<p>If you have to pay a lot of money for remedial classes in math, science and writing courses in college, you probably shouldn’t be in college…at least not yet.</p>
<p>Likewise, if you don’t know what ‘controlling for a variables’ is, or what a standard deviation is, you probably shouldn’t graduate from college. The list goes on. </p>
<p>In any case, everyone should check out the David Brooks oped in the Times today…“History for Dollars.”</p>
<p>
It’s defined by the series for .9/(10^n) starting at n=0, which converges to 1.</p>
<p>The problem with trying to make HS about the liberal arts is that HS kids aren’t intellectually mature enough to handle the kind of good liberal arts program I and others like me think educated people should have. I don’t necessarily think it’s a problem with middle or elementary school, either. Most people just aren’t developed enough by 15 years old to do it.</p>
<p>Of course whether 18/19 is old enough is another story. I hypothesize that that’s catching students at about the right age for undergraduate study. I could be wrong.</p>
<p>We could all make long lists of things people in any major should know before they graduate from college. The problem with that - and why universities don’t do it - is because that would mean a lot of people not graduating.</p>
<p>And yeah, the definition of most real numbers depends on convergence, I think.</p>
<p>“It’s defined by the series for .9/(10^n) starting at n=0, which converges to 1.”
<p>
[Geometric</a> Series – from Wolfram MathWorld](<a href=“http://mathworld.wolfram.com/GeometricSeries.html]Geometric”>Geometric Series -- from Wolfram MathWorld)</p>
<p>Perhaps I misunderstand your concern…?</p>
<p>
.9… is equal to the limit of the series by definition. How else would you definite infinite decimal expansions?</p>
<p>Technical degree gives you an edge in the beginning but it limits your options later on. So, employer sponsored continuing education can keep you updated.</p>
<p>
I think that would actually work provided you can show that 1/9 = .1… . Then 1/9 being the multiplicative inverse of 9 would get you 1 = 9<em>(1/9) = 9</em>(.1…) = .9… (by standard analysis argument).</p>
<p>[0.999</a>… - Wikipedia, the free encyclopedia](<a href=“http://en.wikipedia.org/wiki/0.999]0.999”>0.999... - Wikipedia)…</p>
<p>barium:</p>
<p>i think that to say .9… is the limit of that series is presupposing the conclusion, no different than in the proof you say presupposes the conclusion. Certainly one can easily prove .1… = 1/9 by long division and induction on the number of digits after the decimal. The point is that induction doesn’t work <em>at</em> infinity, just arbitrarily far up there.</p>
<p>noimagination:</p>
<p>Yeah, you did misunderstand my concern. I agree that the series converges to 1. My problem is in saying that the limit of the series and .9… are the same thing. To say that presupposes the conclusion, or at least I see it that way. I prefer proofs that rely on more fundamental notions, like the construction of the real numbers. Wikipedia might even mention that.</p>
<p>The problem with proofs is you have to be clear about how much you can take for granted.</p>
<p>And barium, I think it’s important not to confuse a real number with its representation. .9… is not a number, it’s just the representation of a number, and in a sense it is its own definition.</p>
<p>
It does: [0.999</a>… - Wikipedia, the free encyclopedia](<a href=“http://en.wikipedia.org/wiki/0.999#Proofs_from_the_construction_of_the_real_numbers]0.999”>0.999... - Wikipedia)</p>
<p>It seems to me that in writing 0.999… the ellipsis indicates that we are representing this particular limit - in other words, 9 / (10 ^ increasing powers of n).</p>
<p>EDIT: This is a rather good reason not to use ellipses in formal proofs.</p>
<p>Can you please state the definition of an infinite decimal expansion that you are working with?</p>
<p>For me, an infinite decimal expansion .abcd… is by definition the limit of the series a/10 + b/100 + c/1000 + d/10000 + …</p>
<p>I don’t believe in fractions.</p>
<p>For me, the ellipsis doesn’t mean limit at all, just that there are an infinite number of 9s afterwards. Subtly different from a limit. I can recommend the study of the ball and vase problem to see why. In this instance it just so happens that the two are the same, and therefore perhaps you could use it as a definition, but it does presuppose the conclusion in a proof like this. A better question is why prove it, if your definition is that.</p>
<p>Barium:</p>
<p>like I already said, my position is that .9… Is not a number, just a representation of one. It needs no definition as a limit. The number it represents is just a number, and can be the limit of lots of things. .999… = 1 is shorthand for saying the representations stand for the same number… But there’s just the one #1, no #.999…</p>
<p>
See b@r!um’s post #573. EDIT: And #577…</p>
<p>As for motivation… I don’t find this a very interesting proof. Still, it’s preferable to the previous 28 pages of this thread :)</p>
<p>Barium:</p>
<p>I did define it somewhere. Same meaning as a sequence’s ellipsis… Infinite repetition. Actually, not potentially (or limiting) infinite.</p>
<p>
That’s just notation. You have yet to tell us what the notation of a dot followed by an infinite number of 9s means.</p>
<p>Basically, just because a limit is something doesn’t mean that’s what it actually is, at least not without additional provisos.</p>