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You can divide by zero, but unfortunately it isn't very useful.
The rationals (fractions; numbers like [ieq] \frac{a}{b} [/ieq]) are usually defined such that [ieq] b [/ieq] is not zero, so suppose we get rid of this constraint. Then we can define a number [ieq] \phi [/ieq] such that [ieq] \phi = \frac{a}_{0} [/ieq]. Under the usual rules we can know that therefore [ieq] \phi * 0 = a [/ieq], so thus [ieq] 0 = a [/ieq]. Notice that we could have chosen any [ieq] a [/ieq], so allowing division by zero "forces" us to conclude that there is only one number with zero as its denominator, namely the one with zero as the numerator. At this point we start to realize that this number is pretty anomalous, if it's indeed a number at all.
Other rules go out the window as well. We also have a rule for fractions, called an
equivalence relation, that tells us when two fractions are equal - namely exactly when [ieq] \frac{a}{b} = \frac{p}{q} [/ieq] is true. This rule is cross-multiplying - we say two fractions are equal if and only f [ieq] aq = bp [/ieq]. Suppose we want to check if some rational [ieq] \theta [/ieq] is equal to [ieq] \phi [/ieq]. If [ieq] \theta = \frac{m}{n} [/ieq] then [ieq] \theta = \phi [/ieq] if and only if [ieq] m * 0 = n * 0 [/ieq]. We didn't specify [ieq] m [/ieq] and [ieq] n [/ieq] - this equation is always true. So [ieq] \phi [/ieq] is equal to every other rational. This exposes the degeneracy of the concept of dividing by zero. You can investigate other properties of this "number", but it quickly becomes apparent that it really doesn't belong.
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