What is the name of this type of fraction

[gamow99]

Take the fractions 2/5 and 4/10. 2/5 cannot be simplified further without converting the integers into decimals. I need to know the name of the 2/5 fraction. Maybe it's called irreducible form or something like that. Let me give another example, of 1/3, 3/9, or 33/99 only 1/3 is irreducible because to divide the numerator and the denominator would turn it into a decimal.

 

[andrewkirk]

I would just call the fraction 'irreducible', ie I would say 2/5 is an irreducible fraction and 4/10 is not (we could say it is a 'reducible fraction'). I would not bother adding 'form' because in common usage a fraction is a representation of a (rational) number, rather than a number itself. So that 2/5 and 4/10 are different fractions, both of which represent the same rational number.

Set theoretically, it is convenient to define a rational number as an equivalence class of elements of $\mathbb Z\times\mathbb N$, under the equivalence relation $=_{\mathbb Q}$ such that $(a,b)=_{\mathbb Q} (c,d)$ iff $a/b=c/d$. The equivalence class is a rational number. The elements of the equivalence class are fractions, where the fraction $(a,b)$ is formally written as $\frac ab$, and that fraction is irreducible iff the greatest common factor of $a$ and $b$ is 1.

 

[gamow99]

Thanks. I appreciate you taking the time to help me out.