Editing Division by zero (section)

===Fallacies===
{{further|Mathematical fallacy}}
A compelling reason for not allowing division by zero is that allowing it leads to [[fallacies]]. 

When working with numbers, it is easy to identify an illegal division by zero. For example:

:From <math>0\times 1 = 0</math> and <math>0\times 2 = 0</math> one gets <math>0\times 1 = 0\times 2.</math>  Cancelling {{math|0}} from both sides yields <math>1 = 2</math>, a false statement.

The fallacy here arises from the assumption that it is legitimate to cancel {{math|0}} like any other number, whereas, in fact, doing so is a form of division by {{math|0}}.

Using [[elementary algebra|algebra]], it is possible to disguise a division by zero<ref name="Kaplan" /> to obtain an  [[invalid proof]]. For example:<ref>{{harvnb|Bunch|1982|page=15}}</ref>
{{block indent|em=1.6|text=Let <math>x = 1</math>. Multiply both sides by <math>x</math> to get <math>x = x^2</math>. Subtract {{math|1}} from each side to get <math display=block>x - 1 = x^2 - 1.</math> The right side can be factored, <math display=block>x - 1 = (x + 1)(x - 1).</math> Dividing both sides by {{math|''x'' − 1}} yields <math display=block>1 = x + 1.</math> Substituting {{math|1=''x'' = 1}} yields <math display=block>1 = 2.</math>
}}
This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote {{math|0}} as {{math|1=''x'' − 1}}.