Remembering pi

The technique

π (pi) is usually a number you want to learn by heart (to a certain point) if you are hanging out with math nerds (and even more if you are one yourself).

But there is a very efficient solution for remembering decimals of π : learning them in base 26, with letters replacing digits (a=1, b=2, c=3...).

That way, we have :

π = d.drsqlolyrtrodnlhnqtgkudqgtuirxneqbckbszivqqvgdmelmuexroiqiyalvuzvebmijpqqxlkplrncfwjpbymggohjmmqismssciekhvdutcxtjpsbwhufomqjaosygpowupymlifsfiizrodplyxpedosxmfqtqhmfxfpvzezrkfcwkxhthuhcplemlnudtmspwbbjfgsjhncoxzndghkvozrnkwbdmfuayjfozxydkaymnquwlykaplybizuybroujznddjmojyozsckswpkpadylpctljdilkuuwkqkwtzmelgcohrbrjenrqvhjthdleejvifafqicqsmtjfppzxzohyqlwedfdqjrnuhrlmcnkwqjpamvnotgvyjqnzmucumyvndbpgmzvamlufbrzapmuktskbupfavlswtwmaetmvedciujtxmknvxkdtfgfhqbankornpfbgncdukwzpkltobemocojggxybvoaetmhctt...

Here are 500 decimals, but, since they are in base 26, they are equivalent to more than 700 decimals in base 10.

If you just learn this : "d.drsqlolyrtrodnl" (that is 15 letters), you already know the equivalent of "3.14159265358979323846".

Not only letters are able to convey more information, but they are also much easier to learn : our brain is used to manipulate and remember words. So you can use easy mnemonics, like seeking for things you can pronounce ("loly" is the first one), or seeing that "rodn" could sound like "rodin", a famous sculptor, and also looks like "round"...

"You are cheating !"

If people tend to suggest that this way to remember π is not legitimate, you may answer with these arguments :

• There is no good reason to use base 10 instead of base 26
• There is a simple algorithm to convert it to base 10
• It is more efficient
• You are just jealous

But you probably won't have to, because anybody that is worth reciting π to is probably able to guess these arguments.

How much more efficient ?

You may ask : "but how much digits will I save by learning π in base 26 ?".

The calculation is quite simple : you just have to count how many different numbers can be written with a given number of digits in a given base

In base 10, with 7 digits, you can write 10^7 different number. Same for base 26 with 37 digits : 26^37.

Let's say you learned 15 digits in base 26. How much would it be in base 10 ?

You just have to solve the following equation : 26^15 = 10^x

You can apply a base 10 logarithm on each side of the equation, and you will get :

log_10(26^15) = x

So, learning 15 digits in base 26, is equivalent to learning log_10(26^15) ≈ 21 digits in base 10.