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0.999... = 1 - 1/∞
We talk about infinity all the time in mathematics, teachers use the concept to introduce calculus in a way that people can more easily understand, but using infinity directly is almost universally banned within classrooms.
Nonstandard analysis is a much more intuitive way of understanding calculus, it's the whole "infinite number of infinitely small pieces" concept, but you're allowed to write it down too.
Earlier quoted context omitted.
Sure. Instead of "0.99..." please substitute lim n->inf sum(1..n)(9 times 10^-n). The point I'm making is that the "obvious truth" 0.99... = 1 that we're all talking about depends on the assumption that we're working in the real numbers. I claim that the real numbers are not something intuitively obvious to every sufficiently intelligent person; instead they are kind of weird and technical. I go on to claim, though I…
> I go on to claim, though I'm more unsure of this, that the real numbers are not even the only way to make calculus work. That claim is certainly true. The proof is that calculus (Analysis) exists for the complex number system too. Although I don’t think that’s what you meant and I doubt the complex number system is “more intuitive.” Just out of curiosity, have you heard of real analysis? How do you define calculus?
I'm eager to be corrected if you can tell me something I said that's wrong. I'm not interested in gradually upping the ante with you until it's clear who really has more math background.
Earlier quoted context omitted.
I've had the same experience, even debating this topic with engineers. I think there are actually two hang-ups. 1. People have had it drilled into their heads that humans can't comprehend infinity. It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas), and that tricks such as infinity and the infinitesimal were not rigorous. Mathematical…
Couldn't you formulate a problem for extreme decimal-place accuracy based on the multiplication of errors in a physical process that's repeated in ways that multiply small errors into bigger ones?
Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…
> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…
x = 0.8...
10x = 8.8... = 8 + x
9x = 8
8/9 = 0.8...
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
Yes there is. There is a proof that uses only fundamentals of first year university analysis. When you see
0.99999....
this can be written as an infinite sum
\sum_{i=0}^\infty 0.9 x 10^{-i}.
Truncate the sum and set
S_n = \sum_{i=0}^n 0.9 x 10^{-i}
and now simply use the rules of arithmetic progressions to get the limit out:
0.1 S_n = \sum_{i=0}^n 0.9 x 10^{-i-1}
S_n - 0.1 S_n = 0.9 - 0.9 x 10^{-n-1}
0.9 S_n = 0.9 ( 1 - 0.1^{n+1} )
S_n = 1 - 0.1^{n+1}
Now let n tend to infinity to find the limit, which is 1.
You don't need to imagine infinity to do any of this.
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
when they are done they will have undesrtood.
chuckle
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
Earlier quoted context omitted.
Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the id…
I used to have the same annoyance about group and category theory; learning some about the type-theory-as-foundations work (particularly homotopy type theory and cubical type theory) has helped w/ this; in that setting, you have several distinct but well-developed notions of equality: propositional equality, which group theory and most math cares about, vs judgemental equality, which is the one that's "obviously true…
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.
Computer programmers (and historians!) have a similar problem with dates, and in particular with issues like daylight saving time and time zones. I think a lot of the problem is that again there's no way to talk about a particular instant of time without adopting some necessarily arbitrary and relative nomenclature like “January 17, 1706 at 09:37 local time in Boston”. But when was this _really_? Unfortunately there is no “really”. (“Oh, you mean Ramadan 1117 AH, now I understand.”)