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I don't know if this is a basic question or whatever, but I can't seem to find an answer.

As far as I understand the reciprocal of a number the inverse of that number, that still doesn't clarify why it is needed.

For many years I've only ever done math like if I were a robot. I just did it and never understood what I was doing. So when I went and divided fractions I just used the reciprocal, because "that was the way to do it". I want to understand math at a deeper level, especially subjects like probability, statistics, calculus, and linear algebra. To do that I have to understand the fundamentals however.

Any response is appreciated.

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I think you're asking why the rule for division of fractions, $$\frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \cdot \frac{s}{r},$$ works. And I'm assuming that you're already comfortable with how to multiply fractions.

We need to go back to what division is supposed to achieve in the first place. When we look into that, the answer is that $A\div B$ means something that gives $A$ when we multiply it by $B$ -- or, written in symbols, $A\div B$ means the $X$ that solves the equation $$ X\cdot B = A $$

When our $A$ and $B$ are fraction, the "reciprocal" division rule can be regarded as a trick that happens to produce an $X$ that works. It's easy enough to see that it does work: If we're dividing $\frac pq \div \frac rs$ we need to solve the equation $$ X \cdot \frac rs = \frac pq $$ And indeed setting $X=\frac pq\cdot \frac sr = \frac{ps}{qr}$ does this: $$ \frac{ps}{qr}\cdot\frac rs = \frac{ps\cdot r}{qr\cdot s} = \frac{p\cdot sr}{q\cdot sr} = \frac pq$$ like we want. (I'm also assuming that you're comfortable with cancelling the common factor $sr$ in the middle fraction).

This computation hopefully also gives some ides why it works, at least part way. In $\frac{ps}{qr}$ the $p$ and $q$ are what we want to end up with, and the $s$ and $r$ are there to "neutralize" the $r$ and $s$ we have but want to discard. By making sure that the product has exactly one $r$ and one $s$ on each side of the fraction bar they make sure we can cancel them away.

Writing the solution $\frac{ps}{qr}$ as $\frac pq\cdot \frac sr$ might be best understood as just an easy way to remember what goes where. But this memory trick itself then also serves as motivation for considering the reciprocal to be an interesting operation in its own right in higher algebra.

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Your question isn't completely clear but what I understood is that you don't get why $$\frac{\frac{a}{b}}{\frac{c}{d}}= \frac{a}{b}*\frac{d}{c}$$ the answer it's located in the axioms of the real numbers, a number $b$ it's the reciprocal of a number $d$ if $$ d*b=1$$ now, let's see the definition of fraction $$e/f=e*f^{-1}$$ with $f^{-1}$ the reciprocal of $f$, therefore $$\frac{\frac{a}{b}}{\frac{c}{d}}={\frac{a}{b}}({\frac{c}{d}})^{-1}$$ and since $$\frac{c}{d}*{\frac{d}{c}}=1$$ we have $$\frac{\frac{a}{b}}{\frac{c}{d}}= \frac{a}{b}*\frac{d}{c}$$ our result

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There are already some excellent algebraic answers to this question, but I'd like to provide an answer based on the grade school meaning of division.

When we divide 20 by 4, we're asking for the answer to the question "Given 20 items distributed evenly to 4 piles, how many items are in each pile?"

When we divide 8 by a third, we're asking for the answer to the question "Given 8 items distributed evenly to a third of a pile, how many items are in each whole pile?"

Because each third of a pile has 8, and we want to know how much a whole pile (3 times larger) has, we can use multiplication to compute the answer as 24. Using multiplication of the inverse is just a handy shortcut though, and need not be though of as division itself.

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