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.



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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.

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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.



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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.

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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 − 10<sup>0</sup>    
90 − 10<sup>−1</sup>    
90 − 10<sup>−2</sup>    
90 − 10<sup>−3</sup>    

and that, furthermore, that negative power was the same as the positive power you had to divide tan by to get 180 / π.

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In other words, you can approximate 180 / π as

tan(90º − 10<sup>−n</sup>) / 10<sup>n</sup>


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And hence, approximate π as:

### 180 × 10<sup>n</sup> / tan(90º − 10<sup>−n</sup>) 

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This gives the sequence:

3.1419116870791406...  
3.141595843539755...  
3.141592685494158...    
3.141592653911621...    

which gets closer and closer to π.

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I showed my friend Gerald Yong (featured in my stream on meeting [Murry Gell-Mann](https://thoughtstreams.io/jtauber/murray-gell-mann/)) and we decided to call this Cumusky's Pi after our beloved maths teacher.