Why Do We Use the Reciprocal in Fraction Division?

[Taylor_1989]

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.

 

[Number Nine]

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.

 

[acabus]

My LaTeX always looks so ugly.

[tex]n = \frac{(\frac{a}{b})}{(\frac{c}{d})}[/tex]
[tex]n\left(\frac{c}{d}\right) = \frac{a}{b}[/tex]
[tex]nc = \frac{ad}{b}[/tex]
[tex]n = \frac{ad}{bc} = \left(\frac{a}{b}\right) \times \left(\frac{d}{c}\right)[/tex]

 

[Taylor_1989]

Thanks for the proof abacus, well appreciated makes things clearer for me.

 

[CellCoree]

Sure, I would be happy to provide a proof for why we use the reciprocal in fractions division.

First, let's define what a reciprocal is. A reciprocal of a number is simply another number that, when multiplied together, result in a product of 1. For example, the reciprocal of 2 is 1/2, since 2 x 1/2 = 1.

Now, let's consider the division of two fractions, a/b divided by c/d. This can be written as (a/b) / (c/d).

To solve this division, we can rewrite it as multiplication by the reciprocal of the second fraction. So, (a/b) / (c/d) becomes (a/b) x (d/c).

Using the definition of a reciprocal, we can see that (c/d) x (d/c) = 1, since the c's and d's will cancel each other out. Therefore, (a/b) x (d/c) = (a x d) / (b x c).

This means that when dividing fractions, we can simply multiply the first fraction by the reciprocal of the second fraction to get the answer. This is why we use the reciprocal in fractions division.

In algebra, we can represent this concept as follows:

(a/b) / (c/d) = (a/b) x (d/c) = (a x d) / (b x c)

I hope this proof helps to explain why we use the reciprocal in fractions division. Understanding the reasoning behind mathematical concepts can be just as important as knowing how to solve them.