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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?

Michael
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    It is of course a rational number because it can be reduced to one integer over another. Changing the definition to mention such expressions would complicate the definition without changing which numbers are rational. – Michael Hardy Sep 01 '14 at 01:27
  • Careful with the notation. The set of rational numbers is usually denoted $\mathbb Q$, with $\mathbb R$ being the standard notation for the real numbers, a much larger set. Also, be careful with the descriptions you give. For instance, on the first line, $b$ must be assumed different from $0$. – Andrés E. Caicedo Sep 01 '14 at 01:31
  • Yes, I meant to put $\mathbb{Q}$. My apologies. I neglected the details. – Michael Sep 01 '14 at 01:33
  • It is $\mathbb Q={\frac ab\mid a,b\in\mathbb Z}$. $\mathbb R$ is the real numbers. – Thomas Andrews Sep 01 '14 at 01:33
  • If $a=b$ and $b$ is a rational number, then $a$ is a rational number. Being a rational number is not about its representation, but whether it can be represented that way. – Thomas Andrews Sep 01 '14 at 01:36

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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 $\{ (a,b) : a \in \mathbb{Z}, b \in \mathbb{Z} \setminus \{ 0 \} \}$.

Ian
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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?

Kim Jong Un
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  • Oh yes. I see, take an arbitrary element in B, say $p/q=(a/b)/(c/d)$ which is the same as $(a/b)(d/c)=(ad)/(bc)$. We know the product of two integers is an integer, and we can write $(ad)/(bc)$ as $m/l$ where $m,l\in\mathbb{Z}$ so our sets are the same. – Michael Sep 01 '14 at 01:30
  • Yes, they are :). So you go for the one with the simpler definition (i.e. $A$). – Kim Jong Un Sep 01 '14 at 01:31
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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.

narendra-choudhary
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