Would this count as a paradox?

Question

As we all know, every number (at least real) multiplied by 0 is 0 ($a * 0 = 0$). And on the other hand we have that $$\frac{a}{b}\rightarrow \exists\text{ c such that } a = bc $$ Now, we know that for most cases this is true as long as $b \neq 0,$ because if $a \neq 0$ and $b = 0$ it will not work, because there would be absolutely no number $c$ which satisfies $a = 0 * c$.

But if $a,b = 0$ we would have the equation $0 = 0 * c$, in which it turns out that $c$ can be any number, because every number satisfies this equation.

So if we have $$\frac00 = \frac00,$$

would this count as a paradox?

Which number do you intend to multiply with $0$? Note that $1/0$ is undefined, so it is not a number (so your first statement does not apply). — Thomas, Jan 28 '20 at 19:44
Hmm. It seems like you're being a bit unclear about some things - what do you mean by an expression implying something? What do you even mean by $0/0$ (how do you define it?) It's currently not so much paradoxical as just a bit vague. — Izaak van Dongen, Jan 28 '20 at 19:46
Is your argument $0\times a=0\times b\implies\frac{0\times a}{0}=\frac{0\times b}{0}\implies a=b$? — J.G., Jan 28 '20 at 19:46
It sounds as though you think that $\frac{0}{0}$ is simultaneously equal to every possible real number... and since $\frac{0}{0}$ is the same expression as itself, it should equal itself implying that every possible real number is simultaneously equal to every other possible real number. That is not the case for the real numbers. We have decided that $1\neq 0$. We know that a number is only ever equal to one thing at a time... and so since our logic would suggest that $\frac{0}{0}$ is simultaneously equal to multiple different things, that leads to that $\frac{0}{0}$ must not exist. — JMoravitz, Jan 28 '20 at 19:52
As such, any statement or expression that includes $\frac{0}{0}$ we immediately discard as using invalid syntax involving an undefined term. $\frac{0}{0}=\frac{0}{0}$ is meaningless. — JMoravitz, Jan 28 '20 at 19:54
"we know that that implies the equation $0∗a=0$": you seem to know this. I don't. — , Jan 28 '20 at 21:04

score 1 · Answer 1 · edited Jan 28 '20 at 20:56

1

While $$0\times a=0$$ is true for evey real number, the expression $$a=\frac {0}{0}$$ is not a mathematical statement because $\frac {0}{0}$ is not a number.

Thus $$\frac00 = \frac00$$ does not make sense so it can not be considered a paradox.

edited Jan 28 '20 at 20:56

answered Jan 28 '20 at 19:53

Mohammad Riazi-Kermani

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score 0 · Answer 2 · answered Jan 28 '20 at 20:52

Every math is based on a set of assumptions. If you are using regular math that people generally agree on then you go with Mohammad's answer: division by zero is undefined and you can't use it. Let's say you want to make up your own math where division by zero is defined, and everything else about math is still the same. One of the first things people ask about new sorts of math is whether it's consistent. If you can take your assumptions and apply proper logic to it and come up with something that contradicts any of your assumptions or logical results, then your new system is not consistent. Generally speaking, that means the system is flawed and it's not helpful to apply logic to it to get good results.

Suppose for instance I try to say that 1+1=1. In regular math, it's wrong, but in a made up system, it could be right. One such system is that 1 is the only number. 1+1=1, 1-1=1, 1*1=1, 1/1=1. This system is consistent.

So what if division by zero is defined? In that case, I'm not sure I follow your original argument. Are you saying that 0/0 = 0? so that the zero on the right side of a*0=0 turns into a*0=0/0 and then using a=0/0 turns the whole thing into 0/0 * 0 = 0/0? I guess that all works if 0 is the only number, but you will run into trouble if you start to allow other stuff. For instance, in the real numbers a/a=1. So 0/0=1, but if 0/0 is also 0 then we've hit a contradiction.

It doesn't matter what people have tried to define division by zero as; it's always created inconsistency. That's why they leave it undefined.

Now on to whether it's a paradox.. I didn't find an official definition for mathematical paradoxes, but I did find this general one on google: a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true.

So no; it's just inconsistent.

score 0 · Answer 3 · answered Jan 28 '20 at 21:00

In what way would

$$e=e$$ where $e$ is some mathematical expression count as a paradox ?

Either $e$ has a defined value and $e=e$ obviously holds, or $e$ is a meaningless expression and so is $e=e$.

E.g.

$$3x^2-\pi=3x^2-\pi$$

holds, while

$$+)\sqrt{*^\aleph}=+)\sqrt{*^\aleph}$$ is just meaningless.

Needless to say, $\dfrac00$ does not represent "any real number". It represents nothing.

Would this count as a paradox?

3 Answers3