Losing 'faith' in math?

College Majors Science Majors
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<p>I was recently doing some homework for my physics class and at one point, the simple value of 1/3 came up. And something occured to me. The value of 1/3 in decimal form is .33333 to infinity. Should I decide to add 1/3 to this, I get .66666...., and a final thrid will yield....</p>

<p>.9999... or 1?</p>

<p>when I asked my teacher, he said they were equal!!!</p>

<p>He tried showing an algebraic proof of like 10x-x=9x and frankley that just made me more unsure of whats going on. We start out with x=.9999... and when doing basic algebra we get 1 thus?!?!? .99999=1?!?</p>

<p>I still did not accept this answer and kept pressing, until he told me he took the equality on 'faith' that its true and moved on. This makes my math path seem scary that I might have to just accept some things that we cannot explain, even at the simplest levels. </p>

<p>And it seems this could even stab math in its heart, at least the fundametal theorem of arithmetic which is based on multiplication and division to some degree, yet simple division here in my example seems to fail!</p>

<p>Someone want to talk me off the ledge here before I leap into a business degree.</p>

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<p>Either you accept it or you don’t. I didn’t believe .9999…=1 until I saw this proof on Wikipedia: <a href=“http://upload.wikimedia.org/wikipedia/en/math/6/8/0/680fee112b7c09afa53b3f35eea46f9c.png[/url]”>http://upload.wikimedia.org/wikipedia/en/math/6/8/0/680fee112b7c09afa53b3f35eea46f9c.png&lt;/a&gt;&lt;/p&gt;

<p>A lot of things in math are hard to grasp. I’m a physics major, and so far two classes that I’ve taken are Quantum and Modern Physics. Our professor kept assuring us not to be worried if we don’t understand it, because NO ONE understands it. It reminds me of the quote: “if you think you understand quantum mechanics, you don’t understand quantum mechanics.”</p>

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<p>You shouldn’t take math on faith; that’s the beauty of math. Also, you shouldn’t ask physicists to help explain math; most of them have been winging it since Calculus.</p>

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<p>Take a class in Numerical Analysis and you will start to learn about “error”, especially when computers and calculators perform math. There are very small errors anytime you are not calculating with integers. Computers are not designed to fully compute irrational numbers (infinite decimals).</p>

<p>The key is to have as small an error as possible throughout the calculations. Errors tend to multiply the larger the equation.</p>

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<p>This is true =D</p>

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<p>We prefer the term “Advance plug and chug”</p>

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<p>This actually really caught my attention and threw me off for a little bit. </p>

<p>From how i’ve interpreted it, what’s actually occurring here is that if you have a number of 9s, then 10x will always have one less 9 in the decimals than x will. Try doing x = .9, .999, etc and you will see that. I think it’s kind of like a limit, in the sense that you can approach ever closer, but you can never actually have infinite number of 9s. The limit of that number then would equal 1, but in practice you can never truly reach 1.</p>

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<p>You’re only discovering the tip of the iceberg. .9 repeating IS equal to one. Accept it and move on, there’s absolutely nothing in the rules that says you can’t have the same number represented in two different ways. If two numbers are not equal there must exist some number in between those two numbers. Tell me, what number is between .9 repeating and 1? </p>

<p>If that hurts your head, then the fact that there exist and infinite amount of different sizes of infinity will make your head explode!</p>

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<p>^^ Well your first paragraph helped me to understand it.</p>

<p>Then your second one made me **** my pants. O.o</p>

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<p>It’s true. We only touched the surface of it in my Number Theory class, but there are more real numbers than integers if you think about it. Thus, a larger infinity.</p>

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<p>Think about the set of natural numbers 1,2,3,4,5,6… and just the even numbers 2,4,6,8,10…</p>

<p>How can you transform the first set into the second set. Easy, with the function f(x)=2x where the domain is the set of natural numbers. </p>

<p>Now try to think of ANY function that matches 1,2,3,4,5,6… to all of the real positive numbers. Hell, 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. Your trying to match all of the numbers in 1,2,3,4,… to an infinite set that is bigger (the set of all real numbers from 0 to 1 or from 0 to infinity). </p>

