by yurisagalov on 4/28/20, 6:31 AM with 626 comments
by undecisive on 4/28/20, 2:07 PM
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 have multiple representations - which, in a way is correct. The bar or dot or ... are there to plug a gap, allowing more values to be represented accurately than plain-old decimal numbers allow for. It has the side effect of introducing multiple representations - and even with this limitation, it doesn't cover everything - Pi can't be represented with an accurate number, for example.
But it also exposes a limitation in humans: We cannot imagine infinity. Some of us can abstract it away in useful ways, but for the rest of the world everything has an end.
I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation?
by knzhou on 4/28/20, 8:10 AM
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!
by dwheeler on 4/28/20, 2:35 PM
http://us.metamath.org/mpeuni/0.999....html
Unlike typical math proofs, which hint at the underlying steps, every step in this proof only uses precisely an axiom or previously-proven theorem, and you can click on the step to see it. The same is true for all the other theorems. In the end it only depends on predicate logic and ZFC set theory. All the proofs have been verified by 5 different verifiers, written by 5 different people in 5 different programming languages.
You can't make people believe, but you can provide very strong evidence.
by jl2718 on 4/28/20, 5:24 PM
This is illustrative of what I see as a fundamental problem in mathematics education: nobody ever teaches the rules. In this case, the rules of simple arithmetic hit a dead end for mathematicians, so they invented a new rule that allowed them to go further without breaking any old rules. This is generally acceptable in proofs, although it can have significant implications, such as two mutually exclusive but otherwise acceptable rules causing a divergence in fields of study.
When I was taught this, it was like, “Look how smart I am for applying this obtusely-stated limit rule that you were never told about.” This is how you keep people out of math. The point of teaching it is to make it easy, not hard.
by ginko on 4/28/20, 7:33 AM
1/3 = 0.333..
3 * 1/3 = 3 * 0.333..
3/3 = 0.999..
1 = 0.999..by ping_pong on 4/28/20, 2:43 PM
I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard for me to convince him. Luckily he has many years before it matters!
by klodolph on 4/28/20, 7:22 AM
You’ll see various proofs involving real numbers that must account for the fact that 0.999…=1.0. There are, of course, many different ways to construct real numbers, and often it’s very convenient to construct them as infinite sequences of digits after the decimal. For example, this construction makes the diagonalization argument easier. However, you must take care in your diagonalization argument not to construct a different decimal representation of a number already in your list!
by bytedude on 4/28/20, 7:19 AM
by orthoxerox on 4/28/20, 7:20 AM
by steerablesafe on 4/28/20, 7:41 AM
Arguably the sign symbol ruins it for whole numbers as well, as +0 and -0 could be equally valid representations of the number 0. We just conventionally don't allow -0 as a representation. There are other number representations that don't have this problem.
by heinrichhartman on 4/28/20, 4:24 PM
But for the sake of argument, let's just define numbers as sequences of digits with a mixed in period somewhere:
MyNumber := {
a = (a_1, a_2, ...) -- list of digits a_i = 0 .. 9; a_1 != 0.
e -- exponent (integer)
s -- sign (+/- 1)
}
We can go on and define addition, subtraction, multiplication and division in the familiar way.
Problems arise only when we try to establish desireable properties, e.g.
(1/3) * 3 = 1
Does NOT hold here, since 0.9999... is a difference sequence than 1.000....
So yes, you can define these number systems, and you will have 0.999... != 1. But working with them will be pretty awkward, since a lot of familiar arithmetic breaks down.
by ltbarcly3 on 4/28/20, 7:30 AM
If any two real numbers are not equal, then you can take the average and get a third number that is half way between them. Conversely, if the average of two numbers is equal to either of the numbers, then the two numbers are equal. (this isn't a proof, just a way to convince yourself of this)
What's the average of .9999... and 1?
by sleepyams on 4/28/20, 5:54 PM
Let C be the countable product of the set with ten elements, i.e. {0, 1, 2, ..., 9}. The space C naturally has the topology of a Cantor set (compact, totally disconnected, etc). Furthermore, for example, in this space the tuples (1, 9, 9, 9, ...) and (2, 0, 0, 0, ...) are distinct elements.
The space C can also be described in terms of a directed graph, where there is a single root with ten outward directed edges, and each child node then has ten outward directed edges, etc. C can be thought of as the space of infinite paths on this graph.
A continuous and surjective map from C to the unit interval [0, 1] can be constructed from a measure on these paths. For any suitable measure, this map is finite-to-one, meaning at most finitely many elements of C are mapped to a single element in the interval. For example there is a map which sends (1, 9, 9, ...) and (2, 0, 0,....) to the element "0.2".
