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and 0.3… + 0.3… + 0.3… = 0.9… = 1.0

Maybe there is a difference, but it's intangible.

Maybe it is to the number line what Planck Length is to measures.

As a non-math-guy, I understand and accept it, but I feel like we can have both without breaking math.

In a non-idealized system, such as our physical reality; if we divide an object into 3 pieces, no matter what that object was we can never add our 3 pieces together in a way that recreates perfectly that object prior to division. Is there some sort of "unquantifiable loss" at play?

So yea, upvoting because I too am fascinated by this and its various connections in and out of math.

4 comments

Nothing is broken, just people stumbling on various notations.

0.3... is just the decimal representation of 1/3. So:

0.3... = 1/3

0.6... = 2/3

0.9... = 3/3 (= 1)

But you are assuming 0.3... is the representation of 1/3. We don't have to make this assumption, it's just the one we are usually taught. Math doesn't really break from making different assumptions, quite the opposite.

Let's make some different assumptions, not following high school math: When I divide 1 by 3, I always get a remainder. So it would just be as equally valid to introduce a mathematical object representing this remainder after I performed the infinite number of divisions. Then

1/3 = 0.3... + eps / 3

2/3 = 0.6... + 2eps / 3

3/3 = 0.9... + 3eps / 3

and since 0.9... = 1 - eps, we get 3/3 = 0.9... + eps = 1

It's all still sound (I haven't proven this, but so far I don't see any contradiction in my assumptions). And it comes out where 0.9... is not equal to 1. Just because I added a mathematical object that forces this to come out.

Edit: Yes, I am breaking a lot of other stuff (e.g. standard calculus) by introducing this new eps object. But that is not an indicator that this is "wrong", just different from high school math.

Another easy way to understand it is to extend the idea of remainders to decimals. When we say N / D = Q r R that obviously means N = D * Q + R. For example 13 / 3 = 4 r 1 because 13 = 3 * 4 + 1. Likewise then 1 / 3 = 0.3(...) = 0.3 r 0.1 because 1 = 3 * 0.3 + 0.1, but also 1 / 3 = 0.3(...) = 0.33 r 0.01 because 1 = 3 * 0.33 + 0.01, etc. Hence 3 * 0.3(...) = 0.9(...) = 1.

> if we divide an object into 3 pieces, no matter what that object was we can never add our 3 pieces together in a way that recreates perfectly that object prior to division. Is there some sort of "unquantifiable loss" at play?

When you cut a cake into 3 slices, there is always a little bit of cake stuck the knife.