Why Do We Use the Reciprocal in Fraction Division?

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SUMMARY

The discussion clarifies the mathematical principle that division by a fraction is equivalent to multiplying by its reciprocal. This is established through algebraic proof, demonstrating that dividing the fraction \(\frac{a}{b}\) by \(\frac{c}{d}\) results in \(\frac{a}{b} \times \frac{d}{c}\). The proof emphasizes that division is the inverse operation of multiplication, reinforcing the foundational concept of reciprocals in fraction division.

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Taylor_1989
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Could someone show me the proof to why we use the reciprocal in fractions division. I ask this because it seem we are taught the how in math but never the why. Algebra proof would be best thanks.
 
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It's not a proof, it's a definition. Division is the inverse operation of multiplication, and so dividing by a fraction is the same as multiplying by the inverse of that fraction, which is its reciprocal.
 
My LaTeX always looks so ugly.

n = \frac{(\frac{a}{b})}{(\frac{c}{d})}
n\left(\frac{c}{d}\right) = \frac{a}{b}
nc = \frac{ad}{b}
n = \frac{ad}{bc} = \left(\frac{a}{b}\right) \times \left(\frac{d}{c}\right)
 
Thanks for the proof abacus, well appreciated makes things clearer for me.
 

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