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Let's say we need to prove $$\frac{1}{2} = \frac{2}{4}$$ (as an example) Usually, $\frac{1}{2}$ is manipulated into $\frac{1 \cdot 2}{2 \cdot 2}$, which is $\frac{2}{4}$. So we ended up with what we're trying to prove.

Is it possible to prove this by, say, cross-multiplying: $$\frac{1}{2} = \frac{2}{4} \implies 1 \cdot 4 = 2 \cdot 2 \implies 4 = 4$$ Thus turning the unproven statement into something we know is true? You would have to initially assume that $\frac{1}{2}=\frac{2}{4}$ before doing so.

Thanks!

terran
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UserX2021
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    You can do this only when you can "reverse" your steps and implications. For example, if you want to prove that $-1=1$ and you say $(-1)=1 \implies (-1)^2 = 1^2 \implies 1=1$, hence $-1=1$ then your logic would be flawed because the implication $(-1)^2 = 1^2 \implies -1=1$ is not true, although its reverse is. In your case, you can reverse the implications, hence your proof is not correct but you can reverse the implications to obtain a correct proof. – Sarvesh Ravichandran Iyer Dec 19 '23 at 06:43
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    You can multiply any equation by 0 to get 0=0. – Brady Gilg Dec 19 '23 at 06:47

2 Answers2

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Look carefully at the beginning and end of your post.

"We need to prove $\frac12=\frac24$... [we] would have to initially assume that $\frac12=\frac24$."

In other words, the only way you would know the statement was true at the end would be if you already knew it was true at the beginning. So this does not prove anything. It's a bit like (apologies for possibly controversial example):

"I won the election!"

"Prove it."

"Well for a start, I definitely won the election..."

In your case, although the argument is not correct, you can fix it by reversing all the steps. Note, however, that this is not always possible. For example,

"$x=3$ and $y=4$, therefore $x+y=7$" is correct

but if we try to reverse this argument

"$x+y=7$, therefore $x=3$ and $y=4$" is not correct.

David
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    Yes, that's what I thought. So having a chain of reasoning up to something true ($\frac{1}{2} = \frac{2}{4} => 4=4)$ is not valid, because the steps up to this conclusion are based on the conclusion itself? – UserX2021 Dec 19 '23 at 06:51
  • You need to be a bit careful with how you say this. The implication $\frac12=\frac24\Rightarrow4=4$ is true, and you could conceivably use it as part of a proof that $4=4$ (just supposing you would want to do that!). But you cannot use it to prove $\frac12=\frac24$. – David Dec 19 '23 at 06:54
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Cross-multiplying is the only way to prove this, because the definition of $\Bbb{Q}$ is done by starting from ordered pairs $(a,b)$ of integers where $b\ne 0$ and then defining an equivalence relation saying $$(a,b)\equiv (c,d) \iff ad=bc$$ and $\frac{a}{b}$ is notation for the equivalence class of the pair $(a,b)$. So when you claim that $$\frac{1}{2}=\frac{2}{4}$$ you're claiming that $(1,2)\equiv (2,4)$, and the definition of the equivalence is checking whether $1\cdot 4 = 2\cdot 2$.

Chad K
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