I've just been going through an equation, which is as follows:

$(x+4)^2 = 16$

Lets work through it:

$$x^2 + 8x +16 = 16$$ $$x^2 +8x = 0$$ $$x^2 = -8x$$ $$x = -8$$

However, as i've just found out, $0$ is also a solution, as $0 = -8$. Oh wait, no it doesn't.. And yet, $0^2 + 8 \times 0 + 16 = 16$.

What's going on here though? Does $0$ even matter? It's true that $16 = 16$, but it's not true that $0 =-8$. But maybe $16$ equals $16$, not because $-8^2 - 8 \times 8 = 0$, but just because $16$ equals to itself, independent of $0$.

I think it's interesting to draw a parallel between the falsity of the final solution (when $x = 0$), and the fact that nothing, represented by $0$, literally doesn't seem to exist in the real world.

Is $0$ ever used however? In either pure mathematics for a particular proof, or perhaps within the domain of science? I suppose one use of $0$ would be to do a 'hard reset', for whatever reason, but then again, you could just do that without $0$, I suppose.

Furthermore, maths in supposed to be consistent, and since the consistency of mathematics seems to be disrupted by the usage of $0$, does that mean $0$ is somehow 'dangerous'. My example here was simple, but what if you were say, launching a rocket to space, where lives are on the line. Can you trust $0$?

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    $\begingroup$ Seems strange that you're happy with $-8$ but not with $0$. Anyway, your computation for a different equation would get to $(x-1)^2=-9(x-1)$, giving the exact same phenomenon where $1$ and $-8$, instead of $0$ and $-8$, are the solutions. $\endgroup$ – Kevin Carlson Oct 7 '15 at 0:19
  • $\begingroup$ Remember that I was just talking of the final result. In that, the final equation is the same as the first, but is made false by the use of 0, yet not in the first, giving rise to conflict, which in turn gave rise to this question. $\endgroup$ – Jim Jam Oct 7 '15 at 0:35
  • $\begingroup$ There are lots of equations with more than one solution. For example, $x^2+2=3x$ has two solutions, $1$ and $2$. $\endgroup$ – Akiva Weinberger Oct 7 '15 at 1:09
  • $\begingroup$ It wasn't about that, it was about 0. $\endgroup$ – Jim Jam Oct 7 '15 at 1:10

There is no inconsistency - you just made a mistake. You cannot conclude "$x=-8$" from "$x^2=-8x$". When you try to divide both sides by $x$, you need the hypothesis that $x\not=0$!

How your argument should go is, " . . .so $x^2=-8x$. Now, if $x\not=0$, then we can divide both sides by $x$ to get $x=-8$. So $x$ must be either $0$ or $-8$. Checking the original equation shows that both $0$ and $-8$ work, so those are our solutions."

Don't worry, math is safe. :D There's nothing weird about zero, other than that you can't divide by it. The moral is that, when solving a problem, you need to think about what you're doing - in particular, at each step, you need to understand why what you are saying is true. Otherwise, it's easy to convince yourself that an argument makes sense, or is complete, when in fact it does/is not.

  • $\begingroup$ Mmm... I see your point, but 0 has the strange property of being the only number that we can't divide by (or perhaps we can, it's just that the solution is unknown). I also don't like the way that the final solution can have any two solutions (which are both valid) if (x+4)^2 is anything other than 16, say, 25, then using the same methodology, you obtain 1 and -9. Thanks for your answer by the way :) $\endgroup$ – Jim Jam Oct 7 '15 at 0:23
  • $\begingroup$ @user108262 Well, just because you don't like something doesn't mean it's false. :) An equation defines a set - the set of numbers which make it true. Some equations, like $x+1=2$, define a set containing a single element (=unique solution); some equations, like $x+1=x$, define the empty set (=no solution); and some equations, like $x^2=1$, define a set with more than one solution. Sometimes there isn't a unique solution; that's just the way it is. $\endgroup$ – Noah Schweber Oct 7 '15 at 0:25
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    $\begingroup$ As for the weirdness of zero: yes, zero is a bit weird. But that doesn't mean anything is "wrong" about zero! It's just a property that zero has, that we need to be aware of. Some numbers are prime; that's weird, but that's just a property they have. Etc. $\endgroup$ – Noah Schweber Oct 7 '15 at 0:26
  • $\begingroup$ Towards the confusion re: multiple solutions, think for a moment about other ways I can describe a number: "$x$ is bigger than 3." "$x$ is a positive integer which is a perfect square." "$x$ is the number of ducks my sister is currently throwing pieces of bread at." In each case, you need more information (either about $x$, or about ducks) to figure out exactly what $x$ is. This isn't any different than what's going on with equations - we're just conditioned by early algebra classes to think that equations ought to have unique solutions. Think of equations as information - possibly incomplete! $\endgroup$ – Noah Schweber Oct 7 '15 at 0:29
  • $\begingroup$ After some thought, I think that your are right, I don't think anyone could ever be certain but I think your rationale is sound. To me, what your saying is that we need 0 (or at least desire it) to represent the absence of something, there are (hopefully) 0 people with taste buds in their backside. So, 0 is just a way of representing data, though with different properties. $\endgroup$ – Jim Jam Oct 7 '15 at 0:41

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