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.