Pi Day – Euler’s Infinite Expansion of Pi
Most people know that pi is a number that is infinite and non-repeating i.e. it never ends and there is no repeating pattern to it. Numbers like these are known as irrational or transcendental, depending on the type of equation that can be solved by them. Pi is actually transcendental, though it took a long time for that to be proved mathematically.
Though pi is known by most people as the ratio of a circle’s radius (or diameter) to its circumference, it actually turns up all over the place in mathematics. One of these places is as the solution to infinite series, of which many have been discovered.
A famous result was discovered by the mathematician Leonard Euler, in 1734. It says:
1 + 1/22 + 1/32 + 1/42 + 1/52 + … = pi2/6
In other words, if you take all of the integers and square their reciprocals, then add them up, the answer will be exactly equal to pi squared divided by 6.
A related result can be derived for the even integers:
1 + 1/22 + 1/42 + 1/62 + 1/82 + … = pi2/24
For the odd integers:
1 + 1/32 + 1/52 + 1/72 + 1/92 + … = pi2/8
All of these results can be proved analytically, though you have to know some calculus, as the proofs involve integrals. Rather than do this, I set up some R code to explore the expansions, and see how close they come to the value of pi.
There are a few obvious cautions, both involving infinity:
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We can’t actually add up an infinite number of terms, so we can never get the exact sum for the various series.
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We don’t actually know the exact value of pi (and can’t ever know it), so we can’t actually do the comparisons.
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Computers will run into problems with floating point operations as the numbers become very large or very small.
Nonetheless, even a modest laptop computer can provide some interesting results, as shown below:
The top table shows show:
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the number of terms in each trial of the expansion (from 10 to one hundred million),
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the actual value of pi (as approximated by the R program),
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the value of pi using all integers from1to the upper limit of terms,
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the value of pi using the first “limit” number of even integers,
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he value of pi using the first “limit” number of oddintegers
The second table is similar, but it shows the difference between R’s value of pi and the value computed by the expansions, expressed in parts per billion.
As you can see, the value that most of us routinely use (3.14) is well approximated after about 1000 terms of these series. For those of us that like to be a little more precise and remember the value to five decimal places (3.14159), on the order of 100,000 terms is needed.
Eventually, by 100,000,000 terms in the expansions, the difference between R’s “true” value of pi and our expansion approximation is down to less than 10 parts per billion. Thinking of that in more everyday terms:
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If you drew a circle 1,000 km in diameter, 1 part in a billion is about 1 mm.
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In a circle of 360 degrees, 1 part in a billion is about one-thousandth of an arc-second. The resolution of the Hubble space telescope is about one-twentieth of an arc-second.
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If you had a pumpkin pie that weighed 1 kilogram, 1 part in a billion would be about 1 microgram. A particle of the pie that was one-tenth of a millimeter in diameter would weight about that much.
A couple of interesting results that I wasn’t expecting, are that the sum of odd integers converges faster than the sum of even integers, and even converges faster than the sum of all integers. Also, the “all integers” and the even integers converge at the same rate. So, by 100 million terms, the odd integers have converged to within 3 parts per billion, while the other two are still at about 9 parts per million.
Those results definitely seem counter-intuitive, but I can’t see anything wrong with my R code (given below). It is something to think about, and shows how math sometimes gives results that seem strange, at least at first sight.
Sources:
Inside Interesting Integrals, Paul Nahin, Springer (p 154-55)
Here is some R code that you can run and experiment with, to see for yourself:
