oj, I thought I answered, sry.
By accepting the answer above, we accept 1 to represent any expression of .99...^n
yes we do, that is 'consens' (it is more than consens since it is proofable).
1=0.99...^5=0.99...^12=1
These numbers are not approaching 1 they are 1.
I understand 'consens' (I think, consensus in English?) and the usefulness of it.
I also understand the need to constantly question it.
Once upon a time, there was a 'consensus' that Earth was the flat center of the universe.
I'll try to explain what I mean about having 'better language', in (hopefully) better language.
For the recurring decimals we get from whole divisions of 3, (.3..., .6...), by saying these decimals repeat infinitely, it SEEMS like we're saying we can make this an infinitely smaller fraction (3/10, 33/100,333/1000 ...), which makes it seem to define a moving point.
It's a little more accurate to say these decimals exist in an eternal state that represents a precise division of 1/3.
The idea that we need infinitely repeating numbers to represent a precise division just seems 'messy' to me, and in the case of these particular fractions, represents the problem of dividing by 0. Decimal numbers simply can't show fractions of 3 without them. If we used a number system based on units of 12 instead of 10, we could show these fractions without a repeating (do?)decimal.
I'm not saying we should switch to a base 12 number system, by any means. I'm saying maybe there should be a broader discussion about finding a system that doesn't allow us to show the number one both as 1, and something that (seems to) represent an undefinable quantity smaller than one.
I would say we have the same issue with pi... showing it in decimal numbers is impossible, but the relationship between the diameter and circumference of a circle isn't an impossible, eternally moving relationship, we just can't show it with those numbers.
You are basically explaining why the teachers where always so keen on using fractions instead of decimals, it is in away always the cleaner method. We would never write 0.125 or even 0.25 in college but always 1/4 or 1/8.
It made my head hurt thinking about this, you should be right, but I never looked at fractions in things like hexadecimal or even binary. 0.4 should be exactly 1/3 when your basis is 12.
LOL, I'm glad to hear the my head isn't the only line hurting over this! 😂
Let me try this here for the first time, this thread REALLY deserves it 😄
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