 digits by 1719. At this level, the knowledge of the specific digits has no practical use. It's just mathematical knowledge. Here's an interesting link discussing this.


Euler finalized Euler's formula in 1740 (not 1714), when Fagnano was 58. The 1750 date for Fagnano's publication is his collected works, so he must have proved this well before 1750. I'm trying to figure out exactly how Fagnano proved this, and the reason I think he might have been able to do this without using Euler's formula is that he uses ln((1-i)/(1+i)) instead of ln(-i), which is what it's equal to. If he was going to use Euler's formula to establish this identity, then he would have used ln(-i) (which is what you and, eventually, I used) and not ln((1-i)/(1+i)). So, to me, the fact that he uses ln((1-i)/(1+i)) is a hint that he did NOT use Euler's formula here. The thing is that ln((1-i)/(1+i))=ln(1-i)-ln(1+i) and the MacLaurin series for ln(1+x) was pretty well known by then. You have

main-qimg-e1108a8b46a62c40175367a30b68249e


without using Euler's formula.

It's an interesting discussion. Fagnano's reputation doesn't rest on this formula anyway, as he did a lot of other stuff.

Nice discussion, thanks.
 
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I suspect there's more to this than what meets the eye.

Fagnano's equation is no more or less "useless" than

IaVh3s-Aq_qZrPTIPb23ulgZ9CI0zW3a6f9R0uB-ax5NU359aZ5Sn8RCxlbd_7-8-KCitCE3ShWbiwqOVEVal2dwso3ltso1f7DMff9txuKA5iBfUppDsFCisjeYGM0

or (due to Ramanujan)

Ramanujan.jpg


or (due to Wallis)
maxresdefault.jpg


Formulas like this aren't used to calculate the digits of pi; for example, pi had been calculated to 112 digits by 1719. At this level, the knowledge of the specific digits has no practical use. It's just mathematical knowledge. Here's an interesting link discussing this.


Euler finalized Euler's formula in 1740 (not 1714), when Fagnano was 58. The 1750 date for Fagnano's publication is his collected works, so he must have proved this well before 1750. I'm trying to figure out exactly how Fagnano proved this, and the reason I think he might have been able to do this without using Euler's formula is that he uses ln((1-i)/(1+i)) instead of ln(-i), which is what it's equal to. If he was going to use Euler's formula to establish this identity, then he would have used ln(-i) (which is what you and, eventually, I used) and not ln((1-i)/(1+i)). So, to me, the fact that he uses ln((1-i)/(1+i)) is a hint that he did NOT use Euler's formula here. The thing is that ln((1-i)/(1+i))=ln(1-i)-ln(1+i) and the MacLaurin series for ln(1+x) was pretty well known by then. You have

main-qimg-e1108a8b46a62c40175367a30b68249e


without using Euler's formula.

It's an interesting discussion. Fagnano's reputation doesn't rest on this formula anyway, as he did a lot of other stuff.

Nice discussion, thanks.
Thanks for the insight. I supposed I was a bit harsh on Mr. Fagnano.

Is there a practical application of Fagnano's equation or is it just an interesting abstraction? You cite the various equations that can be used to calculate Pi. Can an expression containing i be used for any practical calculation, like computing Pi?
 
Thanks for the insight. I supposed I was a bit harsh on Mr. Fagnano.

Is there a practical application of Fagnano's equation or is it just an interesting abstraction? You cite the various equations that can be used to calculate Pi. Can an expression containing i be used for any practical calculation, like computing Pi?
I’d be surprised if it’s more than an interesting abstraction. Honestly, this is the first time I’ve ever seen an identity for pi which involves i, which is what makes it so fascinating. As to your last question, I just finished reading a book on quantum physics which points out that if you use complex numbers, Schrodinger’s equation is time-reversing and then a lot of the complexities of quantum physics drop away.
 
The term imaginary numbers always bothered me. There's only one, "i". Every other one is just a multiple of it. There's no r, or z. I think there should be a z, maybe even an f to describe some imaginary friends. Unfortunately this highly accurate Wikipedia page both contradicts and supports my argument. Talk about an asshole.

 
The term imaginary numbers always bothered me. There's only one, "i". Every other one is just a multiple of it. There's no r, or z. I think there should be a z, maybe even an f to describe some imaginary friends. Unfortunately this highly accurate Wikipedia page both contradicts and supports my argument. Talk about an asshole.

Cool, I did not know that it was Descartes who came up with the term “imaginary” numbers. (I had always attributed it to Gauss.)

As the board should know, I taught mathematics at Gallaudet and my mode of communication was American Sign Language. As you would imagine, ASL has its own signs for “real,” “complex,” and “imaginary,” corresponding to their normal English (non-math) usage. (Like, “Boy, that’s a really complex problem,” or “Do you still talk to your imaginary friend?”) Because these words used in the math sense are not descriptive, I would never use these signs in my classes, and would instead spell out the words real, complex, and imaginary. I would tell my students, “Complex (spelled out) numbers are not complex (signed).” It became a running joke with my students.
 
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Cool, I did not know that it was Descartes who came up with the term “imaginary” numbers. (I had always attributed it to Gauss.)

As the board should know, I taught mathematics at Gallaudet and my mode of communication was American Sign Language. As you would imagine, ASL has its own signs for “real,” “complex,” and “imaginary,” corresponding to their normal English (non-math) usage. (Like, “Boy, that’s a really complex problem,” or “Do you still talk to your imaginary friend?”) Because these words used in the math sense are not descriptive, I would never use these signs in my classes, and would instead spell out the words real, complex, and imaginary. I would tell my students, “Complex (spelled out) numbers are not complex (signed).” It became a running joke with my students.
It's a shame that Descartes coined the term "imaginary" for numbers involving i. Many people who haven't spent much time with complex numbers conclude that since they are "imaginary" they are not useful or somehow inferior to "real" numbers. Maybe Descartes should have called these numbers "abstract". Then we could have real numbers and abstract numbers rather than real numbers and imaginary numbers.
 
I’d be surprised if it’s more than an interesting abstraction. Honestly, this is the first time I’ve ever seen an identity for pi which involves i, which is what makes it so fascinating. As to your last question, I just finished reading a book on quantum physics which points out that if you use complex numbers, Schrodinger’s equation is time-reversing and then a lot of the complexities of quantum physics drop away.
I work as a mechanical and nuclear engineer. My electrical engineering buddies use complex numbers all the time to solve electrical engineering problems.
 
It's a shame that Descartes coined the term "imaginary" for numbers involving i. Many people who haven't spent much time with complex numbers conclude that since they are "imaginary" they are not useful or somehow inferior to "real" numbers. Maybe Descartes should have called these numbers "abstract". Then we could have real numbers and abstract numbers rather than real numbers and imaginary numbers.
I have a complex variables textbook which called Gauss’ use of the word “unfortunate.”
 
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