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Law Stack Exchange has a question about a hypothetical law that everyone is meant to be guilty of breaking.

What if a law is literally impossible to follow?

While that's an interesting idea, they used an example law which touched on mathematics that I'm not sure would keep our better mathematicians behind bars.

Every citizen is required to provide the government with a valid solution to the equation 0x = 50, and will otherwise be jailed.

I remember $0 \cdot \infty$ as an indeterminate form, meaning you can't say it is or isn't something without more information. But can't I insist that, whatever $x$ is, it's something that makes this true?

$$0x = 50 \iff \lim_{|y|\to\infty} \left(\frac{50}{y} \cdot y \right) = 50$$

I mean, they didn't even say $x$ had to be a real number. Am I crazy, or are mathematicians about to rule the world?

To make this question a little more answerable: What's the simplest way to show that the law isn't fulfilling it's objective?

candied_orange
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    How did you get that limit expression exactly? I fail to see how that limit is analogous to $0 \cdot \infty$. – PrincessEev Dec 16 '18 at 09:07
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    Also, a personal note: I think you're overthinking the example. There are absolutely topologies of the real numbers in which "nonsense" expressions like $0 \cdot x = 50$ have solutions - indeed, we can even define systems in which division by $0$ is defined (though results in things we do not desire, if just in application). But actually trying to consider the logic of the equation is probably well beyond what the post intended. – PrincessEev Dec 16 '18 at 09:08
  • $0x=50\implies x=50\ \because050=50$ – Shubham Johri Dec 16 '18 at 09:11
  • @EeveeTrainer are you arguing that it's really a different indeterminate form? – candied_orange Dec 16 '18 at 09:19
  • I'm not arguing anything. I'm simply asking for your derivation on the limit expression and its relation to $0 \cdot x = 50$. If it's because "well, no matter how big $x$ gets, $0 \cdot x = 50$, so $x = \infty$" that's another issue altogether. (Namely that in the usual sense of real numbers, $\infty$ isn't a number.) – PrincessEev Dec 16 '18 at 09:22
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    There is no real number $x$ such that $0 \cdot x = 50$. If a law required you to produce such a real number, the law could not be obeyed. – littleO Dec 16 '18 at 09:22
  • Are you working over the reals? Because the equation has a unique solution in the zero ring. – cansomeonehelpmeout Dec 16 '18 at 09:22
  • Like, no one is saying the equation doesn't have solutions - but it needs to be in the appropriate context for such solutions to properly exist. In the usual sense of real numbers, for example, there is no solution. – PrincessEev Dec 16 '18 at 09:24
  • @candied_orange $\infty$ is not a number, and $x$ can't 'equal' $\infty$. You can't assert that you're not guilty of breaking the law by insisting whatever $x$ is, it must satisfy $0x=50$, because you don't know if such an $x$ exists, and its existence has to be demonstrated. – Shubham Johri Dec 16 '18 at 09:24
  • @cansomeonehelpmeout the law made no mention of real numbers. – candied_orange Dec 16 '18 at 09:26
  • Well, if you want to be pedantic, then, sure - since the context of the equation wasn't defined, it absolutely has a solution. We can construct that solution to be whatever we choose by choosing whatever context, mathematical framework, system, we want. (I don't think that'd hold in a court of law though considering the overwhelming majority of people use the usual senses/definitions of real numbers and their topology/operations, to the point that it's pretty much implied in a case like this that we are working in those exact circumstances.) – PrincessEev Dec 16 '18 at 09:28
  • @EeveeTrainer I happen to have gotten that limit from here. Just stuck the 50 in. To make this work you could say I'm using the lim to define what this particular infinity is. – candied_orange Dec 16 '18 at 09:28
  • As far as your limit is concerned, it is the limit of the product of two quantities which only tend to $0$ and $\infty$, but never become equal to them. You also can't express the limit of the product as the product of the limits, since $\displaystyle\lim_{y\to\infty}y$ doesn't converge – Shubham Johri Dec 16 '18 at 09:29
  • That post literally is under the assumption that the laws for the addition/multiplication of natural numbers holds for infinity - it was an argument made to show some of the ridiculousness in letting $\infty$ be a real number. – PrincessEev Dec 16 '18 at 09:31
  • @ShubhamJohri So the problem here is simply my choice of limit? If so I can keep looking for one in the right form. – candied_orange Dec 16 '18 at 09:33
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    One could always artificially introduce a new number (let's call it $\alpha$) and extend the definition of multiplication so that $0 \times \alpha = 50$, but this would be kind of silly (and the resulting number system would not be interesting or useful). Surely the intention was that the law requires citizens to provide a real number $x$ such that $0 \cdot x = 50$. – littleO Dec 16 '18 at 09:35
  • @littleO the intention of the law was to ensure everyone was guilty. Your only hope is to get a judge who sticks to the letter of the law and knows a little math. :) – candied_orange Dec 16 '18 at 09:44
  • @candied_orange No, it is not about the choice of the limit. It is more about how whatever limit of the so-called "$0\cdot\infty$" form equal to $50$ you chose, the quantity inside the limit is only going to tend to $0$, never actually be $0$ – Shubham Johri Dec 16 '18 at 09:44

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Don't confuse the two next equations : $$0 \text{ multiplied by }x=50 \tag 1$$ $$0 \text{ concatenated with }x=50 \tag 2$$ Both are loosely written as $0x=50$. That's the trap.

Of course $(1)$ has no finite solution.

Obviously the solution of $(2)$ is $x=50$ because $$0 \text{ concatenated with }50=050=50.$$

The ambiguous question implicitly refers to $(2)$, not to $(1)$ and the expected answer is $x=50$.

JJacquelin
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    I gave the same answer but someone other than the OP edited the question to make it $0\cdot x$, invalidating my(our) response. – Shubham Johri Dec 16 '18 at 09:36
  • This is not at all what I had in mind and that's wonderful. I've edited the question (twice now) to be faithful with the post on law. The only issue is now my question needs better tags. – candied_orange Dec 16 '18 at 09:55
  • @ Shubham Johri. I fully agree with you. That's a drawback of StackExchange forum : Somebody can modify the wording of a question or of an answer without always fully understanding what really is the meaning. Fortunately it is not often that happens. – JJacquelin Dec 16 '18 at 10:02
  • @ShubhamJohri, et al. I edited the question title and another instance to match what was (at the time) the equation in the yellow "quotation box". I should've examined the question history to see if that notation had previously been changed from OP's original intent. That said ... I prefer the version with the "$\cdot$", as it leaves open the possibility of defining the operation (say, as concatenation, or addition, or the "second operand" function $a\cdot b =b$, or, or, or, ...). :) – Blue Dec 16 '18 at 10:18