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$\frac{a}{b}\neq0 \Rightarrow (a\neq0\land b \neq0)$

At first sight that seems quite obviously true, however, wouldn't $b = 0$ also fit the condition?

$\frac{a}{0}\neq0$

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This statement is not really covered by field axioms. It is sort of like a "junk statement" like saying $1 \in 3$. –  Andrew Salmon Apr 19 '13 at 17:35
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@AndrewSalmon But that is true! $3=\{0,1,2\}$. –  Pedro Tamaroff Apr 19 '13 at 17:39
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4 Answers

up vote 4 down vote accepted

Asumming they are real numbers, because the object $\displaystyle\frac{a}{b}$ does not exists if $b=0$ (because if $b=0$ then $bb^{-1}=b^{-1}b=0\neq 1 \forall b^{-1} \in \mathbb{R}$

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care to explain that $bb^{-1}=b^{-1}b=0\neq 1 \forall b^{-1} \in \mathbb{R}$ part in more layman terms? –  Mr D Apr 19 '13 at 17:57
    
Yes of course. When we say that we divide by a number $a$ what we are actually saying is that exists a number $a^{-1}=\frac{1}{a}$ such that $$a^{-1}a=aa^{-1}=1 $$, and that the division is the multiplication by that $a^{-1}$ number. If the number $0$ is the number such that $ab=0 \forall a in \mathbb{R}$ there cannot be a number such that $00^{-1}=0^{-1}0=1$. This is, we have proved that the division by zero cannot be performed. –  Jorge Apr 19 '13 at 18:00
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The symbol $\frac{a}{0}$ carries no meaning, and thus must be disallowed.

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I think this is too strong. There is no element of $\mathbb{R}$ that satisfies $x = \frac{a}{0}$, but that doesn't mean the symbol carries no meaning. And there are algebraic structures which do have an element $\infty = \frac{a}{0}$, e.g. the real projective line. (I am under the impression that such structures cannot be fields, but I could be wrong.) –  Zack Apr 19 '13 at 20:25
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Maybe better way to put it: $\frac{a}{b}$ denotes the application of the division operator to two values, $a$ and $b$. When $b=0$, assuming we are working in $\mathbb{R}$, this is an invalid operation because one of the operator's inputs is outside the relevant domain. But the symbol $\frac{a}{0}$ still has meaning: its meaning is that invalid operation. –  Zack Apr 19 '13 at 20:36
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For a formula (a sequence of symbols) to have a truth value, it must be a sentence, i.e., it must make sense.

The formula $\displaystyle \frac{a}{0}\neq 0$ isn't a statement because it doesn't make sense. You can't tell wether it is true or false. It is not true, it is not false, for it is not.

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When we write $\frac{a}{b}$, we always mean that $a$ is anything and $b \ne 0$ (since if $b $ is zero, $\frac{a}b$ would be undefined). That is why we still include $b \ne 0 $ besides $a \ne 0$ to remove all the possible ambiguity.

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