<p>Both sets are infinite in size, but clearly there is something strange about them because you can not create a bijective function between them. But look at the set all number 0 to 1 and all numbers from 0 to infinity. Can you create a function between those sets? Yes you can (I’ll let you figure it out). Thus, the set of numbers from 0 to 1 and the set of numbers 0 to infinity ARE the same size. </p>

<p>Now to take it a step further</p>

<p>Consider the set {1,2}. Take the set of all sets of {1,2} which would be {{1}, {2}, {1,2}, {empty set}}. Notice how that set has more elements in it than the set {1,2}. In fact, this is a theorem. If you take the set of all sets of any set, that set will indeed be larger. I won’t prove it, you can find vast proofs online. </p>

<p>Tell me, if you took the set of all sets of </p>

<p>{1,2,3,4,…} what would you get?</p>

<p>How about the set of all sets of the all real numbers between 0 and 1? </p>

<p>The size of the set of all number between 0 and 1 is an infinity bigger than the the size of the set of all positive natural numbers, we already established that since we can’t create a bijective function between them. But what happens when you take the set of all sets of the numbers between 0 and 1? You clearly have to have a bigger set because of the theorem. What size of infinity is that? How about the set of all sets of all sets of the numbers between 0 and 1? And so on ad infinitum…</p>

<p>The notion of a 1 to 1 and onto function between two sets is really at the heart of the theory. A bijective function is essentially just a machine that pairs one set of numbers with another set of numbers. If you put in the entire set of numbers into any such machine you can think of but can’t produce every single element of a set you’re trying to match, then you have two different size sets.</p>

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<p>Pick any two different numbers and you can find another number between them. What’s between .9 repeating and 1? Nothing, so they’re the same number.</p>

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<p>Have you ever worked with limits?</p>

<p>I think you are confused because you don’t know what 0.999… really means. It is defined as the limit of the sequence of partial sums, x(n) = 9/10 + 9/100 + 9/1000 + … + 9/10^n, as n approaches infinity. Doing a bit of epsilon-delta analysis shows that this sequence converges to 1. </p>

<p>It doesn’t matter that 0.999… “never actually reaches” 1. The “infinitely small” error term is built into the definition of a limit, and an infinite decimal expansion is defined as the limit of a sequence.</p>

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<p>^ Which, by the way, is just the mathematically rigorous way of saying, “there is no number between 0.999… and 1, hence they are the same.” (That happens to be an axiom of the real numbers. You can also construct a set of numbers that has an “infinitely small” number epsilon which is greater than 0 but smaller than any positive number. In this set, 0.999… would be different from 1.)</p>

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<p>What was the problem on the homework? And yes, I understand what you mean, those things are weird! Some fractiosn 1/3 2/3 3/3 etc are just unusual. Then again, physics in general is hard to grasp.</p>

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That’s got nothing to do with thirds. Every rational number with a finite decimal expansion can be represented by an infinite decimal expansion as well (e.g. 1 = 0.999…).</p>

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<p>I understand limits, but there is an issue there. Hell back in freshman calc we covered limits and used them to show where an expression will be converging, but not actually equal. Like an expression that has a break in a point and the limit is set at when that break occurs, the limit shows us where we are going, but we will never actually reach that point. Is that the same thing happening here, or is .999… equal to one? I dont think a limit can answer that, but I do have to look at a previous post about geometric series.</p>

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Nice try but no - the value of .999… is defined to be the limit of the sequence 0.9, 0.99, 0.999, … We don’t know how to make sense of an infinite decimal expansion except as a limit.</p>

<p>What you are thinking of is the limit of a function value at a discontinuity. You can apply that here too - but we are looking at the function f(x) = x which is perfectly continuous. Continuity gives us that the function values f(0.9), f(0.99), f(0.999), … converge to f(1) = 1.</p>

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<p>I wonder why it’s so easy to accept that 1/3 = 0.3… but not that 1 = 0.9…? I think like half of the “proofs” I’ve seen that 1 = 0.9… assumes that 1/3 = 0.3…, but once the conclusion is reached, people start to get really confused.</p>

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<p>wait, then what’s



0.10110011100011110000111110000011111100000011111110000000...

???</p>

<p>mind asplode</p>