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Last week someone asked me if I could solve $3x+5 = 3x-5$. I think he just looked up unsolved problems or something like that, but as far as I can tell it has no solution... other than if $x$ was positive and negative $5/3$. So I told him it was unsolvable because you get $0=10$ or $0=-10$, but I started thinking about numbers being positive and negative. Like if you graph a number divided by zero you get an asymptote approaching positive infinity from one side and negative infinity from the other, so while it's unbounded it's also approaching positive and negative infinity simultaneously (as far as I understand it at least). Ultimately, I was wondering if a number could be positive and negative at the same time. Intuitively I don't think that could exist, but I can't find anything on it when I've looked it up and I was hoping someone else would have the answer. Thank you.

[Also the tags probably aren't totally accurate for the question it just seemed like a good way to get some attention.]

Patrick Stevens
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RegularHumanBeing
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    How do you define positive number? And negative number? – Another User Mar 28 '24 at 23:42
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    I was thinking all real numbers. So positive would just be greater than 0 and negative is less than 0. – RegularHumanBeing Mar 28 '24 at 23:46
  • If you wanted to, you could just define a number, say $a$, that is its own negative. However, then $a + 2a = 3a$ and $a + 2a = a - 2a = - a = a$ so $a = 3a$. This would work for any multiple of a, i.e. $...=-2a = -a = a = 2a = 3a = ...$. So I can't see that it would be too useful. (in the sense that doing standard operations on $a$ will just give $a$ again) (Maybe there some definition of a number system that leads to interesting maths though.)

    Doing this reminds me of surreal numbers. Sometimes defining numbers with certain unusual properties gains useful results.

     – psychgiraffe Mar 28 '24 at 23:53
  • If a real number $x$ has $x, -x$ both positive, then $x + (-x) = 0$ would also be positive. – anomaly Mar 28 '24 at 23:57
  • But go to France, and a number can be positif and négatif at the same time. – peterwhy Mar 29 '24 at 00:04
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    If you show that an equation has no solution, then you solved it. It is not unsolvable as said in the question. – Taladris Mar 29 '24 at 00:10
  • Despite in some countries it might be different handled , there is a clear consens among the mathematicians that "positive" rules out $0$. It makes no sense to claim that $0$ is both negative and positive , it is neither. Also , a real number equal to its negative must be $0$ : $x=-x\implies 2x=0\implies x=0$. But as said : $0$ is neither positive nor negative. – Peter Mar 29 '24 at 14:20

1 Answers1

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No.

  • Show that no integer can be both positive and negative. (Precisely how to do this may depend on your definition of "integer"!)
  • Therefore show that no rational number can be both positive and negative.
  • Therefore show that no real number can be both positive and negative (by approximating it by rationals: either all sufficiently good rational approximations to $x$ are positive, or they're all negative, or there are arbitrarily good rational positive and negative approximations, in which case show that the number is $0$).

(There may be easier ways, but this has the least machinery, since it's working directly from the definitions of the relevant objects.)

Patrick Stevens
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  • Thanks for the answers, it definitely makes sense if you think about a number line with positive and negative numbers heading opposite directions, so you'd have to really warp space to get them to overlap. – RegularHumanBeing Mar 29 '24 at 00:13