In my pre-teens, I was interested in programming and 3D graphics and so had one number memorized:
57.29577951...
That's 180 / π, the number of degrees in a radian. I was used to dealing in degrees (I didn't yet appreciate the benefits of radians) but my trigonometric functions in BASIC took angles in radians.
I remember one day in Mr Cumusky's maths class, when I was 13, thinking about the fact that tan 90º goes to infinity.
I wondered what tan 89º was. My calculator gave me:
57.28996163...
I was intrigued at how close that was to 180 / π.
I tried 89.9º and got:
572.9572136...
I was shocked to recognize even more of the digits from 180 / π, albeit with the decimal place shifted over one.
So imagine my even greater shock when I tried tan 89.99º and got:
5729.577893...
which was even closer but again with the decimal place shifted over.
I noted that the sequence
89
89.9
89.99
89.999
is 90 minus
1
0.1
0.01
0.001
or, in other words:
90 − 100
90 − 10−1
90 − 10−2
90 − 10−3
and that, furthermore, that negative power was the same as the positive power you had to divide tan by to get 180 / π.
In other words, you can approximate 180 / π as
tan(90º − 10−n) / 10n
And hence, approximate π as:
This gives the sequence:
3.1419116870791406...
3.141595843539755...
3.141592685494158...
3.141592653911621...
which gets closer and closer to π.
I showed my friend Gerald Yong (featured in my stream on meeting Murry Gell-Mann) and we decided to call this Cumusky's Pi after our beloved maths teacher.