<s>[INST] Why does $\frac{4}{2} = \frac{2}{1}$? I take for granted that $\frac{4}{2} = \frac{2}{1}$. 
Today, I thought about why it must be the case. My best answers amounted to $\frac{4}{2}=2$ and $\frac{2}{1}=2$; therefore $\frac{4}{2}=\frac{2}{1}$.  However, that explanation seems circular:


*

*one can express $2$ as $\frac{2}{1}$. 

*As such, to say $\frac{4}{2}$ equals $\frac{2}{1}$ because both equal $2$, is nearly saying $\frac{4}{2} = \frac{2}{1}$ (the question) and $\frac{2}{1}=\frac{2}{1}$ (trivial, at best). 


So why does $\frac{4}{2} = \frac{2}{1}$?
 [/INST] The set $\mathbb{Q}$ is a group then for all $x\in \mathbb{Q}$, there is a only Inverse element such that $x+(-x)=0$. In this case note that
$$\frac{4}{2}-\frac{2}{1}=0$$
then you conclude that $$\frac{4}{2}=\frac{2}{1}$$
</s>