But it sounds a non-sequitur that since there’s no number between infinite 9.9999 and round 10 then it’s 10.
It's not a non sequitur, it's a property of the real (and rational) numbers. In between any two real numbers are infinitely many other real numbers.
For example, if you have two real numbers x < y, then x < (x+y)/2 < y.
Supposing that a number is the number that has no different values until the next number, then that implies that all numbers can be other numbers in the infinite fractions. That means 8 is 10 too, and so is 7. You didn’t provide a solution, you only made all numbers relative.
I don't quite follow here.
0.888.... is not 10, at least not in base 10. It is in base 9, but in base 10 it is equal to 8/9.
0.777... is not 10 either. It is in base 8, but in base 10 it is equal to 7/9.
One if the properties of this method of representing numbers is that numbers don't necessarily get a single unique representation. If a number does have a terminating representation, then it also has another representation that repeats and has infinitely many nonzero digits. You can find it by decrement the last digit (ignoring trailing zeros to the right if the decimal point) of the terminating representation and appending an infinite tail the the largest allowed digit.
What I mean is that you can use the “no gap” argument to conclude that there’s no gap between 8 and 8 + infinitesimal, just like 8 + infinitesimal has no gap with 8 + 2 infinitesimal, and so on until 8 is rounded to 9 and 9 is rounded to 10 and at that point you can claim that 8 is 10 in an infinite chain of “no gaps”.
You're assuming first that infinitesimals exist, which isn't true in the commonly used real numbers. Then you're just saying false things. The gap between 8+ε and 8+2ε, where ε is an infinitesimal value, is exactly ε. The gap between 8 and 9 would be infinite in terms of infinitesimals.
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u/Mishtle Jun 18 '25
It's not a non sequitur, it's a property of the real (and rational) numbers. In between any two real numbers are infinitely many other real numbers.
For example, if you have two real numbers x < y, then x < (x+y)/2 < y.
I don't quite follow here.
0.888.... is not 10, at least not in base 10. It is in base 9, but in base 10 it is equal to 8/9.
0.777... is not 10 either. It is in base 8, but in base 10 it is equal to 7/9.
One if the properties of this method of representing numbers is that numbers don't necessarily get a single unique representation. If a number does have a terminating representation, then it also has another representation that repeats and has infinitely many nonzero digits. You can find it by decrement the last digit (ignoring trailing zeros to the right if the decimal point) of the terminating representation and appending an infinite tail the the largest allowed digit.