Proving the equality of two fractions

Question

We say two fractions a/b and c/d are equivalent if and only if ad=cb An example on that: 2/4 = 1/2 because 2*2=1*4

But is there another way to prove this operation in general? not by samples of examples

Answer 1

The real question is "what is a fraction?" Once you resolve that, things get easy.

The axioms of real numbers say that for any number $x \ne 0$, there's another number (written $x^{-1}$) with the property that $$ x \cdot x^{-1} = x^{-1} \cdot x = 1 . $$ That number's called the "multiplicative inverse" of $x$.

Now what does $\frac{a}{b}$ denote? It denotes the number $$ a\cdot b^{-1}. $$

OK, so let's suppose that $$ \frac{a}{b} = \frac{c}{d} $$ Translated, that means that $$ ab^{-1} = c d^{-1} $$ We can multiply both sides by $b$ to get $$ a b^{-1}b = c d^{-1} b $$ Then we can use the fact that $b^{-1}b = 1$ to say that $$ a = c d^{-1} b $$ Now we can swap the last two factors and say $$ a = c b d^{-1} $$ and then we can multiply both sides by $d$ to get $$ ad = c b d^{-1}d $$ and finally use the fact that $d^{-1}d = 1$ to write $$ ad = cb 1 = cb $$ So that's a proof that if the fractions are equal, then $ad = cb$. (Hidden in there were applications of the associativity and commutativity of multiplication, and the fact that $1$ is the multiplicative identity, but I wanted to keep things simple.)

Note that my use of $x^{-1}$ didn't have anything to do with exponents per se. It's just a symbol denoting "the unique number which, when multiplied by $x$, gives the number $1$." I could have used some notation like $x'$ instead.

[To really complete things, I should write another proof that if $ad = bc$, and $b$ and $d$ are nonzero, then $\frac{a}{b} = \frac{c}{d}$, but that proof is roughly just the proof I gave, read from bottom to top.]

Answer 2

I'm assuming you want to prove the following proposition: $$\frac{a}{b} = \frac{c}{d} \text{ iff } ad = bc$$ Then you can just do this:
Let $a,b,c,d$ be arbitrary such that $b,d\neq 0$. Then

$\begin{align} \frac{a}{b}=\frac{c}{d} &\Leftrightarrow bd\Big(\frac{a}{b}\Big)=bd\Big(\frac{c}{d}\Big)\\ &\Leftrightarrow d\Big(\frac{a}{1}\Big)=b\Big(\frac{c}{1}\Big)\\ &\Leftrightarrow ad = bc \ \ \ \ \square \end{align} $