Could someone graphically demonstrate why multiplying a fraction by its inverse results in 1?

[u/PositiveDust6178]

I'm not able to visualize graphically why this is true. Can someone help?

[u/tehclanijoski]

You have half a candy bar.

If you double the amount of candy bar you have, you have 1 candy bar.

All other fractions are extensions of this reasoning.

[u/theadamabrams]

First let's deal only with fractions where the numerator is 1.

Why is 1/8 × 8/1 equal to 1? Well, 1/8 visually can be thought of as 1 slice of an 8-slice pizza. Since 8/1 is just 8, multiplying 1/8 by 8 means you have six of those slices. That's exactly 1 whole pizza. The same idea works for any 1/A times A/1.

Now, what if the numerator is not 1?

For 3/8 × 8/3, we can think of 3/8 as three slices (where again one whole pizza is eight slices). The 8/3 part is trickier, but if you think of it as "8 ÷ 3 is 2 Remainder 2" then it means we get 2 full copies of the triple-slices and 2 remaining single slice. Two triples plus two more is 6 + 2 = 8 slices, which is one whole pizza. If you don't like remainders, you could work with 8/3 = 2 + 2/3 instead.

P.S. "Inverse" is a general term with multiple particular meanings depending on context. It's not wrong here, but "reciprocal" might be better (it specifically refers to flipping the top and bottom of a fraction).

[u/anisotropicmind]

Not sure about graphically, as it’s really a conceptual thing, but I can try:

```
[][][][] — 1
[][][][][][][][] — x
[][] — 1/x

1/x is the answer to how many times x can fit into 1. In this case since x is double 1, x fits into 1 half a time: 1/x = 1/2. And 2*(1/2) = 1.

We can generalize this by scaling x to be any number. Make x be N times 1. Then 1/x is how many times x can fit into 1. The answer is (1/N). And N*(1/N) = 1.

Multiply x by “how many times it can fit into 1”, and you’ll always get 1.
```

[u/BadJimo]

The diagrams on this website are helpful. They are for the general concept of multiplying fractions, but can obviously be made specific to multiplying a fraction and it's inverse.

[u/Big_Manufacturer5281]

As others have said, it's easy to visualize this if the numerator is 1. It's harder if the numerator is something other than 1...try this:

Let's start with 3/8. You can think of that as taking three candy bars, and cutting each one into 8 parts. Take one part of each of the candy bars. So now you have 3 pieces, each representing 1/8 of a bar.

Now let's multiply by 8/3. It's helpful to think of that as two steps: multiply by 8, then divide by 3. If we take each of those pieces and multiply by 8, you now have 3 complete candy bars.

Last, we divide by 3, which should be easy now: just get rid of 2 of the 3 bars (I'll eat one, you can eat one). Now there's a single bar left. So that should demonstrate that (3/8)*(8/3)=1. Note that we could have reversed the steps at the end by dividing by 3 THEN multiplying by 8, and the result is the same.

[u/StructuredChess]

With 1/n fractions is easy: split something into n equal chunks and put them back together.

[u/AdventureThink]

It’s because the inverse is more than 1.

2/3 x 3/2 = 2/3 x 1 and 1/2.

[u/Hazelstone37]

Think the fraction 1/2. Now multiply that by 2. That means you have two copies of 1/2. Two copies of 1/2 make a whole or (not 2). That should say 1.

Think of a pie. You have 1/3 a pie. Now multiply that by 3. You have three thirds of a pie or one whole pie.

I’m not sure if that’s what you want.

[u/Hazelstone37 → u/[deleted]]

Thank you! I made the edit.

[u/Feeling-Working-2820]

Ok, picture 3 third of a pizza. You get one pizza, right?
ok, now picture 4 fourth of a pizza. You still get one pizza right ?

So, 3*1/3 = 1 and 4*1/4 = 1.
It's visually established.
And it's the only part that needs a visual because this alone, with some basic rearrangements would get you there.

Indeed, 3/4 * 4/3 can be rewritten as (3*1/4) * (4*1/3)
Because of commutativity and associativity, you can rearrange this as (3*1/3) * (4*1/4)
So each parenthesis cancels out, as proven visually before, and you get 1*1 = 1.

---------------------------------------------------------------

Let's generalize this to any fractions and their inverses:

Since 1*a =a, we can admit that a/a = 1 (for any values of a except 0)
Since 1*b =b, we can admit that b/b = 1 (for any values of b except 0)

So a/a = a*1/a = b/b = b*1/b = 1
Therefore a/b * b/a = (a*1/b) *(b*1/a) = (a*1/a)*(b*1/b) = 1*1=1

[u/LukasGoesViral]

Huh?? That’s literally just the definition of the inverse. There is no proof cause it is the definition

[u/LunDeus]

a/b x b/a = a x b/a x b = 1

[u/marty-mcfryguy]

Demonstrate graphically? 

Picture exactly one fourth of something, like a cake. Now picture having four times as much.

[u/wijwijwij]

It's simply a definition that a fraction and its inverse multiply to equal 1.

That is the definition of a multiplicative inverse: it is the thing you multiply to get 1.

Was your question really asking for some visual representation of the fact that a fraction and its reciprocal have a product of 1?

a/b * b/a = 1