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July 7th, 2010, 01:12 PM  #1 
Member Joined: Mar 2010 Posts: 31 Thanks: 0  Irrational Numbers. Repeating Decimals and Infinity?
This is probably another topic about the same thing that everyone asks, but I haven't yet found a topic covering exactly what I am going to ask, although I am sure it exists, I'm not even sure if this is fundamentally a "Mathematics" question because Math is a system and the system must have some rules to work, but it involves Math so maybe it is. So I haven't used Math for quite some time, and I came across a question the other day that I never thought to ask before. Pi..... I'm not sure I can ask the question in the light that I need to without babbling on but I will try. Pi is a number we use to determine the Circumference of a circle(among other things) given its diameter, right? And it is stated that this magical number is the direction relation of the Circumference divided by its diameter, more or less? But Pi is an irrational or nonfractional number meaning there are no 2 (err conventional? real?) numbers that can be used as a fraction to express Pi. Pi is a nonrepeating, nonfractional decimal that goes on, I suppose for all intents and purposes, to Infinity. That being said, if my diameter is of a fixed unit length(all be it seemingly impossible), then my circumference is Inifinte? So then I ask myself what am I measuring, in the real world I can approximate my circumference by measuring it, using a measurement that is an approximation of a distance, and I can get very close to Pi by measuring a thinner line, thinner and thinner until I get to "0" but 0 is nothing. So I'm looking to measure something that is 0, but not nothing? This not being a math question per say would be that Pi is Pi and accurate to whatever your tolerance maybe in Math or the real world, and therefore I should not be asking, but it leads me to something a little more interesting, and perhaps more confusing. 1/3. Now in an attempt to research this a little bit just for my own understanding as bleak as that quest appears to be, I came across 2 things, 0.9...(is that right for a repeating decimal?) and division by 0. Both are equally confusing but understandable in terms of the math system as something multiplied by nothing can't be anything, although is still raises question for me. All explanations for why 0.9... is 1 appear valid, such as x = 0.9... 10x = 9.9... 9x then = 9 so x = 1. Although it seems to be just a question to confuse people and cause a stir, I am torn on the subject as 1/3 will never be 0.34. 1/3 came into question for me when I was revising how to convert a repeating decimal into fractional form, and was asking myself what a repeating decimal is, it goes on forever, ok. Then i suppose stupidly I imagined taking a metre rule and attempting to cut it into 3 equal lenths. In the real world 3 equal lengths is accurate to whatever tolerance or margin of error you allow, and how good your equipment is but they will never be equal, it seems. I tried to think of what 1/3 is and probably broke a rule or something but anyway, it appears that 1/3 or 0.3... can be viewed as many fractions added together. 0.3 first of all is 3/10, plus 0.03 or 3/100 plus .003 or 3/1000 and so on forever. So if I attempted to cut this meter rule, the 3 sections would have to get infinitely bigger but still never reach 34 centimetres. The same goes for 0.9...(although I may be missing something altogether) it looks like it is 1 by a rule governing repeating decimals, but if I apply the same thinking to it, it will get infinitely bigger but never reach 1. So what am I looking at? I am not debating math as such because the rules work for most things, so it is a pretty good system, however some things it seems are refused because they don't fit the system, which is fine, its just not a perfect system that explains everything, so how do we explain these things? Well I guess we can't or we probably wouldn't see cannot divide by 0 for starters. It seems to me that these questions must have(although I can't exactly assume it) something to do with our existence and the fabric of the universe. I cannot really begin to grasp to idea of infinity because the systems I use everyday do not govern it as such, but it obviously exists, maybe something multiplied by nothing, or 0 but not nothing, can equal anything. Wouldn't that mean that the entire math system itself would be flawed? How can I say 1/3 of something when it cannot be expressed as something I can comprehend giving the rules I am using. But then again how can I say 1 of anything from a measuring standpoint being that I cannot measure something infintely small, but it must exist, somehow. Anyway even tho I am more confused than ever, what is the standpoint of the math world on such things? besides the fact that you cannot divide by 0 because it doesn't make sense. Infintely small cannot be expressed as a number so is it 0? but its not nothing...? 1/3 of something appears to be infinitely large but still smaller than 0.34? 
July 7th, 2010, 03:27 PM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Irrational Numbers. Repeating Decimals and Infinity? Quote:
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Pi is a particular number. Numbers that are close to pi are not the same as pi. Certainly there are cases when you don't need much precision, but don't confuse that with an inexactness of definition. I don't know why you wrote "1/3". Quote:
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What in the world are you talking about? Quote:
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From my perspective, numbers like 0.23 or 210423.7 are aberrant, "weird" numbers. Almost all real numbers are nonterminating in base 10 (or, for that matter, irrational), but there are certain exceptions like those I listed. Why you'd want to restrict yourself to that tiny, thin subset is beyond me. Quote:
 
July 7th, 2010, 09:21 PM  #3  
Member Joined: Mar 2010 Posts: 31 Thanks: 0  Re: Irrational Numbers. Repeating Decimals and Infinity?
