[math-fun] Symbolic Calculation
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter: sqrt(8) and it will output: 2*sqrt(2) Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator! However, I then decided to test it: ln(640320^3+744)/sqrt(163) Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered! Hmm... I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!" On this topic, there is this comic: http://xkcd.com/217/ Sincerely, Adam P. Goucher
Your Casio is not the first "symbolic hand calculator"; MuMath was really the first: http://en.wikipedia.org/wiki/MuMATH At 09:46 AM 5/6/2011, Adam P. Goucher wrote:
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter:
sqrt(8)
and it will output:
2*sqrt(2)
Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator!
However, I then decided to test it:
ln(640320^3+744)/sqrt(163)
Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered!
Hmm...
I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!"
On this topic, there is this comic:
Sincerely,
Adam P. Goucher
Yes, but MuMath *actually* performs symbolic calculation (or that's the impression I got, anyway), whereas the Casio must perform an inverse lookup to convert from a decimal approximation to a symbolic expression. Sincerely, Adam P. Goucher
Your Casio is not the first "symbolic hand calculator"; MuMath was really the first:
http://en.wikipedia.org/wiki/MuMATH
At 09:46 AM 5/6/2011, Adam P. Goucher wrote:
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter:
sqrt(8)
and it will output:
2*sqrt(2)
Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator!
However, I then decided to test it:
ln(640320^3+744)/sqrt(163)
Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered!
Hmm...
I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!"
On this topic, there is this comic:
Sincerely,
Adam P. Goucher
You know that the Casio performs an inverse lookup for sure? As an aside, there is an interesting trend in symbolic algebra that's been going on for 35 years or so: doing less traditional algebra & more numerical calculations. E.g., instead of using bignums to compute the inverse of an integer determinant, compute the determinant mod p for enough p's; using "black box programs" to compute polynomials & computing the coefficients (if you really want them) by interpolating enough point values. Part of this trend is driven by necessity: modern computers are far better at numerical calculations than symbolic calculations, & part is driven by good mathematics & computational complexity. Mathematicians & physicists have a long history of doing numerical calculations & then guessing the correct symbolic formula. In some cases (Newton springs to mind), the data are then fudged to make the symbolic formula look even better. At 10:20 AM 5/6/2011, Adam P. Goucher wrote:
Yes, but MuMath *actually* performs symbolic calculation (or that's the impression I got, anyway), whereas the Casio must perform an inverse lookup to convert from a decimal approximation to a symbolic expression.
Sincerely,
Adam P. Goucher
Your Casio is not the first "symbolic hand calculator"; MuMath was really the first:
http://en.wikipedia.org/wiki/MuMATH
At 09:46 AM 5/6/2011, Adam P. Goucher wrote:
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter:
sqrt(8)
and it will output:
2*sqrt(2)
Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator!
However, I then decided to test it:
ln(640320^3+744)/sqrt(163)
Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered!
Hmm...
I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!"
On this topic, there is this comic:
Sincerely,
Adam P. Goucher
I'd buy a handheld Plouffe Inverter. On Fri, May 6, 2011 at 10:29 AM, Henry Baker <hbaker1@pipeline.com> wrote:
You know that the Casio performs an inverse lookup for sure?
As an aside, there is an interesting trend in symbolic algebra that's been going on for 35 years or so: doing less traditional algebra & more numerical calculations. E.g., instead of using bignums to compute the inverse of an integer determinant, compute the determinant mod p for enough p's; using "black box programs" to compute polynomials & computing the coefficients (if you really want them) by interpolating enough point values.
Part of this trend is driven by necessity: modern computers are far better at numerical calculations than symbolic calculations, & part is driven by good mathematics & computational complexity.
Mathematicians & physicists have a long history of doing numerical calculations & then guessing the correct symbolic formula. In some cases (Newton springs to mind), the data are then fudged to make the symbolic formula look even better.
At 10:20 AM 5/6/2011, Adam P. Goucher wrote:
Yes, but MuMath *actually* performs symbolic calculation (or that's the impression I got, anyway), whereas the Casio must perform an inverse lookup to convert from a decimal approximation to a symbolic expression.
Sincerely,
Adam P. Goucher
Your Casio is not the first "symbolic hand calculator"; MuMath was really the first:
http://en.wikipedia.org/wiki/MuMATH
At 09:46 AM 5/6/2011, Adam P. Goucher wrote:
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter:
sqrt(8)
and it will output:
2*sqrt(2)
Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator!
However, I then decided to test it:
ln(640320^3+744)/sqrt(163)
Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered!
Hmm...
I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!"
On this topic, there is this comic:
Sincerely,
Adam P. Goucher
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Henry Baker wrote:
You know that the Casio performs an inverse lookup for sure?
