I don"t recognize if this is a simple question or whatever, but I can"t it seems to be ~ to find an answer.

As much as I understand the mutual of a number the station of the number, the still doesn"t clarification why the is needed.

For many years I"ve only ever done rememberingsomer.com favor if i were a robot. I just did it and also never understood what i was doing. So as soon as I went and divided fountain I just used the reciprocal, due to the fact that "that was the means to perform it". I want to understand rememberingsomer.com in ~ a depth level, particularly subjects like probability, statistics, calculus, and also linear algebra. To perform that I have to understand the fundamentals however.

Any solution is appreciated.


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asked may 20 "19 at 2:32
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I think you"re questioning why the dominance for division of fractions,$$fracpq div fracrs = fracpq cdot fracsr,$$works.And I"m assuming the you"re currently comfortable with how to multiply fractions.

You are watching: Why do you multiply by the reciprocal

We must go ago to what department is claimed to achieve in the first place. As soon as we look right into that, the prize is that $Adiv B$ means something that offers $A$ once we multiply it by $B$ -- or, composed in symbols, $Adiv B$ method the $X$ the solves the equation $$ Xcdot B = A $$

When ours $A$ and also $B$ space fractions, the "reciprocal" division rule have the right to be concerned as a trick that happens to create an $X$ the works. It"s easy sufficient to watch that the does work: If we"re dividing $frac pq div frac rs$ we must solve the equation$$ X cdot frac rs = frac pq $$And indeed setting $X=frac pqcdot frac sr = fracpsqr$ does this:$$ fracpsqrcdotfrac rs = fracpscdot rqrcdot s = fracpcdot srqcdot sr = frac pq$$like us want. (I"m likewise assuming the you"re comfortable v cancelling the typical factor $sr$ in the center fraction).

This computation hopefully likewise gives part ides why that works, in ~ least component way. In $fracpsqr$ the $p$ and $q$ space what we desire to finish up with, and also the $s$ and $r$ room there to "neutralize" the $r$ and also $s$ us have yet want come discard. Through making certain that the product has exactly one $r$ and one $s$ on every side the the fraction bar lock make certain we can cancel them away.

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Writing the systems $fracpsqr$ together $frac pqcdot fracvphantompsr$ can be best understood as simply an easy way to remember what walk where. However this storage trick itself then also serves together motivation for considering the reciprocal to be an exciting operation in its own right in higher algebra.