Tao Analysis. Definition of positive rational numbers.

Question

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!

Answer 1

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.

Answer 2

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?

1. 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.

2. 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 $F_2.$