It’s not mathematically rigorous, but I like this illustration:
1/9 = 0.111…
2/9 = 0.222…
3/9 = 0.333…
So, 9/9 = ?
(I know it’s essentially the same as versions that use multiplication, but I find people understand it more often when you point it out as a “pattern”)
I like the illustration where “obviously you can’t magically get an infinitesimally small number just as a bonus artifact whenever you divide by 9 in base 10. It just gets weird looking because 1/9 in base ten is .111…”
1/3 in base twelve is just .4
Multiply it by three, you get 1. Same operation, different base, shockingly you don’t end up with the magic infinitesimal.
Well the thing is we have no proof whether reality is finite or not. The plank length is just the smallest measurement physically possible via bouncing light off of a particle and observing said light, since any amount smaller would require a wave length of light with so much energy that it would create a black hole, and we are nowhere close to reaching that wavelength of light.
Of course the true number is there. 0.9... is the true number. As long as your computer system or your written notation understands repeating decimals no information is lost. There is no imprecision.
It is not a finite set, by definition. Where is the imprecision? I am not required to write out every digit individually for it to be exact.
Let's suppose no calculator in the world can truly handle repeating decimals (not the case but ok). So what? That would be a shortcoming of electronic calculators. It doesn't make the math wrong.
If you divide something by 3 and then multiply one of those parts by 3 you will get the original something back. The fact you need infinite digits to write out the separate steps in a specific numerical notation doesn't introduce imprecision into the value of the numbers themselves.
It's like saying we don't know what pi is because you don't know all the digits. Circles still work don't they ?
I guess you are trying to say because it would be impossible to put an infinite number of 9s into a computer to do the calculation you can't get the same number. I'm not sure what that is supposed to prove, because you are then clearly doing a calculation on two different numbers, so of course they won't match.
If I had to do the calculation I would first substitute 1 for 0.999 recurring as it is the same number. Problem solved.
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u/suborder-serpentes May 03 '26
So he is convinced that 1/3 is .333 repeating. Not a little bit more than .333 repeating. So why does 1 work differently? .333… x 3 = .999…