Tao Analysis. Definition of positive rational numbers. Tao Analysis. Definition of positive rational numbers.

Definition $4.2.6.$ A rational number $ x$ is said to be positive iff we have
$x = a/b$ for some positive integers $a$ and $b$. It is said to be negative iff
we have $x = −y$ for some positive rational $y$ ($i.e., $$x$ = $(−a)/b$ for some
positive integers $a$ and $b$).

My question: Does not $x = a/b$ for some negative integers $a$ and $b$ satisfy? as in $x = -2/-3$. So why does this definition use iff?  Thank you!
 A: The original definition, made more verbose, is the following:

A rational number $x$ is said to be a positive rational number if and only if there exists some positive integer $a$ and some positive integer $b$ such that $x=\frac ab$. A rational number $x$ is said to be a negative rational number if and only if there exists a positive rational $y$ such that $x=-y$.

The key point to highlight here is "there exists". This means that the definition only "cares" about positive integers, and such a counter-example is not relevant.
If I have to explain this further, please tell me.
A: 

Definition $4.2.6.$ A rational number $ x$ is said to be positive iff we have
$x = a/b$ for some positive integers $a$ and $b$.

Why does this definition use iff?

Every definition in mathematics uses "if and only if"; try to come up with one that doesn't!

Does $x = a/b$ for some negative integers $a$ and $b$ not satisfy?


*

*The above definition can be written $$\forall x{\in}\mathbb
    Q\;\left(x>0\iff\exists a{,}b{>}0\:\:x=\frac ab\right).$$
$\implies$ direction:

*

*Now, if $x$ equals $\dfrac{-8}{-3}$ or $\dfrac{-24}{-9}$ or $\dfrac{16}{6}$ (so, $x$ is positive), then
$x=\dfrac{\mathbf8}{\mathbf3},$ that is, $$x=\frac ab\text{ for
}\mathbf{some}\text{ pair of positive integers }a\text{ and }b;$$
here, we can choose $(16,6)$ instead of $(8,3),$ but $(-8,-3)$ won't
work.

$\impliedby$ direction:

*

*If $x$ equals $\dfrac{-8}{-3}$ (so, $x=\dfrac{\mathbf8}{\mathbf3},$ that is, $x=\dfrac ab\text{ for
}\mathbf{some}\text{ pair of positive integers }a\text{ and }b$),
then $x$ is positive.



*A less technical explanation.
Each positive rational number $r$ can be expressed as an non-simplified fraction
$F_1$ of integers with numerator and denominator both negative;
when these two negative signs are dropped, we obtain a non-simplified fraction
$F_2$ of integers with numerator and denominator both positive;
it is $F_2,$ not $F_1,$ that the above definition is using to define
$r$ as positive.
This alternative definition is equivalent to the above one:


Definition $4.2.6.$Alt $\;\;$ A rational number $ x$ is said to be positive iff we have $x = a/b$ for some negative integers $a$
and $b.$


If we adopt this definition, then we need to choose $F_1,$ not
$F_2.$
