<p>so true. Physicists can be lazy at math 
i plug and chuck all the time without pts taken off…and it’s neat
/shrug
decimal numbers are awesome also</p>
<p>By the way, there seems to be some misconception that a limit “never reaches” the end. The truth is that a limit IS equal to what it’s evaluated to be. Since an infinite decimal expansion is defined in terms of a limit, it IS equal to what the limit evaluates as.</p>
<p>Another thing to consider is that a finite sequence is basically just a truncation of an infinite sequence that is eventually constant. Similarly, a finite series is just a truncation of an eventually constant sequence of partial sums. In both these cases, the sequence clearly does, on an intuitive level, reach its limit.</p>
<p>The misconception seems to come from the existence of functions that have point discontinuities or may be undefined at a certain point. In this case lim f(x.n) != f(lim x.n). That’s not because the limit “never reaches” the end, but rather they are two separate limit processes.</p>
<p>I think it’s a fair misconception to have; if the limit of f(x) = x^2/x is 0 as x approaches 0 while f(x) != 0, it seems a fair question to ask why the limit of 0.999…'s unambiguously being 1.0 requires that the real value of the actually infinite mathematical object 0.999… should be 1.0. It happens that in this case 0.999… should equal 1.0 for other reasons - other proofs have been advanced in this thread - but let’s be realistic here: if this particular thing were as clear as some of us math guys like to think it is, the number of people this confuses into giddiness would be significantly lower than it is. I think one problem math guys can have sometimes is that we are happy with sufficient proof…</p>
<p>MUAHAHAHA.
The FIRST thing that came to mind when reading the OP was “geometric series”! I’m actually gonna go look over that right now…
Geometric series was the first thing to help me understand when decimals ‘majikally’ changed into numbers.</p>
<p>The symbols we use to represent numbers are arbitrarily defined. .999… = 1 because the definition of a repeating decimal says it does.</p>
<p>
</p>
<p>You’re drawing a distinction that doesn’t exist. .999… is defined to be the limit of the sequence .9, .99, .999, etc. The symbol .999… represents no “actually infinite mathematical object” apart from that limit.</p>
<p>Who cares…</p>
<p>Think about something that could actually be useful in life.</p>
<p>Think about in in quasi-practical terms. However close you want to get between .999… and 1, you can always get closer. It’s effectively 1.</p>
<p>Wat about this proof?
x = 0.999999999 (infinite)
So: 10x = 9.999999999999 (infinite)
10x - x = 9.999999999999 - x
9x = 9
x = 1
Therefore : 1=0.9999999 (infinite)</p>
<p>Sent from my SCH-I400 using CC App</p>
<p>Good point Frosh2013.</p>
<p>Also:</p>
<p>"try to think of ANY function that matches 1,2,3,4,5,6… to all of the numbers from 0 to 1</p>
<p>Don’t try for too long, because it can’t be done."</p>
<p>f(x)=x/[infinity] ?</p>
<p>f(1)=1/[infinity]
f(1)=0.0…1</p>
<p>f(2)=2/[infinity]
f(2)=0.0…2</p>
<p>Ok, just noticed that .0…1 should =0, based on the fact that if no real numbers are between them, they are equal. Oops.</p>
<p>Math can be counterintuitive sometimes. Can you believe that given a sequence {xn} in a topological space X with the trivial topology, the sequence converges to every point in X?</p>
<p>“Math can be counterintuitive sometimes. Can you believe that given a sequence {xn} in a topological space X with the trivial topology, the sequence converges to every point in X?”</p>
<p>lolwut?</p>
<p>Def. A sequence {pn} converges to a point p if for every neighborhood U of p there exists N such that n>N => pn is in U.</p>
<p>Def. Given X, the trivial topology on X is {X,empty}</p>
<p>Prop. Given topological space X with the trivial topology, any sequence {pn} in X converges to all points of X.</p>
<p>Proof. Let x be an arbitrary point in X. X is the only neighborhood of x. As the sequence {pn} is always in that neighborhood, {pn} converges to x.</p>
<p>@cowgill- Everything’s fine as long as you realize that math may explain a lot of things, but many a times it doesn’t = the real world.</p>
<p>
It depends on how you think of 0.999…; that should be obvious. Consider the string x = 0.9^n over symbols 0, ., 9, where n is the cardinality of the set of natural numbers. That’s an infinite string, a well-defined mathematical object, and I think that it’s perfectly reasonable for lay people to confuse the entity with its representation. The distinction between the thing and how it’s represented lies at the core of modern mathematics and is by no means a “distinction that doesn’t exist”.</p>
<p>@nwcrazy: Agree with this 100%. If I’ve learned anything from math, it’s that knowing when to leave it alone is part of what distinguishes people who are good at math from people who are great at math. Not that I fall into either of these categories, but everybody can benefit from realizing that math doesn’t really have all the answers.</p>
<p>Well I see my thread is still alive and kicking, please understand I am currently in Lin alg and diff eqs. So if there are answers to my next set of questions in future classes, please help me out.</p>
