Can't find a particular example: two numbers very close together

[drpizza]

I'm looking for a particular (actually, there are probably several) of two numbers that are "extremely close together" - by which I mean they share the first 15 or so significant digits before they differ from each other. I've seen at least one really excellent example of something like e^2pi vs. some square root or something.

i.e.
simple expression equals something like
29.140293501160298310107002522...
and a different simple expression equals something like
29.140293501160298310107086610...
(the first 24 digits are identical; then they are different.)

I thought I had seen in on the Wolfram site, possibly in a discussion of extra precision, but have been searching quite unsuccessfully.

Thanks.

 

[HallsofIvy]

I'm looking for a particular (actually, there are probably several) of two numbers that are "extremely close together" - by which I mean they share the first 15 or so significant digits before they differ from each other. I've seen at least one really excellent example of something like e^2pi vs. some square root or something.

i.e.
simple expression equals something like
29.140293501160298310107002522...
and a different simple expression equals something like
29.140293501160298310107086610...
(the first 24 digits are identical; then they are different.)

I thought I had seen in on the Wolfram site, possibly in a discussion of extra precision, but have been searching quite unsuccessfully.

Thanks.

??What exactly do you want? If by "extremely close together" you mean, as you say, "they share the first 15 or so significant digits before they differ from each other", just write a decimal point, whatever 15 digits you want, then write another decimal point, the same 15 digits, followed by a "1". Here's an example:
0.134323434343433 and 0.1343234343434331.

Perhaps I am missing something?

 

[matt grime]

I think the OP wants some examples of this phenomenon that aren't artificial, or involve constants that mean something, like e, or pi, or the golden ratio. There are some. One that is accurate to 5 decimal places:

the fine structure constant v. 1/137, the both start 0.00729...one i definitely recall being mentioned in this forum is someone asking how you prove that exp{some expression} equalled pi to the power something. Not very helpful for seraching. Also, it wasn't true - they weren't equal but they were close.

Anyone want a bit of a laugh might like to read this (crank) page http://ebtx.com/ntx/ntx33a1.htm which I came across whilst trying to google for a 'near algebraic relation pi e' or something.

 

[drpizza]

Yes, what I'm referring to is what Matt Grime mentioned... there was some example, and if I recall correctly, it had pi to some power and some other expression. The context of that example was showing how double-precision numbers or some such other tool were needed when using some particular piece of computer software.

 

[CRGreathouse]

Yes, what I'm referring to is what Matt Grime mentioned... there was some example, and if I recall correctly, it had pi to some power and some other expression. The context of that example was showing how double-precision numbers or some such other tool were needed when using some particular piece of computer software.

Try
http://mathworld.wolfram.com/AlmostInteger.html

 

[mathwonk]

how about 10^(-16) and zero?

or [e^(-pi)]^(pi)^2 and zero?

 

[drpizza]

Here's one similar to the one specific example that I thought was fairly well known:

pi^(e sqrt(163)) = 2622537412640768743.99999999999925
At first it looks like the value is going to be an integer. Twelve 9's before there's something else.

The reason I was looking for the particular example was to show my students that within the limits of their calculators, just because two things look like they're equal, doesn't guarantee that it's necessarily so. I'm trying to get some of them to develop more of a curiosity toward problems that appear strange, or more of a desire to seek more evidence than just numerical evidence. i.e. why should we prove a trig identity; the two graphs look like they overlap perfectly? (of course, if we graphed two functions and they clearly didn't overlap, that'd probably be enough for us to conclude that there was no identity.)

Something I did today was putting points on the circumference of a circle and dividing a circle into regions by drawing secants. With 1 point, there's no line so, there's 1 region, of course. With 2 points, there are 2 regions. 3 points, 4 regions; 4 points, 8 regions. 5 points: up to 16 regions. So, follow the pattern...
1,2,4,8,16, how many regions for 6 points? Weird, I keep trying, but I always end up with less than 32. Can we draw a circle 6 points on the circumference connected in such a way that the circle is broken into 32 regions? Is it possible? From this, I'm trying to get them more curious and willing to work toward solutions which may end up being more difficult than they expected up front.

 

[CRGreathouse]

Here's one similar to the one specific example that I thought was fairly well known:

pi^(e sqrt(163)) = 2622537412640768743.99999999999925

Yes, I think that's one of the examples used in the MathWorld site I linked to.

 

[Eighty]

These almost-counter examples to Fermat's theorem were in Simpsons:
$$(1782^{12} + 1841^{12})^{1/12}\approx1921.99999995587$$
$$(3987^{12} + 4365^{12})^{1/12}\approx4472.00000000706$$

 

[CRGreathouse]

See also http://en.wikipedia.org/wiki/Mathematical_coincidence .

 

[robert Ihnot]

drpizza: pi^(e sqrt(163)) = 2622537412640768743.99999999999925
At first it looks like the value is going to be an integer. Twelve 9's before there's something else.

This example, I think, was a conjecture by Ramanujan that it was actually an integer. It must be very difficult to calculate that by hand.

 

[Zurtex]

Here's some really good ones: http://mathworld.wolfram.com/PiApproximations.html

I think my favorite is:

$$(\pi + 20)^i \approx -1$$

Also if it's not on there, another good one is:

$$\left(9^2 + \frac{19^2}{22}\right)^{\frac{1}{4}} = 3.14159265258265\ldots$$

 

[Alkatran]

$$e^{\pi i} = -1$$

 

[Gib Z]

Here's one similar to the one specific example that I thought was fairly well known:

pi^(e sqrt(163)) = 2622537412640768743.99999999999925
At first it looks like the value is going to be an integer. Twelve 9's before there's something else.

Would you think I was quite a calculating machine if I said that $$2622537412640768744 = 640320^3 +744$$?

I happen to be able to say that because what drpizza just said comes from the fields of elliptic curves and complex multiplication which gives the following approximation for pi, correct to 30 digits:

$$\frac{ \log_e (640320^3 +744)}{\sqrt{163}}$$.

To the OP, why not try ...

$$\Gamma(z) \approx \sqrt{\frac{2 \pi}{z} } \left( \frac{1}{e} \left( z + \frac{1}{12z- \frac{1}{10z}} \right) \right)^{z}$$

Its a great approximation, accurate to more than 8 places for z with real part >8. :D
If you want a specific example that everyone can try, let z=70 (since 69! is the largest a calculator screen can display).