The point is that all decimal expansions of elements of [0, 1] can be described like this, and we can instead think of the unit interval not as being composed of numbers _instrinsically_, but more like some kind of mathematical object that _admits_ decimal expansions. The unit interval itself can be described in other ways mathematically, and is not necessarily tied to being represented as real numbers. Hope this helps someone!
by cjfd on 4/28/20, 7:29 AM
by calibas on 4/28/20, 4:00 PM
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.
by russellbeattie on 4/28/20, 8:05 AM
I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.
by JJMcJ on 4/28/20, 5:20 PM
There are approaches to mathematics that avoid infinite constructions, and a "strict finitist" would not assign 0.999... a meaning.
The stunning success of limit based mathematics makes finitism a fringe philosophy.
Remember, class, for every epsilon there is a delta.
by traderjane on 4/28/20, 8:03 AM
https://www.youtube.com/watch?v=WabHm1QWVCA
I mention him because I would think he sympathizes with those who have concern over the meaning of this kind of notation.
by sv_h1b on 4/29/20, 8:03 PM
However as a representation of physical world, there is a caveat. What we understand is physical world appears and behaves discretely, because at planck scale (approx. 10^-35) the distances seem to behave discretely.
Although common people don't know/ understand planck scale, they do grasp this concept intuitively. What they are really saying is that in physical world there's some small interval (more precisely, about[1 - 10^-35, 1]) which can't be subdivided further, based on our current knowledge.
Same thing applies to planck time (approx. 5 * 10^-43) too.
So people are arguing two different things - the pure maths concept, or the real world interpretation.
by sebringj on 4/28/20, 3:29 PM
by gigatexal on 4/28/20, 7:41 PM
by edanm on 4/28/20, 7:58 PM
The problem isn't that you can't come up with axioms to convince people you have a proof - the problem is with people not understanding that 0.99999.... is not a number - it's one representation of an abstract entity called a number.
The problem is, the maths required to actually define the concept of a number is fairly complicated, so it's hard to explain to someone why all of these axioms make sense in the first place.
by fluganator on 4/28/20, 8:53 PM
Generally the proofs of .9...=1 rely on the fact there is no number that exists that can be between .9.. and 1 and therefore .9... is equal to 1.
.9... is the least upper bounds of the set. My question is if .9... was removed from the set what would be the new least upper bounds. Another way of asking the question is if we define it in this context doesn't any set bounded by a real number have a least upper bounds and aren't all real numbers equal to each other?
Thanks!
by jdashg on 4/28/20, 10:58 PM
To me, 0.9999 indicates a directional limit, which can't necessarily be evaluated and substituted separately from its context.
by rs23296008n1 on 4/28/20, 7:29 AM
What gets broken? What consequences do we hit?
by j-pb on 4/28/20, 7:48 AM
by jefftk on 4/28/20, 2:33 PM
by shrimpx on 4/28/20, 4:36 PM
by vfinn on 4/28/20, 8:19 AM
by zests on 4/28/20, 3:39 PM
Is 0.999... = 1? Yes, because we define decimal numbers to behave that way. Why do we define them to behave that way? That's the real question.
by fourseventy on 4/28/20, 5:27 PM
The result of 1 minus 0.999... is 0.000 with zeroes that go to infinity. And I think its easier to reason that 0.000 with repeating zeroes forever is in fact equal to zero.
by j7ake on 4/28/20, 8:01 AM
by 2OEH8eoCRo0 on 4/28/20, 3:22 PM
by ttonkytonk on 4/28/20, 2:43 PM
by juanmacuevas on 4/28/20, 6:44 PM
by berkeleynerd on 4/28/20, 8:08 AM
by alpple on 4/28/20, 4:34 PM
by flerchin on 4/28/20, 5:05 PM
by heavenlyblue on 4/28/20, 4:17 PM
by novacole on 4/28/20, 5:37 PM
by clevbrown on 5/1/20, 10:20 AM
by grensley on 4/28/20, 5:41 PM
by amelius on 4/28/20, 2:20 PM
0.9999... < 1
by seyz on 4/28/20, 2:35 PM
by ristos on 4/28/20, 2:04 PM
Given ε = 1/∞ then: ε = 0
Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)
by smlckz on 4/28/20, 2:06 PM
At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following:
1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}}
1 + 2 + 4 + 8 + 16 + ... [somehow] = -1
by upofadown on 4/28/20, 1:57 PM
How about we prove that an infinite number of 9s is impossible?
Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines.
Any process that can not be completed in a finite amount of time can not complete and can not have a valid result based on that completion. Any process that can not be completed in an infinite amount of time is also bogus, but is in a sense even more bogus.
Added: Note that this is different than the case where we are asked to contemplate infinity with respect to continuous functions. By defining the number of 9s as a discrete (integer) value it opens things up to a discrete argument. These pointless navel gazing exercises always end up as a war of what everyone things things are defined as.