Let’s consider the number x = …999; in other words, now we have infinitely many 9s to the left of the decimal point.
My gut response (I can’t reasonably claim to know math above basic algebra) is:
Infinite sequences of numbers to the right of the decimal point are in some circumstances an artifact of the base. In base 3, 1⁄3 is 0.1 and 1⁄10 is 0.00220022..., but 1⁄10 “isn’t” an infinitely repeating decimal and 1⁄3 “is”—in base 10, which is what we’re used to. So, heuristically, we should expect that some infinitely repeating representations of numbers are equal to some representations that aren’t infinitely repeating.
If 0.999… and 1 are different numbers, there’s nothing between 0.999… and 1, which doesn’t jive with my intuitive understanding of what numbers are.
The integers don’t run on a computer processor. Positive integers can’t wrap around to negative integers. Adding a positive integer to a positive integer will always give a positive integer.
0.999… is 0.9 + 0.09 + 0.009 etc, whereas …999.0 is 9 + 90 + 900 etc. They must both be positive integers.
There is no finite number larger than …999.0. A finite number must have a finite number of digits, so you can compute …999.0 to that many digits and one more. So there’s nothing ‘between’ …999.0 and infinity.
Infinity is not the same thing as negative one.
All I have to do to accept that 0.999… is the same thing 1 is accept that some numbers can be represented in multiple ways. If I don’t accept this, I have to reject the premise that two numbers with nothing ‘between’ them are equal—that is, if 0.999… != 1, it’s not the case that for any x and y where x != y, x is either greater than or less than y.
But if I accept that …999.0 is equal to −1, I have to accept that adding together some positive numbers can give a negative number, and if I reject it, I just have to say that multiplying an infinite number by ten doesn’t make sense. (This feels like it’s wrong but I don’t know why.)
I think you mean “they must both be positive” here, but 0.999… isn’t guaranteed to be an integer a priori.
Aside from that, everything you’ve said is basically correct. But… well, there’s something pretty interesting going on with infinite decimals to the left. For numbers that don’t exist they sure do have a lot of interesting properties. This might be worth a top-level post.
My gut response (I can’t reasonably claim to know math above basic algebra) is:
Infinite sequences of numbers to the right of the decimal point are in some circumstances an artifact of the base. In base 3, 1⁄3 is 0.1 and 1⁄10 is 0.00220022..., but 1⁄10 “isn’t” an infinitely repeating decimal and 1⁄3 “is”—in base 10, which is what we’re used to. So, heuristically, we should expect that some infinitely repeating representations of numbers are equal to some representations that aren’t infinitely repeating.
If 0.999… and 1 are different numbers, there’s nothing between 0.999… and 1, which doesn’t jive with my intuitive understanding of what numbers are.
The integers don’t run on a computer processor. Positive integers can’t wrap around to negative integers. Adding a positive integer to a positive integer will always give a positive integer.
0.999… is 0.9 + 0.09 + 0.009 etc, whereas …999.0 is 9 + 90 + 900 etc. They must both be positive
integers.There is no finite number larger than …999.0. A finite number must have a finite number of digits, so you can compute …999.0 to that many digits and one more. So there’s nothing ‘between’ …999.0 and infinity.
Infinity is not the same thing as negative one.
All I have to do to accept that 0.999… is the same thing 1 is accept that some numbers can be represented in multiple ways. If I don’t accept this, I have to reject the premise that two numbers with nothing ‘between’ them are equal—that is, if 0.999… != 1, it’s not the case that for any x and y where x != y, x is either greater than or less than y.
But if I accept that …999.0 is equal to −1, I have to accept that adding together some positive numbers can give a negative number, and if I reject it, I just have to say that multiplying an infinite number by ten doesn’t make sense. (This feels like it’s wrong but I don’t know why.)
I think you mean “they must both be positive” here, but 0.999… isn’t guaranteed to be an integer a priori.
Aside from that, everything you’ve said is basically correct. But… well, there’s something pretty interesting going on with infinite decimals to the left. For numbers that don’t exist they sure do have a lot of interesting properties. This might be worth a top-level post.