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It always seemed to me any number X divided by zero would simply be X, since we're dividing by nothing, so then the original number wouldn't be altered. Why isn't this true? Can this ever be true?

Jack M
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Amy
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    You could equally well argue that if you multiply by nothing then the original number wouldn't be altered. – littleO May 19 '14 at 12:50
  • What is $1 \times 0$, then? On the one hand, $1 \times 0 = (1 \times 0)/0 = 1$. On the other hand, $1 \times 0 = (1 \times 0)/1 = 0$. Which is correct? – Ben Grossmann May 19 '14 at 12:54
  • You could define $1/0$ as anything you want, but any such definition will not be useful, and we'd lose the useful properties of division, specifically, that $\left(\frac{a}{b}\right)\cdot b = a$ when $\frac a b$ is defined and that $\frac{ab}{ac} = \frac{b}{c}$ when the latter is defined and $a\neq 0$. – Thomas Andrews May 19 '14 at 12:56
  • There are some circumstances where you can divide by zero. For example, in the extended complex plane, we adjoin a symbol $\infty$ to $\Bbb C$ and impose a few restrictions so that for any $a\in\Bbb C\backslash{0}$, $\frac{a}{0}=\infty$. [See Priestley's "Introduction to Complex Analysis: Second Edition," page 19, for details.] – Shaun May 19 '14 at 13:03

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Division means seperating a number into equal parts. So dividing by 2 should give you two equal parts, and dividing by 1 would give you one part, which is the number itself. Therefore, it is not possible to divide something into zero equal parts.

Think of a cake, you can slice it into 2 or 3 equal parts, you could leave it as 1 part, but it isn't possible to slice it into zero equal parts.

Tymric
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Division is the inverse of multiplication. If $X\div Y=Z$, then $Z\times Y=X$. If you decide $X\div 0=X$, then $X\times0=X$ !