Why we turn the fraction upside down when dividing?

Hello everyone, I have a doubt.

Why a/b times c/d = a/b divided by d/c?

Why multiplying by the reciprocal is the same as dividing that fraction? I get the mechanics, I get that the division is the inverse of multiplication, but I can't really visualize it in my mind and I don't get WHY that is true.

Sorry if this question sounds dumb, but I really love the way we're learning math here at Brilliant, and I would love if someone could provide some visual aid or any guidance, really.

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Interesting thought! Unfortunately, it might not be something you can exactly visualize though. Think of it this way: When we do $10~\div~2$ for example, we visualize it by imagining that we have $10$ objects, and $2$ people who want to share it. That way, we physically split up the $10$ objects evenly into $2$ groups. But we if tried to do the same with $10~\div~\frac{1}{2}$ , our visualization would fail, since it doesn't make sense to have $\frac{1}{2}$ a person, or $\frac{1}{2}$ groups of objects.

Problems like this arise with even simpler operations like subtraction. For quite some time, mathematicians would simply refuse to subtract a larger number from a smaller one, since doing this would give a number smaller than $0$ , which to them was impossible! We of course today use negative numbers to get around this, but the concept still lacks a physical representation, since we can never have less than $0$ things.

Hope this helps! Maybe someone has a different suggestion...

David Stiff - 1 month, 4 weeks ago

In China no one in middle school has this problem anymore. Maybe it is hard to visualise how multiplying by reciprocal is, but we can prove this: $a\div b=a\times 1\div b=a\times \frac{1}{b}. ~~~~~\text{ (Reciprocal)}$ $a\div\frac{b}{c}=a\div(b\div c)=\frac{a}{b} \div (1\div c)=\frac{a}{b}\times c=\frac{ac}{b}.$

Jeff Giff - 1 month, 3 weeks ago

Hello @Jeff Giff! Thanks for this explanation. Can I ask you why a/b divided by 1 divided by c = a/b times c? Thanks!

Davide Taraborrelli - 1 month, 3 weeks ago

Oh! Umm... since a number’s reciprocal is one divided by it, $c$ and $\frac{1}{c}$ are reciprocals. So $\frac{1}{c}$ and $c$ are reciprocals of each other. Therefore $\frac{1}{\frac{1}{c}}=c.$

Jeff Giff - 1 month, 3 weeks ago

@Jeff Giff – Hi @Jeff Giff! Sorry for the delay I'm writing to you. I just wanted to let you know that I appreciated your explanation! Thank you

Davide Taraborrelli - 1 month, 1 week ago

$\blue{a}\div \red{b}=\blue{a}\times \red{b}^{\purple{-1}}$ $\Rightarrow \blue{\dfrac{a}{b}}\div \red{\dfrac{c}{d}}=\blue{\dfrac{a}{b}}\times \left(\red{\dfrac{c}{d}}\right)^{\purple{-1}}$ $=\blue{\dfrac{a}{b}}\times \red{\dfrac{d}{c}}=\dfrac{\blue{a}\times\red{d}}{\blue{b}\times\red{c}}$

Zakir Husain - 1 month, 3 weeks ago

Hello everyone! Thank you tons for clarifying this idea for me and offering some proof. That indeed helped. I am still trying to wrap my head around those as I am still at the beginning of my math journey, but I do appreciate your help! Thanks!

Davide Taraborrelli - 1 month, 3 weeks ago

And negative numbers mean the opposite meaning!
I suppose nobody would use negative numbers like ‘I have -2 cakes’ or ‘he is -1.8 meters tall’. Instead, negative numbers are used in daily life to represent things of the opposite meaning, for example ‘I walked -1 meters south’ is equivalent to ‘I walked a meter north’.

Jeff Giff - 1 month, 3 weeks ago

You're right, the only sensical way to think of negative numbers is as a direction, not as a quantity. But then we still have a slight problem, since now it would seem we've redefined what a "number" is. To most, it would seem that a number must be a quantity, since their original function was to count things.

David Stiff - 1 month, 3 weeks ago

Finally, to visually represent a negative number, there’s always the number axis or a negative pointing vector maybe :)

Jeff Giff - 1 month, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$