[-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.

## 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 $\sqrt{2}$:

## 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].

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