There was a mathematical expression that made the email rounds a few years ago. It was sort of a trick. The expression was something like 5 / ( sqrt(13) + 1/7) and it evaluated to something like 10.0000000000005776893… if you had an accurate enough computation means, which looks like exactly 10 on calculators and computer programs that provide around 12 digits of precision. The point was to challenge the victim to show mathematically that the expression equalled 10, whereas it didn’t look like it should equal any integer, and the victim would check with his calculator and conclude that oddly enough the stated equality was correct and then go nuts trying to prove it.
Only, now, I am going nuts trying to find where I put it. Does anybody know this one? WHAT is that expression???
Could this be what you’re thinking of:
e[sup]pi*sqrt(163)[/sup] = 262537412640768743.99999999999925…
Cabbage, that isn’t the one that I saw, but it is certainly an example of the same kind of thing. In fact it’s the only example I have, now that you have given it to me. Thanks!
I’m pretty sure the one I saw evaluated to 10 plus about 1e-12, which made it more impressive (not many digits before the zeroes).
I don’t have an arbitrary precision calculating program that includes the exp() funciton - did you verify the evaluation of your expression?
Yeah, I verified the example I gave, it’s a pretty well known approximation.
It’d be virtually impossible to come up with the specific one you saw without having seen it before. There are ways to make these things up by yourself, relying on the fact that you can approximate any real number with rationals.
For a basic example, take sqrt(13), which is about 3.60555, (so I’ve approximated it with a rational). So from that I can play around with it a little and get 5757/250 - 5*sqrt(13) = 5.00024… just from doing a little algebra with my original approximation.
Not as nice as yours, since the numbers are so big, and it’s not as close to 5 as we’d like it to be, but I guess you could play around with that idea some and see if anything comes up. You could also try Newton’s method and continued fractions as well, I suppose, if you’re familiar with those, to get an approximation to work with.
Sorry I can’t be of any more help.
I think I know an equivalent test. Try:
80/81
This number, given enough precision will evaluate to something like 0.98765432109876543210987654321… and it’s the easiest to remember irrational number of its type.
I hope that’s right, might be .987654321987… with no zero in the repeating sequence, but I don’t have access to anything with higher precision math at the moment that I can check it.
Some calculators will choke after .987654 since they only carry 6 digits after the decimal point. To test that, multiply by 10s, i.e. 100x will probably evaluate to 98.76540 on a badly designed calculator.
If it’s the ratio of two integers, it’s a rational number, not an irrational number.
According to my 100 digit calculating utility, 80/81 evaluates to:
0.98765432098765432098765432098765432098765432098765
432098765432098765432098765432098765432098765432
so that’s pretty neat. It does skip the ones, though.
I’m pretty sure the one I am looking for had perhaps 5 small integers in it, maybe including 5 and 13. I think it was a fraction, and there was a square root in the denominator.
Shoot. I even had it taped to the wall for a while, but that was about ten offices ago…
This page on “almost integers” may be of interest:
http://br.crashed.net/~akrowne/crc/math/a/a155.htm
It doesn’t seem to include what you’re describing, unfortunately (it has some very interesting ones, however), but it looks like it could be a good lead to find what you’re looking for.
The following contains sqrt(13) and is quite close but not equal to 10:
1/(130 - 36*sqrt(13) - 65/649)
How about (3 + x[sup]3[/sup])/x where x=(sqrt(13) - 3)/2 ?
Since, as pointed out, 80/81 is rational, you don’t need a 100 digit calculator to find its decimal expansion; it will repeat.
yabut, the whole point is to prove that your average pocket calculator doesn’t carry more than 8 or 9 digits of precision so the best you’ll ever get is .987654321.
Here’s a cute one: sqrt(sqrt(2143/22)))
I took the one Manlob used, which is 1/(130 - 36*sqrt(13) - 65/649) and edited the last part, which is the easiest part to change. However, I don’t think this is what you’re looking for… but it does contain 12 zeros after the decimal.
1/(130 - 36*sqrt(13) - 6500000005936/64900000000000) = 10.000000000000900529658812589672760818115884998753062101439096510910635821539150369673119868097135678822905063041783724424166337255849668649424474369073306133947393150786962985088591119926049582880110
You could always of course, get more and more zeros just by changing the last fraction. FTR, the evaluated expression is to 200 decimals.
This is something interesting about 80/81.
I have a TI-83 Graphing Calculator, which I thought was fairly precise. I multiplied 81 by .987654321 and it gave me 80. Which is weird, I expected it to give something like this: 80.000000001. I could probably trick a couple people with this.
manlob:
I liked that last one :-).
cute.
-luckie
Actually, it gives just that. Using Mathematica, I get 80.000000001 followed by at least a million zeros.