<p>So everyone is saying that 0.9999… is equal to one, showed me an algebraic proof and such. My next question then is, what is the value of the decimal number directly next to the number one? is it 0.999999999…, well, isnt that supposed to be the value of one itself by the proofs provided thus far? </p>
<p>I also understand that there cannot be sub infinities, ie 0.0000…0001. If thats true, whats the value of the number directly next to the number one on the other side of the number line? 1.0000000000…0001? If I cant have that as valid number due to their being no sub inifities, does that number not exist? If any of these values exist, then we really do not have an infinite set of numbers repeating do we?</p>
<p>EDIT: The previous 2 posters I believe gave me the best answer to the question, which is not so much a proof, but more of a reality I need to accept.</p>
<p>The answer to this new round of questions - what’s the “next” number after/before 1 on the number line - is, as you expect, that the notion of the “next” number is undefined… not just for real numbers, but even the rationals (if you’re using the standard > relation for ordering). The proof is one that, I believe, has already been given: assume you had the “next” biggest number after 1, x. You can get a number closer to 1 but bigger than it by averaging 1 and x; that is, (1+x)/2 is closer to 1 than x but still bigger than 1. Our assumption that there exists an x which is the “next” number after 1 must have been incorrect.</p>
<p>To make it a little more palatable, we can name lots of other mathematical questions that just don’t have answers (well, the answer is that there isn’t one of what you’re looking for). The greatest positive integer less than 1, the biggest or smallest integer, a (finite) number x such that x + 1 = x, etc. These things don’t exist by the definitions we give to related concepts. That doesn’t mean that the results can’t be confusing, or surprising. In fact, all of mathematics proceeds (or can proceed) from definitions and axioms, which are simply “definitions” of how people accept things to be. While theorems requiring fewer definitions and axioms and relatively more elaboration can seem more genuinely worthwhile, every proof can (in theory, I believe) be formalized to the point where the only facts that are used are definitions and axioms. Every mathematical truth is therefore a direct result of definitions that human beings create, in a sense.</p>
<p>To address your second question, we do indeed have an infinite set of numbers, and the proof I outline above is sufficient to show that. In fact, we can prove that there are infinitely many rational numbers, let alone reals. Start with a set S(0) = {x, y} where x and y are any two distinct rationals. Define S(i+1) to be the set you get from averaging all pairs of (possibly non-unique) elements of S(i). The size of S(0) is 2, S(1) is 3, S(2) is 6, S(3) is 21, etc. By continuing this process indefinitely, you can get an arbitrarily large number of numbers between x and y, all of which are rational numbers. We can imagine taking this process to “infinity”, in the sense that we consider the set of all numbers which could possibly ever be in this set; the set is infinite, but it doesn’t even contain all the rational numbers between x and y!</p>
<p>The set of rationals is actually what’s called a countably infinite set; the set of reals is infinitely bigger than the set of rationals. We can “enumerate” the rational numbers, hence count them, but as you correctly point out, the standard > relationship is not adequate to the task (what number comes after 1 if rationals are ordered by > ?). Consider this ordering of numbers, however:</p>
<p>1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, 3/2, 4/1, …</p>
<p>If we continue this process indefinitely, not only will we eventually list every rational number, but we can answer questions like “what number comes after 1”. In this enumeration, 1/2 comes after 1.</p>
<p>EDIT: Note that you cannot enumerate the real numbers, so questions like “what’s the next real number after 1” literally do not have an answer.</p>
<p>You’re confusing which number is exact and which is the approximation. Clearly (1/3)*3 is the 1, this is the clearest form. Now .333333… is the infinite decimal expansion that converges to 1/3. Or phrased differently, the infinite sum of 3/(10^n) where n=[1,inf). If you don’t believe that is 1/3 at first glance, I understand and in any finite case you’re right it is not and thus 3 times that quantity is not 1. </p>
<p>It’s all in that …, in that triple dot we’re saying that for any number between it and 1/3 we can add terms so that that number is no longer between them. Since we’re working with real numbers, they are continuous, that is between any two unique numbers there exists another number. Since no number can exists between .333… and 1/3, they must be equal. Keep the faith</p>
<p>We learned this in my calculus class the other day. It makes sense most when I think about it this way:</p>
<p>1-.9999… = .000000…1</p>
<p>You can’t have infinitely many zeroes, and THEN a one after it. It just doesn’t make sense. It’s just zero. So .999… must be 1.</p>