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The confirmed world record for memorizing the digits of $\pi$ goes to a Chinese graduate student named Lu Chao, who claims he has memorized up to 100,000 digits (although for the record breaking attempt, only got to just shy of 68,000).

Memorizing the digits of $\pi$ seems to be a thing people actually do for fun, and while I'm not here to judge their definition of fun, I am curious how one could memorize so many digits of what seems to be a random string of numbers. I know $\pi$ isn't random (quite the opposite), but there doesn't seem any correlation between digits, and it boils right down to rote memorization than any kind of actual maths involved.

Is there a mathematical approach to calculate a given digit of $\pi$, and this is the method these people use? I have trouble memorizing seven digit phone numbers, so it comes to me as a surprise that people can memorize tens of thousands of a digit sequence without error.

gator
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    Pretty sure it's just rote memorization. –  Feb 09 '14 at 21:25
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    Make up a story to connect the digits. This really makes it surprisingly easy. Go like this: there was $3$ people in $1$ house with $4$ rooms and they had to share $1$ bath. This was unfortunate because everybody needed to use it $5$ times a day... and so o.. – J.R. Feb 09 '14 at 21:26
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    Memory olympians generally use things like mind palaces (seen in Sherlock for example). Some savants have more innate mental procedures, like forming landscapes where the digits are landforms (seen in Born on a Blue Day for example). – anon Feb 09 '14 at 21:27
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    @TooOldForMath, that may be a good approach for a small order of digits, but when you get into the 10,000-100,000 mark, it becomes akin to memorizing a several hundred page book. – gator Feb 09 '14 at 21:27
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    There are methods to memorize anything. With respect to $\pi$, I know there are methods in which you associate numbers with sounds, then you make up a text with those sounds (a poem for instance), then memorize the poem and translate back to digits of $\pi$ in your head. – Git Gud Feb 09 '14 at 21:27
  • Possible duplicate – NasuSama Feb 09 '14 at 21:27
  • what do you mean $\pi$ is not random? the decimal expansion for $\pi$ is considered, if I am not wrong, indeed random – Ant Feb 09 '14 at 21:30
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    @riista I agree. But this is an easy, effortless way to memorize a few hundred of them, I tried it and it works. – J.R. Feb 09 '14 at 21:31
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    I know 25 digits by heart without specific memorizing techniques (e.g. stroies), but in that short string there are still many "funny" patterns (such as $141$, $535$, $8979$, $323$, $626433$) that make memorizing easy (isn't any random enough sequence full of funny patterns?). For the remaining 999975 digits, one needs real memorizing techniques that have nothing to do with $\pi$ specifally but could help memorize Shakespeare's collected works as well (and perhaps more - literature has less entropy than its mere length suggests) – Hagen von Eitzen Feb 09 '14 at 21:37
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    $$\pi\simeq3.\color{blue}{14\ 15}\ 926\ \color{blue}{53\ 58}\ \color{red}{979}\ \color{red}{323}\ 84\ \color{red}{626}\ 43\ \color{red}{383}\ \underbrace{2795}{\begin{align}|2+7|=9\|2-7|=5\end{align}}\ \underbrace{02884}\text{Even Digits}\ 6\ \color{red}{93}\ 9\ \color{red}{93}...$$ $$e\simeq2.7\ \underbrace{1828\ 1828}{\begin{align}\text{Repeating}\\text{Sequence}\end{align}}\ \underbrace{45\ 90\ 45}{45+45=90}...$$ – Lucian Feb 09 '14 at 22:13
  • It doesn't get anywhere near 100,000 (or even 1,000) digits, but people have made up sentences or poems in which the number of letters in the words are the digits of $\pi$. One such mnemonic begins "Now I, even I, would celebrate in rhymes inept the great immortal Syracusan ..." – Andreas Blass Feb 09 '14 at 22:18
  • There's an easy way to memorize any number that it will be important to remember, but it only works if you have good aural skills and know a little something about music. Take a major scale in any key that you can sing comfortably. The scale will comprise 9 notes: low "ti" through "do" a minor ninth above. Assign a number (0 through 9) to each of these notes sequentially. Any sequence of numbers then becomes a melody: you don't have to memorize the numbers, only the melody. This is how I remember everything from important phone numbers to passwords; it works beautifully with pi. : ) – Ryan Apr 15 '14 at 01:11
  • Also, if you give the melody context by immuring it in some harmonic framework, it becomes even easier to remember the number. – Ryan Apr 15 '14 at 01:12

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How to memorize things is not a mathematical question, BUT something in the posting is a mathematical question.

If there were patterns in the decimal expansion of $\pi$ that can be undrestood by understanding some mathematical idea, then understanding of those could help one memorize the digits.

Here's the relevant mathematical finding: Such patterns do emerge in some things like the continued-fraction expansion of $e$ and some yet more elaborate things. But so far no one has identified such patterns in the decimal expansion of $\pi$. It behaves statistically like a sequence of random digits in which all digits are equally likely to appear at any point, and the digits that came before a particular digit do not alter the probability that any particular digit will appear. And it is conjectured (no one has been able to prove it so far) that $\pi$ is a "normal number", i.e. one that has just those statistical properties in the long run. The short run may be another question, but I don't think any sensible patterns appear there either.

Consequently mathematics can't offer any help here other than that you're essentially just memorizing random digits. However, no one can yet rule out the possibility that some future discovery will reveal some patterns that could be used.

I'll have to leave it to others to cite the relevant literature.

Michael Hardy
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