[WorldMaker]

> The issues people have with 0.333… and 0.999… is due to two things: not understanding what the notation means and not understanding sequences and limits.

Also a possible third thing: not enjoying working in a Base that makes factors of 3 hard to write. Thirds seem like common enough fractions "naturally" but decimal (Base-10) makes them hard to write. It's one of the reasons there are a lot of proponents of Base-12 as a better base for people, especially children, because it has a factor of 3 and thirds have nice clean duodecimal representations. (Base-60 is another fun option; it's also Babylonian approved and how we got 60 minutes and 60 seconds as common unit sizes.)

[tsimionescu]

You get the same problem with 0.44... + 0.55... - I don't think that makes it any easier to anyone who is confused. It's more likely just that 0.33... and 0.66... are very common and simple repeating fractions that lead to this issue.

[WorldMaker]

Sure, I was just pointing out that Base you use for your math does affect how common repeating digits are, based on the available factors in that base.

In Base-12 math, 1/3 = 0.4 and 2/3 = 0.8. With the tradeoff that 1/5 is 0.2947 repeating (the entire 2947 has the repeating over-bar).

Base-10 only has the two main factors 2 and 5, so repeating fractions are much more common in decimal representation, making this overall problem much more common, than compared to duodecimal/dozenal/Base-12 (or even hexadecimal/Base-16). It's interesting that this is a trade-off directly related to the base number of digits we want to express rational numbers in.