The expression: ln(640320^3+744)/sqrt(163) is an *approximation* for pi, accurate to 31 decimal places. As the calculator prints the symbol for pi when given this expression as input, it must be performing an inverse lookup. Sincerely, Adam P. Goucher
As an aside, there is an interesting trend in symbolic algebra that's been going on for 35 years or so: doing less traditional algebra & more numerical calculations. E.g., instead of using bignums to compute the inverse of an integer determinant, compute the determinant mod p for enough p's; using "black box programs" to compute polynomials & computing the coefficients (if you really want them) by interpolating enough point values.
Part of this trend is driven by necessity: modern computers are far better at numerical calculations than symbolic calculations, & part is driven by good mathematics & computational complexity.
Mathematicians & physicists have a long history of doing numerical calculations & then guessing the correct symbolic formula. In some cases (Newton springs to mind), the data are then fudged to make the symbolic formula look even better.
At 10:20 AM 5/6/2011, Adam P. Goucher wrote:
Yes, but MuMath *actually* performs symbolic calculation (or that's the impression I got, anyway), whereas the Casio must perform an inverse lookup to convert from a decimal approximation to a symbolic expression.
Sincerely,
Adam P. Goucher
Your Casio is not the first "symbolic hand calculator"; MuMath was really the first:
http://en.wikipedia.org/wiki/MuMATH
At 09:46 AM 5/6/2011, Adam P. Goucher wrote:
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter:
sqrt(8)
and it will output:
2*sqrt(2)
Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator!
However, I then decided to test it:
ln(640320^3+744)/sqrt(163)
Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered!
Hmm...
I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!"
On this topic, there is this comic:
Sincerely,
Adam P. Goucher
The expression:
ln(640320^3+744)/sqrt(163)
is an *approximation* for pi, accurate to 31 decimal places. As the calculator prints the symbol for pi when given this expression as input, it must be performing an inverse lookup.
I'm not sure this constitutes proof. I'd love to get my hands on that calculator to see what sort of expressions it can reduce to symbols, and what it can't. And how accurate the result has to be. I mean, certainly it doesn't have a lookup table that handles, for instance, sqrt(102305609)? -tom
Tom (and everyone else), sqrt(998) returns the value of 'sqrt(998)', whereas sqrt(1001) returns the value of '31.63858404', so the lookup table for square roots seems to be limited to 1000. As for how precise the results need to be, it is convinced by: ln(640320^3+744)/sqrt(163) but not by: sqrt(sqrt(9^2+19^2/22)) both of which, incidentally, were discovered by Ramanujan. Sincerely, Adam P. Goucher ----- Original Message ----- From: "Tom Rokicki" <rokicki@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Cc: "Henry Baker" <hbaker1@pipeline.com> Sent: Friday, May 06, 2011 7:00 PM Subject: Re: [math-fun] Symbolic Calculation
The expression:
ln(640320^3+744)/sqrt(163)
is an *approximation* for pi, accurate to 31 decimal places. As the calculator prints the symbol for pi when given this expression as input, it must be performing an inverse lookup.
I'm not sure this constitutes proof.
I'd love to get my hands on that calculator to see what sort of expressions it can reduce to symbols, and what it can't. And how accurate the result has to be.
I mean, certainly it doesn't have a lookup table that handles, for instance, sqrt(102305609)?
-tom
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* Henry Baker <hbaker1@pipeline.com> [May 06. 2011 20:25]:
You know that the Casio performs an inverse lookup for sure?
As an aside, there is an interesting trend in symbolic algebra that's been going on for 35 years or so: doing less traditional algebra & more numerical calculations. E.g., instead of using bignums to compute the inverse of an integer determinant, compute the determinant mod p for enough p's; using "black box programs" to compute polynomials & computing the coefficients (if you really want them) by interpolating enough point values.
Note this method (compute mod enough coprime moduli, then CRT) is 1) exact 2) bloody fast
Part of this trend is driven by necessity: modern computers are far better at numerical calculations than symbolic calculations, & part is driven by good mathematics & computational complexity.
Computers could be good at both. Humans (programmers) just do not happen to double in coding capacity every 18 months. Machines have left humans 20 years behind when it comes to exploiting the full potential of the machines. That gap is widening. Also the invention of algorithms tends to be time consuming...
Mathematicians & physicists have a long history of doing numerical calculations & then guessing the correct symbolic formula. In some cases (Newton springs to mind), the data are then fudged to make the symbolic formula look even better.
[...]
cheers, jj
On 6 May 2011, at 19:53, Joerg Arndt wrote:
* Henry Baker <hbaker1@pipeline.com> [May 06. 2011 20:25]:
You know that the Casio performs an inverse lookup for sure?
