A Largest Natural Number From 0.999... = 1
This is a ramification of the contents of the thread: 0.999...= 1 started by @jorndoe. Thanks jorndoe
First off, the simplest infinity that we know of is the natural numbers: N = {1, 2, 3,...}. The infinite set N is generated by the simple iteration of adding one to the preceding number as so: 1; hen 1 + 1 = 2; then 2 + 1 = 3; and so on and so forth.
What I'm particularly concerned about is the ratio between consecutive elements in the set N. The ratios look like below:
1) 1; there is no ratio here as there is no natural number that precedes 1
2) 2; the ratio is 1 : 2 = 0.5
3) 3; the ratio is 2 : 3 = 0.666...
4) 4; the ratio is 3 : 4 = 0.75
5) 5; the ratio is 4 : 5 = 0.8
6) 6; the ratio is 5 : 6 = 0.833..
7) 7; the ratio is 6 : 7 = 0.851742...
8) 8; the ratio is 7 : 8 = 0.875
9) 9; the ratio is 8 : 9 = 0.888...
10 ) 10; the ratio is 9 : 10 = 0.9
11) 11; the ratio iss 10 : 11 = 0.90...
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3000) 3000; the ratio is 2999 : 3000 = 0.9996...
37896544) 37896543; the ratio is 37896543 : 37896544 = 0.9999999736123695078896904169075
As you can see as the numbers get larger the ratio between a natural number x and its successor x + 1, given by x : (x + 1) approaches, in the limit, 0.999...
But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1
In other words, there will come a point in the sequence of natural numbers where a natural number x and its successor will have the relationship x : (x + 1) = 0.999... but since 0.999... = 1, x : (x + 1) = 1 and that means x = x + 1 which basically means there's a natural number which will not increase in size when you add 1 to it. We can't say that x is infinite because if it is then x : ( x + 1) = infinite : infinite which is undefined and can't equal 0.999... Ergo x must be a finite natural number but since adding 1 doesn't get us a number larger by 1, it follows that there is a largest natural number.
First off, the simplest infinity that we know of is the natural numbers: N = {1, 2, 3,...}. The infinite set N is generated by the simple iteration of adding one to the preceding number as so: 1; hen 1 + 1 = 2; then 2 + 1 = 3; and so on and so forth.
What I'm particularly concerned about is the ratio between consecutive elements in the set N. The ratios look like below:
1) 1; there is no ratio here as there is no natural number that precedes 1
2) 2; the ratio is 1 : 2 = 0.5
3) 3; the ratio is 2 : 3 = 0.666...
4) 4; the ratio is 3 : 4 = 0.75
5) 5; the ratio is 4 : 5 = 0.8
6) 6; the ratio is 5 : 6 = 0.833..
7) 7; the ratio is 6 : 7 = 0.851742...
8) 8; the ratio is 7 : 8 = 0.875
9) 9; the ratio is 8 : 9 = 0.888...
10 ) 10; the ratio is 9 : 10 = 0.9
11) 11; the ratio iss 10 : 11 = 0.90...
.
.
.
3000) 3000; the ratio is 2999 : 3000 = 0.9996...
37896544) 37896543; the ratio is 37896543 : 37896544 = 0.9999999736123695078896904169075
As you can see as the numbers get larger the ratio between a natural number x and its successor x + 1, given by x : (x + 1) approaches, in the limit, 0.999...
But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1
In other words, there will come a point in the sequence of natural numbers where a natural number x and its successor will have the relationship x : (x + 1) = 0.999... but since 0.999... = 1, x : (x + 1) = 1 and that means x = x + 1 which basically means there's a natural number which will not increase in size when you add 1 to it. We can't say that x is infinite because if it is then x : ( x + 1) = infinite : infinite which is undefined and can't equal 0.999... Ergo x must be a finite natural number but since adding 1 doesn't get us a number larger by 1, it follows that there is a largest natural number.
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