Short version(for people who somehow don't know what this is about):
3/3 = 1
1/3 = .333r
1/3 x 3 = .333r x 3
3/3 = .999r
1 = .999r
...I hate that, and I think it is wrong:
Please try to do this problem(in your head or on paper):
3 x .999r
Show your work like you were doing an assignment for a math class.
If the above proof is correct then the answer should be 2.999r. The trouble, as you may have found out when attempting to write it out, is that you can't get that answer.
There are two methods for multiplying two numbers, first we try the basic way: Multiply 3 by the right most digit of the other number, then the next digit to the left and so on. Impossible, the idea behind and infinitely repeating decimal is there is no end, so there is no digit to the right of all others.
Now for the other method: Multiply 3 by the left most, then the next digit to the right and so on. We start with 3 x .9 = 2.7, then 3 x .09 = .27(+2.7) = 2.97, 3 x .009 = .027(+2.97) = 2.997...
Keep going, no matter how long you work at it you will never finish, because it is an infinitely repeating decimal.
Now try this: 3 x .333r. You run into the same problem as above, you either can't start or you can't finish. The only way you can pull out an answer like .999r is on the assumption that the infinite calculation will produce an infinite series of 9s. The issue with that is 3 x .333r is a part of the proof at the beginning of the post, and when an assumption is made in a proof it needs to then be proven or the proof itself is invalid.
To clarify: I am claiming that adding/subtracting/multiplying/dividing with any repeating decimal is impossible without making an assumption that cannot be proven.
Tired now, have fun with whatever I just wrote(I am probably wrong for some obscure reason).
3/3 = 1
1/3 = .333r
1/3 x 3 = .333r x 3
3/3 = .999r
1 = .999r
...I hate that, and I think it is wrong:
Please try to do this problem(in your head or on paper):
3 x .999r
Show your work like you were doing an assignment for a math class.
If the above proof is correct then the answer should be 2.999r. The trouble, as you may have found out when attempting to write it out, is that you can't get that answer.
There are two methods for multiplying two numbers, first we try the basic way: Multiply 3 by the right most digit of the other number, then the next digit to the left and so on. Impossible, the idea behind and infinitely repeating decimal is there is no end, so there is no digit to the right of all others.
Now for the other method: Multiply 3 by the left most, then the next digit to the right and so on. We start with 3 x .9 = 2.7, then 3 x .09 = .27(+2.7) = 2.97, 3 x .009 = .027(+2.97) = 2.997...
Keep going, no matter how long you work at it you will never finish, because it is an infinitely repeating decimal.
Now try this: 3 x .333r. You run into the same problem as above, you either can't start or you can't finish. The only way you can pull out an answer like .999r is on the assumption that the infinite calculation will produce an infinite series of 9s. The issue with that is 3 x .333r is a part of the proof at the beginning of the post, and when an assumption is made in a proof it needs to then be proven or the proof itself is invalid.
To clarify: I am claiming that adding/subtracting/multiplying/dividing with any repeating decimal is impossible without making an assumption that cannot be proven.
Tired now, have fun with whatever I just wrote(I am probably wrong for some obscure reason).
