# Name for $f(a,b) = c/d$

What is the a name for functions of the form $f(a_1/b_1,\ldots,a_n/b_n) = c/d$ where $a_1,\ldots,a_n,b_1,\ldots,b_n,c,d \in Z$ and all the denominators are not zero.

I was thinking about calling them rational functions; but that name is already taken :).

• Maps $\mathbf{Q}^n\to\mathbf{Q}$. – Unit Sep 1 '15 at 5:35
• Thanks, this is a clearer notation. I would like to know whether there is an English name for these functions. – Willem Sep 1 '15 at 5:54
• What special properties differentiate your function from $\mathbb{R}^n \rightarrow \mathbb{R}$, and how are they going to be useful? and finally is it called Rational Number Function? – Ashutosh Gupta Sep 1 '15 at 5:59
• Thank you. I will take that as an answer. I am no mathematician; but I was thinking about the problem of finding optimal values for such functions (e.g. for which arguments is f at its highest value). [The optimal value can be guaranteed to be found - assuming there bounds placed on the arguments] I assumed there is lots of work done in this area; I just did not know how to find it. So I asked the question here. – Willem Sep 1 '15 at 6:17
The restriction to rational (instead of real) arguments will just make any treatment harder. As the rationals are dense, it can very well be that your function has a maximum in the reals (e.g. at $\sqrt{2}$), but none in $\mathbb{Q}$, and for practical purposes it makes no difference.
• You are correct - for practical purposes there are no differences. A rational number function cannot have an answer of $\sqrt{2}$ by definition. However a rational number function may have something that approximates that value. I am thinking about rational number functions are not well suited to linear programming; where the relationships are not linear; and the derivatives are not known: en.wikipedia.org/wiki/Derivative-free_optimization – Willem Sep 1 '15 at 16:33