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Goal: I need help to intuitively understand division by a fraction.

Background: I've read this and this. I understand that division is "even distribution" so, with integers, $\frac 6 2 = 3$ could mean that 6 friends get 2 chocolates each.

Observation: When the denominator is >0, the solution is less than the numerator (e.g., $\frac 3 4$ and $\frac 4 3$), but when 0 < denominator < 1, the solution is larger than the numerator (e.g., $\frac {1} {\frac 1 2}$ = 2). Graph of y=$\frac 1 x$.

Problem: "Dividing" (as understood by "into smaller parts") by a fraction doesn't seem possible: you end up multiplying, never dividing--and yes, I understand that division by a fraction is multiplication by that fraction's multiplicative inverse (Khan Academy).

Question: Since 3--2 (three minus negative two) = 3+(-(-2))--i.e., addition not subtraction (source)--is division by a fraction similar? In other words, is division by a fraction not division at all, but multiplication?

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The definition of division is: $ a \div b$ is $c$ where $a = b \times c$. In the case of positive integers $b$ you can interpret this as even distribution of an amount $a$ into $b$ equal parts. This is because if $b$ is a positive integer, $b \times c$ is the result of adding together $b$ copies of $c$.

If $b$ is a fraction $m/n$, where $m$ and $n$ are positive integers, you can think of $b \times c$ as the result of adding together $m$ copies of $c/n$, i.e. you split $c$ into $n$ equal pieces, and you add together $m$ of those pieces (or pieces equal to one of those if $m$ is greater than $n$). Thus $a/b$ would be an amount $c$ that if you divide into $n$ equal pieces and add $m$ of those pieces together you get $a$.

However, when you go beyond fractions (e.g. to real numbers), even this doesn't make sense. Again, the definition is in terms of multiplication, the interpretation is secondary.

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  • $\begingroup$ Thank you! Just to clarify (at least for me), the second paragraph refers to re-organizing a/(m/n) = c as m * (1/n) * c = a $\endgroup$ – mellow-yellow Mar 10 '17 at 15:38

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