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Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you defi
by knzhou 6y ago
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 "arithmetic proofs" are really showing that if we want "..." to have certain extremely reasonable properties, then we must choose to define it in such a way that 0.999... = 1. Other definitions would be possible (for example, a stupid definition would be 0.999... = 42), just not useful.
What probably causes the flame wars over "..." is that most people never see how "..." is defined (which properly would require constructing the reals). They only see these indirect arguments about how "..." should be defined, which look unsatisfying. Or they grow so accustomed to writing down "..." in school that they think they already know how it's defined, when it never has been!
- ssivark 6y ago> 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. Sure, but the point of "elementary school" arithmetic is not "elementary" arithmetic, as a mathematician would define it :-) The goal is to teach people to reason by matching patterns. Deductive/Inductive reasoning can slowly proceed from that, as they try to frame their intuition for patterns into increasingly more general abstractions.
- agumonkey 6y agoAnd I thought mathematicians would reject normal arithmetic operation over the domain of `N...` elements. They're always so ultra rigorous to classify what is or is not, what is defined, the domain .. but then they let an infinite sequence be treated as any finite number.
- saagarjha 6y ago> but then they let an infinite sequence be treated as any finite number Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.
- agumonkey 6y agocertain cases, so define this special one but frankly it doesn't make my brain happy (but who am I)
- topaz0 6y agoIt's not a special case. It's the standard general definition of the limit of a sequence. The limit of a sequence a_1,a_2,...,a_n,... exists if and only if there exists some L (the limit) such that, for any (arbitrarily small) epsilon > 0, there exists some N with the property |a_n - L| < epsilon for every n>N. In this case a_n is n 9s after the decimal point (or a_n = 9*sum_{i=1}^n (.1)^i).
- tomtomtom1 6y agoif we say that infinitesimals exist. that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0. what are the properties that we would lose?
- MaxBarraclough 6y agoThe obvious one seems terrible enough, that division is no longer the inverse of multiplication: (1/3)*3 != 1
- mFixman 6y agoIf we say that infinitesimals exist, it still happens that 1 = 0.999…. It just happens that 0.999 ≠ 1 - 𝛚. 0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.
- tomtomtom1 6y agowouldn't 0.999.. be equal to 1 - 10^w since it's only a countably finite series of nines.
- mFixman 6y ago0.999...9 with a countable finite amount of nines is clearly less than 1. 0.999... with infinite nines is equal to 1.
- tomtomtom1 6y agosorry I typoed I meant, countably infinite. by w I meant the ordinal.
- ulucs 6y agoNonstandard analysis exists (with infinitesimal and infinite numbers) , but 1/3 and 9/9 is the same there. The problem is that the numbers 0.333... and 0.999... don't really exist.
- 6y ago
- baryphonic 6y agoThe way '...' is used here is perfectly consistent with being defined as a geometric series where the ratio between successive elements is 1/10 and the start term is the final digit. Geometric series always converge when the absolute value of the ratio of less than 1. I should note that when I learned about rational & irrational numbers in elementary school (I think third or fourth grade), we used a "bar" notation where we'd put a bar over the last digits in a decimal expression that repeated forever (i.e. it corresponded exactly to a geometric series with r = (1 / 10)^k where k is the number of digits under the bar, though we didn't know about that at the time). Our teachers explained that the difference between a rational and irrational number was that there would be no pattern you could ever find in an irrational number that would allow us to use the bar, which is surprisingly accurate for grade school arithmetic.
- CogitoCogito 6y ago> 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're basically just defining it to be 1. I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Unless you're making use of limits, completeness, or something equivalent I don't see how a proof could possibly make any sense.
- JoeAltmaier 6y agoYes that proof depends upon the representation in text of rational numbers (a dot and a series of digits). Try it in hexadecimal - it becomes opaque nonsense. Without some mathematical basis for 0.9... X 10 being something, there's a dangerous dependency on the representation that makes many folks uneasy.
- ginko 6y agoI don't see how it would be significantly different in other bases. In hexadecimal it'd be 0x0.ffff.. * 0x10 = 0xf.fff.. The rules are essentially the same.
- JoeAltmaier 6y agoOh right! My mistake. Was stuck in my head with "10 == nine plus one" instead of your insight, multiply by '10' in whatever base. Anyway it's still an artifact of representation. If rationals were represented as fractions, then it is unrepresentable. 1/1 X 10 == 10/1?
- judofyr 6y agoYou only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1.