I strongly agree with you, and while the people replying aren’t wrong, they’re arguing for something that I don’t think you said.
1/3 ≈ 0.333… in the same way that approximating a circle with polygons of increasing side number has a limit of a circle, but will never yeild a circle with just geometry.
0.999… ≈ 1 in the same way that shuffling infinite people around an infinite hotel leaves infinite free rooms, but if you try to do the paperwork, no one will ever get anywhere.
Decimals require you to check the end of the number to see if you can round up, but there never will be an end. Thus we need higher mathematics to avoid the halting problem. People get taught how decimals work, find this bug, and then instead of being told how decimals are broken, get told how they’re wrong for using the tools they’ve been taught.
If we just accept that decimals fail with infinite steps, the transition to new tools would be so much easier, and reflect the same transition into new tools in other sciences. Like Bohr’s Atom, Newton’s Gravity, Linnaean Taxonomy, or Comte’s Positivism.
Decimals require you to check the end of the number to see if you can round up, but there never will be an end.
The character sequence “0.999…” is finite and you know you can round up because you’ve got those three dots at the end. I agree that decimals are a shit representation to formalise rational numbers in but it’s not like using them causes infinite loops. Unless you insist on writing them, that is. You can compute with infinities just fine as long as you keep them symbolic.
That only breaks down with the reals where equality is fundamentally incomputable. Equality of the rationals and approximate equality of reals is perfectly computable though, the latter meaning that you can get equality to arbitrary, but not actually infinite, precision. You can specify a number of digits you want, you can say “don’t take longer than ten seconds to compute”, any kind of bound. Once the precision goes down to plank lengths I think any reasonable engineer would build a bridge with it.
…sometimes I do think that all those formalists with all those fancy rules about fancy limits are actually way more confused about infinity than freshman CS students.
Eh, if you need special rules for 0.999… because the special rules for all other repeating decimals failed, I think we should just accept that the system doesn’t work here. We can keep using the workaround, but stop telling people they’re wrong for using the system correctly.
The deeper understanding of numbers where 0.999… = 1 is obvious needs a foundation of much more advanced math than just decimals, at which point decimals stop being a system and are just a quirky representation.
Saying decimals are a perfect system is the issue I have here, and I don’t think this will go away any time soon. Mathematicians like to speak in absolutely terms where everything is either perfect or discarded, yet decimals seem to be too simple and basal to get that treatment. No one seems to be willing to admit the limitations of the system.
The system works perfectly, it just looks wonky in base 10. In base 3 0.333… looks like 0.1, exactly 0.1
Noone in the right state of mind uses decimals as a formalisation of numbers, or as a representation when doing arithmetic.
But the way I learned decimal division and multiplication in primary school actually supported periods. Spotting whether the thing will repeat forever can be done in finite time. Constant time, actually.
The deeper understanding of numbers where 0.999… = 1 is obvious needs a foundation of much more advanced math than just decimals
No. If you can accept that 1/3 is 0.333… then you can multiply both sides by three and accept that 1 is 0.99999… Primary school kids understand that. It’s a bit odd but a necessary consequence if you restrict your notation from supporting an arbitrary division to only divisions by ten. And that doesn’t make decimal notation worse than rational notation, or better, it makes it different, rational notation has its own issues like also not having unique forms (2/6 = 1/3) and comparisons (larger/smaller) not being obvious. Various arithmetic on them is also more complicated.
The real take-away is that depending on what you do, one is more convenient than the other. And that’s literally all that notation is judged by in maths: Is it convenient, or not.
That does very accurately sum up my understanding of the matter, thanks. I haven’t been adding on to any of the other conversation in order to avoid putting my foot in my mouth further, but you’ve pretty much hit the nail on the head here. And the higher mathematics required to solve this halting problem are beyond me.