As an aside, there is an interesting trend in symbolic algebra that's been going on for 35 years or so: doing less traditional algebra & more numerical calculations. E.g., instead of using bignums to compute the inverse of an integer determinant, compute the determinant mod p for enough p's; using "black box programs" to compute polynomials & computing the coefficients (if you really want them) by interpolating enough point values.
Note this method (compute mod enough coprime moduli, then CRT) is 1) exact 2) bloody fast
Where can I find details of (compute mod enough coprime moduli, then CRT) ? (that someone with only UK "A" level math can understand) (I mean for example - what's CRT here ?)
Knuth's Art of Computer Programming, among others. I would hope that it is accessible to UK "A" levels! At 12:37 PM 5/6/2011, David Makin wrote:
On 6 May 2011, at 19:53, Joerg Arndt wrote:
* Henry Baker <hbaker1@pipeline.com> [May 06. 2011 20:25]:
You know that the Casio performs an inverse lookup for sure?
As an aside, there is an interesting trend in symbolic algebra that's been going on for 35 years or so: doing less traditional algebra & more numerical calculations. E.g., instead of using bignums to compute the inverse of an integer determinant, compute the determinant mod p for enough p's; using "black box programs" to compute polynomials & computing the coefficients (if you really want them) by interpolating enough point values.
Note this method (compute mod enough coprime moduli, then CRT) is 1) exact 2) bloody fast
Where can I find details of (compute mod enough coprime moduli, then CRT) ? (that someone with only UK "A" level math can understand) (I mean for example - what's CRT here ?)
On Fri, May 6, 2011 at 3:37 PM, David Makin <makinmagic@tiscali.co.uk> wrote:
(I mean for example - what's CRT here ?)
http://en.wikipedia.org/wiki/Chinese_Remainder_Theorem -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Thanks Mike and Tom :) On 7 May 2011, at 01:06, Mike Stay wrote:
On Fri, May 6, 2011 at 3:37 PM, David Makin <makinmagic@tiscali.co.uk> wrote:
(I mean for example - what's CRT here ?)
http://en.wikipedia.org/wiki/Chinese_Remainder_Theorem -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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* David Makin <makinmagic@tiscali.co.uk> [May 07. 2011 07:25]:
On 6 May 2011, at 19:53, Joerg Arndt wrote: [...]
Where can I find details of (compute mod enough coprime moduli, then CRT) ? (that someone with only UK "A" level math can understand) (I mean for example - what's CRT here ?)
%% Must have: Henri Cohen: A Course in Computational Algebraic Number Theory, Springer-Verlag, (1993). List of errata URL: http://www.ufr-mi.u-bordeaux.fr/~cohen/ %% there is a newer edition %% Must have: Joachim von zur Gathen, J\"{u}rgen Gerhard: Modern Computer Algebra, Cambridge University Press, second edition, (2003). List of errata URL: http://www-math.upb.de/mca/ %% Also excellent: Paulo Ribenboim: The Little Book of Bigger Primes, second Edition, Springer-Verlag, (2004). %% Also excellent: Richard P.\ Brent, Paul Zimmermann: Modern Computer Arithmetic, Cambridge University Press, (2010) Online: http://arxiv.org/abs/1004.4710 %% Also, from the implementation perspective: Documentation of GMP %% Lesser, but might be quite accessible: http://www.jjj.de/fxt/#fxtbook section 39.4, pp.772-774
[...]
cheers, jj
Part of this trend is driven by necessity: modern computers are far better at numerical calculations than symbolic calculations, & part is driven by good mathematics & computational complexity.
Computers could be good at both. Humans (programmers) just do not happen to double in coding capacity every 18 months. Machines have left humans 20 years behind when it comes to exploiting the full potential of the machines.
That gap is widening.
Also the invention of algorithms tends to be time consuming...
However as we also learn more about our own abilities even given the advance in raw power of computer hardware we're still just as far as ever from being able to use a computer to replicate the majority of what the human brain is capable of :)
I tried the same thing, worked for me as well. In fact your approximation is quite a bit more precise than it need to be, you can completely remove the +744 and it still works. On an unrelated note, just saw that Balkan Questions have gone online if anyone's interested. On Fri, May 6, 2011 at 5:46 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter:
sqrt(8)
and it will output:
2*sqrt(2)
Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator!
However, I then decided to test it:
ln(640320^3+744)/sqrt(163)
Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered!
Hmm...
I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!"
On this topic, there is this comic:
Sincerely,
Adam P. Goucher
participants (8)
-
Adam P. Goucher -
ben elliott -
David Makin -
Henry Baker -
Joerg Arndt -
Mike Stay -
Thane Plambeck -
Tom Rokicki