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Some theories suggest there is only one electron, one proton, etc, and we just see it many times.
Are you saying every 1 is a different entity, or are they all the same 1 used a lot of times?
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Maths query
- Thread starter Doggy Styles
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Some theories suggest there is only one electron, one proton, etc, and we just see it many times.
Are you saying every 1 is a different entity, or are they all the same 1 used a lot of times?
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That's correct. They are two decimal representations of the same real number.
The proof comes from infinite geometric series. Mathwords: Infinite Geometric Series
which is: a/(1-r)
a + ar + ar^2 + ar^3 + ... = a/(1-r)
0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 9/10 + 9/10(1/10)^1 + 9/10(1/10)^2 + 9/10(1/10)^3 + ...
Which in a/(1-r) we can say a = (9/10) and r = (1/10) (where r in the range: -1 < r < 1)
Substituting values in:
[(9/10)]/[1-(1/10)]
(9/10)/(9/10) = 1
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So would 0.1 recurring also be the same as 1?
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Don't be silly. There's many numbers between 0.1 and 1. For example, 0.2.
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I know that was a joke (at least I hope to God it was!) But for any reader not too comfortable with recurring decimals
All you have to do is note that if x = 0.111 .. then 10 * x = 1.111 .., so subtracting the first from the second gives 9 * x = 1, or equivalently x = 1/9
This is similar to the geometric series method explained in a previous post, and the same approach works for any recurring decimal.
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Ah yes, well done. You ahem spotted the deliberate mistake.
Anyway I was thinking or rather not thinking about whether the rule you mentioned would work from the opposing side with 1.1r and 1 but now I have been at my desk for a few hours and got a little more sleep it occurs to me that obviously there are numbers between 1.1r and 1 - eg 1.01.
I am not sure if the phase '1.0 recurring apart from the very very very very last one which is 1' has a mathematical shorthand though.
That's feeble
There are no integer values between 1 and 2; doesn't make them the same thing.
By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.
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That rather negates the principle of limits, summation of converging infinite series, etc.
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How many more fecking times?
The infinite sequence {0.9, 0.99, 0.999, ...} "gets increasingly close to 1 but never reaches it" (to use your words).
But "by its very definition" 0.999 .. is not that sequence but defined as the _limit_ of that sequence, which is 1.
The persistently muddled thinking (and it isn't just you) around these sequences and limits is astounding!
How many more fecking times?
The infinite sequence {0.9, 0.99, 0.999, ...} "gets increasingly close to 1 but never reaches it" (to use your words).
But "by its very definition" 0.999 .. is not that sequence but defined as the _limit_ of that sequence, which is 1.
The persistently muddled thinking (and it isn't just you) around these sequences and limits is astounding!
The key word here being "limit"
This is the value beyond which the sequence can not progress, it is NOT the value that the sequence reaches. They are not the same thing.
