Math question (part 4)

Water Cooler
#1

This isn’t a joke or riddle. It’s a real question for all the math geniuses (which I am not). Well, actually three questions.

How can you tell if a number is divisible by 7? I know how to tell with 2 - you just see if it’s an odd or even number. 3 is also easy–if the sum of the digits is divisible by 3, then the whole number is divisible by 3. What’s the rule for 7?

Also, how can you “back in” to a fraction? EG - I’m looking at my calculator, and it says “97.3571428471”. What is the original fraction? The numerator and denominator? (For the record, it’s 1363/14.) Is there some kind of scientific way to do that?

Also, how does one calculate pi (in real life)? I know it’s 3.14159… And people have apparently calculated it out to the 5 billionth decimal place or something. But on some very micro level, is it really possible to get accurate calculations for circumference and diameter? Accurate enough to calculate pi out to the 5 billionth decimal place?

A couple of things that have been bothering me lately. Any answers will help me sleep tonight.

#2

Seven’s not particularly elegant.

If you know the full sequence of repeating digits, it’s pretty easy.

In your example, the repeating digits are: 714285 (note that there’s a typo in your original number). The sequence is 6 digits long. So, if you multiply the number by 106 (essentially, moving the decimal point 6 places to the right, so that the repeating sequences line up), then subtract the original number from that, the repeating digits will all cancel, leaving you with an easy problem to solve. Here goes:

x = 97.35714285714385 . . .
1,000,000x = 97,357,142.85714385 . . .
999,999x = 97,357,142.85714385 . . . - 97.35714285 . . . = 97,357,045.5

Multiplying by 10 (to get rid of the decimal):

9,999,990x = 973,570,455
x = \frac{973,570,455}{9,999,990}

From there, you simply divide out any common factors to get a fraction in lowest terms. Here, both numerator and denominator are divisible by 714,285.

For an easier one, try 4.715151515 . . .

x = 4.7151515 . . . = 4.7\overline{15}
100x = 471.5151515 . . . = 471.5\overline{15}
99x = 471.5\overline{15} - 4.7\overline{15} = 466.8

Multiplying by 10 (to get rid of the decimal):

990x = 4,668
x = \frac{4,668}{990} = \frac{778}{165}

This is usually done with infinite series known to converge to something involving \pi. For example:

\frac{\pi}{4} = \sum_{k=1}^\infty \frac{\left(-1\right)^{k+1}}{2k - 1} = 1 - \frac13 + \frac15 - \frac17 + \cdots
\frac{\pi^2}{6} = \sum_{k=1}^\infty \frac{1}{k^2} = 1 + \frac14 + \frac19 + \frac1{16} + \cdots
\pi = \sum_{k=0}^\infty \left[\frac{1}{16^k}\left(\frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)\right]

The first two converge extremely slowly; the last one converges extremely quickly.

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#3

What, Buffon’s Needle not good enough for you to estimate pi??? You kids and yer fancy, dancy infinite series!!! :unamused:

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#4

I wasn’t sure that we could trust that Greenman would be safe around sharp objects.

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#5

We could start him off with frozen hot dogs!!!

http://www.excelhero.com/blog/2014/11/calculate-pi-by-throwing-hotdogs.html

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#6

I’m getting the impression that you’re playing us here.

I have an inkling that you slept just fine without even reviewing this thread.

#7

If there aren’t hotdogs on his kitchen floor between taped lines, I will be very disappointed. :unamused:

#8

Sleepless nights still ensue. The hot dog thing kinda blows the mind.