# Diophantine equations with irrational coefficients

I was going through the book "Rational Points on Elliptic Curves by Silverman and Tate" and the rational line was defined like this.

"We call a line a rational line if the equation of the line can be written with rational numbers, that is, if it has an equation $$ax + by + c =0$$ with $a, b$ and $c$ rational."

Now, my question is :

Why do mathematicians study about the Diophantine equations with only integer(rational) coefficient? Why haven't they worked on the equations having irrational coefficients(in more than one variable)? Since, Diophantine Equation in one variable is nothing but a polynomial equation in $n$ degree and a lot of work has been done on that.

• I guess that if the coefficients were irrational, it wouldn't be a diophantine equation anymore. – C. Oliveira Feb 18 '18 at 18:00
• In your quote, they were talking about a rational line, so with your kind permission, they'll restrict that to equations with rational coefficients. Concerning the term "Diophantine equation", there are dictionaries explaining what that means. Please, don't misuse MSE for such trivial requests! – Professor Vector Feb 18 '18 at 18:15
• @ProfessorVector The wolfram link for Diophantine equation mathworld.wolfram.com/DiophantineEquation.html says in very first line that A Diophantine equation is an equation in which only integer solutions are allowed. But for the definition of Linear Diophantine Equation it restricts the coefficients to integers. So, If you want to call it by some other name you can(The equation with irrational coefficients).The whole point of asking the question was that there is no literature available with consideration of irrational coefficients. What's the reason behind it? – SARTHAK GUPTA Feb 18 '18 at 19:43
• I think the reason is that the number of things in the literature is finite, while the number of potential problems is countable, but infinite. Mathematicians are adding to the first set according to how interesting and important they find a problem. You're welcome to do the same. But a problem isn't important or interesting just because it isn't in the literature, yet. – Professor Vector Feb 18 '18 at 20:02