I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin.
I have mainly 3 questions regarding this article. I am writing all these $3$ question in one question because they are related to each other.
Question $1$
The author defined the equality of two positive rational numbers by the rule:
$\dfrac{a}{b}=\dfrac{a'}{b'}\ \ \ \mbox{if}\ \ \ \ \ \ ab'=a'b. \tag{2}$
Here $a,a',b,b'$ are Natural numbers.
The problem is that this definition can be proved by definition of the product of two positive rational numbers. In the article the definition of product of two positive rational numbers is give after the definition$2$.
In the definition$5$ the author defines the product of two positive rational numbers as: $$\dfrac{a}{b} \cdot \dfrac{a'}{b'}=\dfrac{aa'}{bb'} \tag{5} $$
We can use this definition to prove definition$2$. To do so let's first consider three ratioanal numbers $\dfrac{a}{b}$ , $\dfrac{b'}{a'}$ and $\dfrac{a'}{b'}$ s.t. $a,a',b,b'$ are Natural numbers.
$\dfrac{a}{b}\times \dfrac{b'}{a'}=\dfrac{ab'}{ba'}$ by definition$5$
If $ba'=ab'$
Then $\dfrac{ab'}{ba'}=1$ hence $\dfrac{a}{b}\times \dfrac{b'}{a'}=1$
Multiplying both sides with $\dfrac{a'}{b'}$ we get :
$\dfrac{a}{b}\times \dfrac{b'}{a'}\times \dfrac{a'}{b'}=1\times \dfrac{a'}{b'}$
$\implies \dfrac{a}{b}=\dfrac{a'}{b'}$
So definition$5$ implies that if $ab'=a'b$ then $\dfrac{a}{b}=\dfrac{a'}{b'}$ which is nothing but the definition$2$.
- "The definition$2$ can be proved from the definition$5$ so why the author quote Equation$2$ as a definition?"
- "What if we reject definition$2$?"
Question $2$
Author has quoted an important result in equation$3$ as:
"We identify certain rational numbers with integers by regarding $$a=\dfrac{a}{1}\tag{3}"$$
How this relation is identified? Uptill equation$3$ the author had not defined the multiplication of two positive rational numbers. Does equation $1$ and $2$ imply equation$3$. Or equation$3$ is in itself a definition(this doesn't seem to be the case)?
The best I could understand this is as:
$\dfrac{a}{1}=x$(say) is a rational number which satisfies the relation $a=x\times 1$. We have not defined the rule for multiplication of $x$ with $1$ when $x$ is a rational number. But we see that $x=a$ is also a solution of relation $a=x\times 1$ because $a=a\times 1$.
So if $\dfrac{a}{1}$ is just the solution of $a=x\times 1$ then $a=\dfrac{a}{1}$ is an identity not an equality because for every positive number $a$ , $a$ is equal to $\dfrac{a}{1}$, that is we should have $a \equiv \dfrac{a}{1}$ not $a=\dfrac{a}{1}$.
Question $3$
Why we defined different operations on rationals number this way, that is the rules described in the article from equation $(4)$ to equation$(13)$.
e.g. The multiplication of two positive rational numbers is as: $\dfrac{a}{b} \cdot \dfrac{a'}{b'}=\dfrac{aa'}{bb'} $. An applied Mathematician may answer this by saying:
" Let $\dfrac{a}{b}$ and $\dfrac{a'}{b'}$ be the magnitude of the length and breadth of a rectangle then $\dfrac{aa'}{bb'}$ represents its area that's why we defined multiplication of two positive rational numbers this way."
On the other hand a pure Mathematician may answer this as:
"Let $x=\dfrac{a}{b}$ and $y=\dfrac{a'}{b'}$ be the two positive rational numbers s.t. $a,a',b,b'$ are natural numbers. By defining the multiplication of two positive rational numbers this way(as defined in equation$5$) we recognize that the commutative law and the associative law will hold good. That's why we defined it to be this way.
- In reality who, how, when and why defined these definitions/rules(equation$1$ to equation$13$)?
- Are the reason behind these definition to be of this kind pure mathematical or applied mathematical or both?
Please give me a link or tell me further reference on the historical perspective on these definitions. I want to study the history of these things in detail.