# Why f beta score define beta like that?

This is the F beta score:

The Wikipedia article states that $F_\beta$ "measures the effectiveness of retrieval with respect to a user who attaches β times as much importance to recall as precision".

I did not get the idea. Why define $\beta$ like that? Can I define $F_\beta$ like this:

And how to show β times as much importance?

Letting $\beta$ be the weight in the first definition you provide and $\tilde\beta$ the weight in the second, the two definitions are equivalent when you set $\tilde\beta = \beta^2$, so these two definitions represent only notational differences in the definition of the $F_\beta$ score. I have seen it defined both the first way (e.g. on the wikipedia page) and the second (e.g. here).
The $F_1$ measure is obtained by taking the harmonic mean of precision and recall, namely the reciprocal of the average of the reciprocal of precision and the reciprocal of recall:
Instead of using weights in the denominator that are equal and sum to 1 ($\frac{1}{2}$ for recall and $\frac{1}{2}$ for precision), we might instead assign weights that still sum to 1 but for which the weight on recall is $\beta$ times as large as the weight on precision ($\frac{\beta}{\beta+1}$ for recall and $\frac{1}{\beta+1}$ for precision). This yields your second definition of the $F_\beta$ score:
Again, if we had used $\beta^2$ instead of $\beta$ here we would have arrived at your first definition, so the differences between the two definitions are just notational.