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Version 3 |
| Suppose that we construct a number system where we can divide by zero, what are the consequences? |
Suppose that we construct a number system where we can divide by zero, what are the consequences? |
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| Dividing by $x$ means there exists an inverse $\frac{1}{x}$ |
Dividing by $x$ means there exists an inverse $\frac{1}{x}$ |
| so that $x\cdot\frac{1}{x}=1$. So if we could divide by 0 then we would have |
so that $x\cdot\frac{1}{x}=1$. So if we could divide by 0 then we would have |
| a number we would call $\frac{1}{0}$. |
a number we would call $\frac{1}{0}$. |
| We know $0\cdot x=0$ for any $x$, so $0\cdot \frac{1}{0}=0$. However |
We know $0\cdot x=0$ for any $x$, so $0\cdot \frac{1}{0}=0$. However |
| $0\cdot\frac{1}{0}=1$ because $x\cdot\frac{1}{x}=1$ for all $x$. Therefore $0=1$. |
$0\cdot\frac{1}{0}=1$ because $x\cdot\frac{1}{x}=1$ for all $x$. Therefore $0=1$. |
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| The problem is worse than forcing $0=1$. Indeed, $x\cdot 1=x$ for all $x$, so |
The problem is worse than forcing $0=1$. Indeed, $x\cdot 1=x$ for all $x$, so |
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| \[x=x\cdot 1=x\cdot \left(0\cdot \frac{1}{0}\right)=(x\cdot 0)\cdot \frac{1}{0}=0\cdot \frac{1}{0}=0.\] |
\[x=x\cdot 1=x\cdot \left(0\cdot \frac{1}{0}\right)=(x\cdot 0)\cdot \frac{1}{0}=0\cdot \frac{1}{0}=0.\] |
| So indeed every number is then 0. |
So indeed every number is then 0. |
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| \emph{Therefore, in a number system were we can add, subtract, multiply and divide (a field), we do not allow division by 0 because it would force the number system to only contain 0.} |
\emph{Therefore, in a number system were we can add, subtract, multiply and divide (a field), we do not allow division by 0 because it would force the number system to only contain 0.} |
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| There are various thought experiments that can also be used to explain the omission of division by zero, for instance: |
There are various thought experiments that can also be used to explain the omission of division by zero, for instance: |
| \begin{quote} |
\begin{quote} |
| How could you slice a pizza into slices that have 0 width? |
How could you slice a pizza into slices that have 0 width? |
| \end{quote} |
\end{quote} |
| Since this would correspond to division by zero, it seems intellectually impossible. There is however an unfortunate side effect from such thinking. |
Since this would correspond to division by zero, it seems intellectually impossible. There is however an unfortunate side effect from such thinking. |
| For example: |
For example: |
| \begin{quote} |
\begin{quote} |
| How could you slice a pizza into slices of width -7, width is always a positive |
How could you slice a pizza into slices of width -7, width is always a positive |
| quantity. |
quantity. |
| Furthermore, how could you actually slice a pizza into slices of width $\sqrt{7}$ or $e^\pi$, etc. Yet we can actually divide by each of these |
Furthermore, how could you actually slice a pizza into slices of width $\sqrt{7}$ or $e^\pi$, etc. Yet we can actually divide by each of these |
| numbers. |
numbers. |
| \end{quote} |
\end{quote} |
| This is an example of how mathematical abstraction benefits the understanding |
This is an example of how mathematical abstraction benefits the understanding |
| of a problem. |
of a problem. |
