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 Rational or not? (Posted on 2015-05-29)
If I enter the expression

```35+42√2
```
into my graphing calculator and then ask it for the fractional (rational) equivalent it cannot do so. Since the number is irrational it displays the first few decimal digits:

```94.39696962
```
However, if I enter the expression

```55+42√2
```
into the same graphing calculator and then ask it for the fractional equivalent it displays:
```37751
-----
330
```
But this second number I entered is clearly not rational either.

What's going on?

 No Solution Yet Submitted by Jer Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 solution | Comment 1 of 5
42*sqrt(2) ~= 59.39696961967

Add 35 and it still has two digits before the decimal point and so is 94.39696961 or 94.39696962 when truncated or rounded, respectively, to 10 significant digits. The 1 or the 2 breaks the apparent repetition of 969696.

When 55 is added, rather than 35, there are now 3 digits to the left of the decimal, so the last digit to the right of the decimal is lost: 114.3969696. Whoever programmed the calculator's rational recognition routine apparently thought three repetitions of a pair of digits was enough to justify the assumption that the repetition continued forever and calculated the appropriate numerator and denominator for that continuation, reduced to lowest terms.

It's also an indication that the calculator works in binary coded decimal: groups of 4 bits don't go any higher than 9 and the groups (half-bytes or nybbles) are considered as decimal digits.

 Posted by Charlie on 2015-05-29 10:09:45

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