.999... =1

College Majors Engineering Majors
41

<p>you say 0.33333… approaches a number. Do you know the meaning of the word approaches here? Because you are saying that 0.33333… “gets closer and closer” to 1/3, and that is so not how it works. 0.33333… is one specific value, and that value can be expressed as the fraction 1/3. 0.33333… does not “fill up” a progress bar on its way to 1/3, nor does it get “closer and closer” by walking along the number line towards 1/3. It is the value 1/3, expressed in decimal notation. Tell me, do you also consider “infinity” to be a number which keeps getting bigger and bigger? Because from what you’re saying, I believe this to be your interpretation, which is wrong. “Infinity” is not a number that keeps increasing, it is a concept.</p>

<p>You must distinguish the value of the number from the way we write the number. The value of 1.3 and 0.33333… is the same. The way we write them is different.</p>

42

<p>i can’t believe there are engineers who are saying</p>

<p>1/3 != .333… </p>

<p>it’s not an approximation. it’s <em>exactly</em> the same number</p>

43

<p>this thread is painful, did you all skip the section on convergence of infinite series in calculus? just because there is an infinite number of things doesn’t mean we can’t make an exact calculation, in fact that is EXACLTY what calculus is based on. the number .9999…=1, it is simply a matter of convergence. this doesn’t mean .99=1, or .999=1, or .99999999999999999999999=1, it only means .999…=1. it isn’t an approximation, it is a well defined mathematical expression that is exactly true.</p>

44

<p>@xjis: I’m shocked when reading this thread, too. One can easily see that 0.9999…=1 after taking Calc BC in high school.</p>

<p>That’s why we now have a lot of “students who are interested in engineering but lack talents in Math” on this board.</p>

45

<p>I think noimagination summed it up the best. Clearly, norris and systemshock need to read into what he is saying. </p>

<p>And Hawkwings, great job in explaining it with detail.</p>

46

<p>I see what everyone is saying about it being equal to 1. I mean you can’t subtract anything from 1 to get .999…, so it must be equal. The limit equals 1 but I guess I don’t get how they can be equal when .9999… is not actually a number.</p>

47

<p>I dont know dude, 0.3333… is an infinite series which is also a limit. Of course, engineers would say that 0.9999… = 1. But pure math doesnt say so, you better go ask your calculus teacher what a limit is in the first place. Or rather,</p>

<p>lim (x-> 1) x = 1 but its actually not, it approaches to one</p>

<p>OR</p>

<p>1/0 is undefined BUT
lim(x->0) 1/x = infinity.</p>

48

<p>who cares?</p>

49

<p>^smartest thing said on this entire thread.</p>

50

<p>i guess noimagination is right…</p>

51

<p>So let’s say we take an airplane and put it on a treadmill…</p>

52

<p>Let me clear something up here, all jokes aside.</p>

<p>0.999… is <em>not</em> a limit, it’s a well-defined representation of a certain real number. The question that we have is whether or not the representation 0.999… applies to the same real number as the representation 1.000…</p>

<p>Rigorously speaking, I don’t like the proofs like:
1/3 = 0.333…
3/3 = 3(1/3) = 3(0.333…) = 0.999…
That sort of presupposes a little too much about the algebra of real numbers to make the proof too useful…</p>

<p>Also, proofs involving limits of summations really don’t say anything about the actual number 0.999…, just about limits of sequences of rational numbers approaching the number. Mathematical induction typically cannot be trusted at infinity, and I don’t see why summations need be any more special.</p>

<p>For my money, the most useful proof that 0.999… represents the same real number as 1.000… relies on the definition of the real numbers… perhaps the one using Dedekind cuts, for instance. Proofs based on simple properties of the real numbers are pretty straightforward.</p>