# -----------------------------------------------------------
# R program to estimate pi from Euler's formula,
# that uses an infinite series.
# ---------------------------------------------------------
ls()
options(digits = 11)
# the dataframe is intialized with the first, short expansion.
limit <- 10
# All integers
pi_est_all <- seq(1, limit, 1)
pi_est_all$a <- 1/(pi_est_all*pi_est_all)
pi_all <- sqrt(6*sum(pi_est_all$a))
pi_all
# Even integers
pi_est_even <- seq(2, limit*2, 2)
#pi_est_even
pi_est_even$a <- 1/(pi_est_even*pi_est_even)
pi_even <- sqrt(24*sum(pi_est_even$a))
pi_even
# Odd integers
pi_est_odd <- seq(1, limit*2+1, 2)
#pi_est_odd
pi_est_odd$a <- 1/(pi_est_odd*pi_est_odd)
pi_odd <- sqrt(8*sum(pi_est_odd$a))
pi_odd
pi_init <- cbind(limit, pi, pi_all, pi_even, pi_odd)
pi_init
pi_est2 <- pi_init
pi_est2
# Loops through, with other limits for the expansion. This increases the limit
# by a factor of 10, for each trip through the loop. The limits and increments
# can be changed. Computer will run out of memory at some point.
limit <- 100
while(limit < 1000000000)
{
# All integers
pi_est_all <- seq(1, limit, 1)
pi_est_all$a <- 1/(pi_est_all*pi_est_all)
pi_all <- sqrt(6*sum(pi_est_all$a))
pi_all
rm(pi_est_all)
# Even integers
pi_est_even <- seq(2, 2*limit, 2)
#pi_est_even
pi_est_even$a <- 1/(pi_est_even*pi_est_even)
pi_even <- sqrt(24*sum(pi_est_even$a))
#pi_even
rm(pi_est_even)
# Odd integers
pi_est_odd <- seq(1, 2*limit+1, 2)
#pi_est_odd
pi_est_odd$a <- 1/(pi_est_odd*pi_est_odd)
pi_odd <- sqrt(8*sum(pi_est_odd$a))
#pi_odd
rm(pi_est_odd)
pi_est <- cbind(limit, pi, pi_all, pi_even, pi_odd)
pi_est2<-rbind(pi_est2, pi_est)
limit <- limit * 10
}
# Convert to a data frame and print results.
pi_est_df <- as.data.frame(pi_est2)
pi_est_df
ls()
And, here are some earlier blogs about pi, where it shows up in other areas of math, namely geometry, statistics and number theory.
https://dodecahedronbooks.blogspot.com/2020/03/pi-day-2020-pi-and-normal-distribution.html
https://dodecahedronbooks.blogspot.com/2019/03/pi-day-2019-shooting-arrows-at-target.html
https://dodecahedronbooks.blogspot.com/2018/03/pi-day-2018-prime-numbers-and-pi.html
https://dodecahedronbooks.blogspot.com/2017/03/pi-day-floor-pie-and-floor-pi.html
https://dodecahedronbooks.blogspot.com/2016/03/pi-day-31416-some-eerie-pi-coincidences.html
https://dodecahedronbooks.blogspot.com/2015/03/pi-day-31415-pi-and-science-fiction.html
So, now that you have done some math, you should read a science fiction book, or even better, a whole series. Book 1 of the Witches’ Stones series even includes a reference to pi.:
Kati of Terra
How about trying Kati of Terra, the 3-novel story of a feisty young Earth woman, making her way in that big, bad, beautiful universe out there.
http://www.amazon.com/gp/product/B00811WVXO
http://www.amazon.co.uk/gp/product/B00811WVXO
The Witches’ Stones
Or, you might prefer, the trilogy of the Witches’ Stones (they’re psychic aliens, not actual witches), which follows the interactions of a future Earth confederation, an opposing galactic power, and the Witches of Kordea. It features Sarah Mackenzie, another feisty young Earth woman (they’re the most interesting type – the novelist who wrote the books is pretty feisty, too).
https://www.amazon.com/dp/B008PNIRP4
https://www.amazon.co.uk/dp/B008PNIRP4
The Magnetic Anomaly: A Science Fiction Story
“A geophysical crew went into the Canadian north. There were some regrettable accidents among a few ex-military who had become geophysical contractors after their service in the forces. A young man and young woman went temporarily mad from the stress of seeing that. They imagined things, terrible things. But both are known to have vivid imaginations; we have childhood records to verify that. It was all very sad. That’s the official story.”
https://www.amazon.com/dp/B0176H22B4
https://www.amazon.co.uk/dp/B0176H22B4
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