hi every1~!

 

I was just curious that what is the point of studying only integar solutions of a problem. One reason that comes up to my mind is that some irrational numbers have infinite numbers after decimal, so therefore it would be a problem to write an exact solution. But are there any more reasons that prevent us playing with other numbers?

Irrationals do not have infinite numbers after their decimal places (there is no such thing as an infinite number in a decimal representation system). They *all* have non-repeating decimal expansions, however. There is no problem writing an exact solution to any equation, by the way. It is just your preference for thinking that 2 is exact but sqrt(2) is not.

 

The point is that rational numbers, and integers are more natural than the real numbers, they are what has fascinated mankind throughout the centuries.

More than that, the study of the solution of Diophantine equations pretty much inspired the creation of modern day algebra - rings of integers, ideal classes, stuff like that.

 

For example, an equation like [imath]x^2 + 5 = y^3[/imath] is not easy to solve by the classical methods of say, reduction modulo n or infinite descent. However, if we apply some basic algebra theory - in this case, the theory of prime ideals - then it's easy to show that there are only finitely many solutions and determine what those solutions are.