RE: The Beauty of Math
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Wouldn't the graph of y=x^2 look very different for x=1 and x=.9...?
Sorry to keep picking at this, I'll stop if you ask me to.
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Wouldn't the graph of y=x^2 look very different for x=1 and x=.9...?
Sorry to keep picking at this, I'll stop if you ask me to.
its ok, keep asking if you are still in doubt. If you want to look at the graph for x-->x² then we only focus on one point x=1=0.99... . At this point the result or the y-coordinate is 1, because 1²=1=(0.99...)².
This is where things get hard for me, and I feel there needs to be a better way to define what's actually happening here.
By accepting the answer above, we accept 1 to represent any expression of .99...^n. While the result of all those expressions still approach infinitely close to 1, they approach at different rates, while the solution offered by substiting 1 doesn't approach 1 at all.
oj, I thought I answered, sry.
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! 😂
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Thank you for this inspiring post. I've done some more reading on infinitesimals and other types of numbers that I hadn't heard of in my calculus class. This led me down a very interesting rabbit hole, and I really can't thank you enough.
happy to have inspired you to look some more into math, there is some really fascinating stuff. I never heard of infinitesimals to be honest, but did you know there are different kinds of infinities?
Different kinds of infinites? I know now 😁. The infinitesimals led me to surreal and hyperreal numbers, then nature reminded me that she doesn't care how well I grasp math, by sending a downpour to ruin some of my spring work... looks like my 'light' reading on mathematical theories will have to wait until this winter!
unlucky, all you need is pen & paper, but it is quite essential :D