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.

Note by Davide Taraborrelli
3 months, 1 week ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Interesting thought! Unfortunately, it might not be something you can exactly visualize though. Think of it this way: When we do 10 ÷ 210~\div~2 for example, we visualize it by imagining that we have 1010 objects, and 22 people who want to share it. That way, we physically split up the 1010 objects evenly into 22 groups. But we if tried to do the same with 10 ÷ 1210~\div~\frac{1}{2}, our visualization would fail, since it doesn't make sense to have 12\frac{1}{2} a person, or 12\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 00, 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 00 things.

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

David Stiff - 3 months, 1 week ago

Log in to reply

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 - 3 months, 1 week ago

Log in to reply

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÷b=a×1÷b=a×1b.      (Reciprocal)a\div b=a\times 1\div b=a\times \frac{1}{b}. ~~~~~\text{ (Reciprocal)} a÷bc=a÷(b÷c)=ab÷(1÷c)=ab×c=acb.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 - 3 months, 1 week ago

Log in to reply

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 - 3 months, 1 week ago

Log in to reply

Oh! Umm... since a number’s reciprocal is one divided by it, cc and 1c\frac{1}{c} are reciprocals. So 1c\frac{1}{c} and cc are reciprocals of each other. Therefore 11c=c.\frac{1}{\frac{1}{c}}=c.

Jeff Giff - 3 months, 1 week ago

Log in to reply

@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 - 2 months, 3 weeks ago

Log in to reply

a÷b=a×b1\blue{a}\div \red{b}=\blue{a}\times \red{b}^{\purple{-1}} ab÷cd=ab×(cd)1\Rightarrow \blue{\dfrac{a}{b}}\div \red{\dfrac{c}{d}}=\blue{\dfrac{a}{b}}\times \left(\red{\dfrac{c}{d}}\right)^{\purple{-1}} =ab×dc=a×db×c=\blue{\dfrac{a}{b}}\times \red{\dfrac{d}{c}}=\dfrac{\blue{a}\times\red{d}}{\blue{b}\times\red{c}}

Zakir Husain - 3 months, 1 week ago

Log in to reply

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 - 3 months, 1 week ago

Log in to reply

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 - 3 months, 1 week ago

Log in to reply

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

Jeff Giff - 3 months, 1 week ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...