## Sunday, April 14, 2013

### Approximating irrational numbers using intervals

Every irrational number can be progressively approximated using rational intervals. Here is a sequence of intervals that approximate $\sqrt{2}$:
[-Infinity Infinity]
[0 2]
[4/3 5/3]
[11/8 10/7]
[24/17 27/19]
[65/46 58/41]
[140/99 157/111]
[379/268 338/239]
[816/577 915/647]
[2209/1562 1970/1393]
[4756/3363 5333/3771]
[12875/9104 11482/8119]
[27720/19601 31083/21979]
[75041/53062 66922/47321]
[161564/114243 181165/128103]
[437371/309268 390050/275807]

This sequence of intervals is produced by the continued fraction representation for $\sqrt{2}$ which is 1,2,2,2,2,2,2,... with an infinite sequence of twos.

## Saturday, April 13, 2013

### Interval analysis

The set of all rational intervals is countable. We can develop analysis in terms of this intervals firstly through interval arithmetic:
(= (add-intervals [1 2] [3 4]) [4 6])
(= (multiply-intervals [1 2] [3 4]) [3 8])

We can express any continuous real function as a function of intervals using its set of extrema. For example, the square of the interval [-2,2] is [0,4] and the square of the interval [2,3] is [4,9].