Built on Facts

Pi day!

Whew! Busy week for me, hence the missing post or two. Traditionally Saturdays on this blog tend to be non-physics fare – either links, commentary on something nonscientific, or whatever. Today I think it will be the celebration of Pi Day!

Pi Day: As you know from the SB main page, today is pi day. 3/14, representing the famous number 3.14159265359…

You first meet is as the ratio of the circumference of a circle to its diameter. As you go on in math, you meet it pretty much everywhere else too. Seriously everywhere.

There’s also a Pi Approximation Day, on 7/22. Which of course only makes sense if you write your dates in the 22/7 format. 22/7 is approximately pi, because 22/7 = 3.14286…

Let’s look a little more closely at that concept. To approximate a number as a fraction, we’re saying there are two numbers a and b such that a/b is close to our number. You would start with a low number for b and work your way up with the desired approximation. Why don’t we start with b = 1? In that case, clearly 3/1 = 3 is our best approximation.

Now it turns out that as we try higher values of b, we won’t be able to find a closer approximation to pi until we hit b = 7. Then we’ll find that a = 22, leading to the famous 22/7 approximation. But we can take b higher: can we (say) find some number a such that a/13 or is a better approximation? Nope. b = 13 (which I picked as an example for no particular reason) won’t do it. In fact 22/7 is a very good approximation. There’s no a that will give a better approximation for any b until we reach b = 106.

With a little effort, we find that the next rational approximation to pi that’s better than 22/7 is 333/106. And further it will turn out that the next approximation to improve upon that doesn’t have to make its own b much higher: 355/113.

So what are all these approximations buying us compared to the actual value of pi? Let’s see:

Approximation: 3/1 = 3
Percent Error: 4.5

Approximation: 22/7 = 3.14286…
Percent Error: 0.04

Approximation: 333/106 = 3.14151…
Percent Error: 0.0026

Approximation: 355/113 = 3.1415929…
Percent Error: 0.0000085

And just for kicks, the next one after that:

Approximation: 103993/33102 = 3.14159265301…
Percent Error: 0.000000018

Going that high kind of defeats the entire point of a rational approximation, which is to make the math easier. And it’s worth pointing out that in some sense 22/7 and 355/113 are especially good approximations because you have to make a quite sizable jump in b before you can find a better approximation. If you’d like to take a look at the machinery of this calculation, check out the theory of continued fractions. It’s a beautiful subject but not one that can really be done justice in a single Pi Day news post.

Enjoy, and go eat a pie!

Comments

  1. #1 Uncle Al
    March 14, 2009

    Baby Boomer early retirement will implode Social Security and Medicare tightly near 2015. 3/14/15 would be good; a year later would be satisfactory.

    Some repeating sequences in pi (places after the decimal). Interested readers are invited to find the other five.

    777777777777 from 368,299,898,266-th
    999999999999 from 897,831,316,556-th
    111111111111 from 1,041,032,609,981-th
    888888888888 from 1,141,385,905,180-th
    666666666666 from 1,221,587,715,177-th

  2. #2 Carl Brannen
    March 14, 2009

    In addition to continued fractions, one of my chess buddies recently pointed out continued square roots. For example,
    let f(a) = x = sqrt( a + sqrt( a + sqrt( a + …) ) )

    then x^2 = a + x, and one has that x^2 – x – a = 0 and so
    f(a) = (1 + – sqrt( 1 + 4a ) ) / 2.

    Therefore, one may have
    1 = sqrt( 0 + sqrt( 0 + sqrt( 0 + …) ) )

  3. #3 Kobra
    March 14, 2009

    Happy pi day!

  4. #4 DaleP
    March 15, 2009

    Uncle Al said:
    Baby Boomer early retirement will implode Social Security and Medicare tightly near 2015.

    No implosion will happen within the 75 year planning horizon. A minor reduction in benefits might happen in 40 years. Please make true statements.

  5. #5 Uncle Al
    March 15, 2009

    70 million Baby Boomers entitled to average benefits $1300/mo plus average $400/mo Medicare expenses = $1.4 trillion/year sudden outflow with their previously SS taxed income contributions reduced to zero. There is no Social Security fund backup – it was 100% embezzled as T-bills (backed by the Social Security fund). Incoming is outgoing, except there will be no incoming for quadrupled outgoing.

    What social strata did 3+ millon incremental firings since November 2008 impact? 50+ aged adults kicked the Hell out of their fat salaries, benefits, pensions, and nest eggs. They are unhirable given massive youth oversubscription to cheap employment. Like an imploding fission warhead, a little Rayleigh-Taylor instability won’t matter at all as shock wave hits dense implosion tamper and that accelerates across free space onto the delta-phase plutonium/0.8 wt-% gallium alloy pit. The implosion front is *already* retired. All that remains is a few shakes of patience while the wavefont chirps.

  6. #6 DaleP
    March 15, 2009

    #5
    Your rant is practically poetry, but is a gross mis-estimate of the future. 70 million will not retire at once. It will be many years before they are all retired. The “Trust Fund” is as it always was, a promise to pay, based on taxing power. You may have the opinion that the Congress may vote not to pay and that the President will sign that bill, but I think it is unlikely. Your political opinion doesn’t make it fact.

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