# What does “rational approximation” in Wolfram Alpha mean?

I'm trying to figure out a nicer rational expression of the following:

$$\frac{1}{14.5}$$

http://www.wolframalpha.com/input/?i=1%2F14.5

But Wolfram Alpha only gives "rational approximation" $\frac{2}{29}$.

Is this the exact analytic expression or is it really some sort of "approximation"? And if it's an approximation, then why is this number approximated?

• $\dfrac{1}{14 \dfrac{1}{2}} = \dfrac{1}{\dfrac{29}{2}} = \dfrac{2}{29}$ – Moo Nov 19 '16 at 18:01
• Any terminating decimal can be written as a rational. I imagine that Wolfram might have computed the floating point result of the above, which does not have an exact representation, hence the 'approximation'. – copper.hat Nov 19 '16 at 18:02
• $\frac{1}{14.5}$ is technically not a rational since $14.5$ is not an integer, hence wolfram gives the form $\frac{2}{29}$, which is a rational. – mrp Nov 19 '16 at 18:04
• @mrp I don't think that's correct. How can a number have a property if you write it one way and not have a property if you write it another way? It's the same number. A number is either rational or it's not, it doesn't matter how you write it. – user223391 Nov 19 '16 at 18:38

## 1 Answer

Wolfram-alpha treats all decimals as approximation, and all integers as exact values. (Cf. Mathematica)

Therefore, $1/14.5$ is not the exact quantity $2/29$, but a number very close to $2/29$.

If you want Wolfram-alpha to treat your numbers as exact quantities, use integer quotients.