Is a complex fraction considered part of the rationals?

Question

I have always been taught that $\mathbb{Q}=\{ \frac{a}{b}|\,\,a,b\in \mathbb{Z},\, \,b\neq0\}$. Is this definition of the rationals limited? Could it also be true that a complex fraction, i.e. $\frac{\frac{a}{b}}{\frac{c}{d}}$ is also a rational number? I know it can be reduced to an integer over another, but as is, would it be considered a rational number?

Answer 1

The sets $A=\{m/n:m,n\in\mathbb{Z},n\neq 0\}$ and $B=\{p/q:p,q\in\mathbb{Q},q\neq 0\}$ are the same. Can you verify this?

Answer 2

We don't really care about the representation of the rationals that much. We just care about the algebraic and order properties. Technically it is not wrong to include $\frac{\frac{a}{b}}{\frac{c}{d}}$ as a distinct rational number from $\frac{ad}{bc}$. But because non-reduced fractions have exactly the same algebraic and order properties as the corresponding reduced fraction, it is justified to not include them.

This is somewhat like how we consider the rational numbers to be the appropriate quotient of $\{ (a,b) : a \in \mathbb{Z}, b \in \mathbb{Z} \setminus \{ 0 \} \}$.

Answer 3

Any number that can be expressed in form of $ \frac{a}{b}$ where ${a,b \in \mathbb{Z} \cap b\neq 0}$ is called a rational number.

And this complex fraction $\frac{\frac{a}{b}}{\frac{c}{d}}$ is in fact same as $\frac{ad}{bc}$

Given that $a,b,c,d \in \mathbb{Z}$ and $b,c,d \neq 0$ , both products $ad,bc \in \mathbb{Z}$ .

Hence this complex fraction is a rational number.