I appreciate the response, and I'm sure you realised that I wasn't arguing anything err.. per se, just asking some questions in an unintentionally elaborate way because I am unsure of the questions...? Quote:
I suppose due to the fact that if with a set diameter, I have to mulitply by pi, accurate to an indefinite number of decimal places. If pi goes on forever then somewhere surely I have to round, therefore the circumference is an approximation? This is not an argument of the fact but simply a misunderstanding of pi using base 10? Quote:
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So all in all after having to rewrite this because I took so long I had to log in again, this is not an argument, just a lack of understanding, which is why I am even posting, so where did I get?  
July 8th, 2010, 05:52 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Irrational Numbers. Repeating Decimals and Infinity? Quote:
(Also, assuming that (mathematically perfect) circles exist in the real world....) Quote:
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or as any number of different things. Similarly, I can write 1 as "1" or or . Quote:
It's very usual to calculate exactly with nonterminating decimals. Sometimes the final result is even an integer! Quote:
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But there is no "reaching" in 0.999.... You add up *infinitely* many terms, all at once, in writing the "..." (or the vinculum, if you use that instead of dots). Quote:
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July 8th, 2010, 02:01 PM  #5  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Irrational Numbers. Repeating Decimals and Infinity? Quote:
 
July 8th, 2010, 06:20 PM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Irrational Numbers. Repeating Decimals and Infinity? Quote:
But calculus can be taught using the extended real numbers instead of these fake infinities to much the same effect. So depending on your position, the calculus infinity is not a number at all, just a notational convention, or it represents the extended real . Ball's in your court.  
July 8th, 2010, 06:24 PM  #7  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Irrational Numbers. Repeating Decimals and Infinity? Quote:
Also, what about the concepts of infinity introduced by Cantor? Aren't they "real"? EDIT: Many branches of mathematics deal explicitly with something called "infinity", whether it be "real", "fake" or what have you. How can you then say that in mathematics there is no such thing as infinity?  
July 9th, 2010, 01:31 PM  #8  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Irrational Numbers. Repeating Decimals and Infinity? Quote:
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With a "Cantorian" set theory, there is also no number called "infinity"; infinite in this sort of system is an adjective to describe sets (or cardinalities... or ordinalities) which satisfy a certain property. But there is not a unique set which is "infinite", or even a single "canonical" infinity which captures everything set theorists are interested in about infinite sets. Quote:
It happens a lot that by confusing the two statements "x is infinitely large" (what mathematicians mean when they say "x is infinite"), and "x has an infinitely long decimal expansion", people confuse the two concepts, and decide really stupid things. (If I had a dime for every time I've heard someone say ".999... is infinite, and 1 is not, so they can't be equal!", I would be able to buy a couple of drinks.) Quote:
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For having very little formal training in math (as you say), you've come surprisingly close to some of the key ideas. Quote:
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So, we want to say that the terms of a sequence get "close" to some value. E.g., (0.9,0.99,0.999,...) gets close to 1, or, as you suggest may be the case, (0.3,0.33,0.333,...) gets close to 0.34. Of course, just being close is not good enough there wouldn't be heated arguments about 0.999... and 1, if mathematicians just said they were "close". Obviously, we can't just take the last term in the sequence, because there is no last term, so we need the terms in (0.9,0.99,...) to get arbitrarily close to 1. I.e., we need to be such that if you give me a nonzero distance (any distance, however big or small you choose, just nonzero), then I can find a term such that that and all subsequent terms are closer to 1 than that distance I.e., no matter how small you try to make my gap, if I keep going down the sequence, I'll eventually only see numbers that are inside of that gap. When this happens, we say that the sequence converges to whatever value so (if it fits the definition), (0.9,0.99,0.999,...) converges to 1. It turns out the definition does fit: if you give me a distance from 1 (say 0.0001), then if I go far enough in the sequence (in this case, to the 5th term, 0.99999), then the difference between the terms in the sequence and 1 will be less than that distance away. Now, lets look at 0.333... and 0.34. First, the "sequence version" of 0.333... is (0.3,0.33,0.333,...). Now, you claim that this sequence converges to 0.34. I call you a liar. My job is to find a distance from 0.34 that is "too small" for the terms of your sequence to fit into. Without batting an eyelash, I say "0.001". This means you want terms that are bigger than 0.339, and less than 0.341. This is a problem: no term in the sequence is even 0.334, so none will every be as big as 0.0339, so you'll never even make it in the gap I've made for you. Changing gears a bit, you correctly noticed that a decimal expansion for a number is just a shorthand for a sequence (of partial sums) which converges to that number, and you correctly understood that no term in the sequence is exactly the correct number (or at least, not when the number has an infinite decimal expansion). Turning now to pi, you get the same sort of thing: you need to go to the "end" of the sequence (which doesn't exist) to get something with value exactly pi. Analysis contains the tools that you need to use to get all the way "to the end". Quote:
Anyway, regarding division by 0: the reason it "doesn't make sense" to divide by 0 is even deeper than it not being mathematically welldefined the squareroot of a number is not welldefined, but it makes sense to "take the square root", you just end up needing to do more than one thing: one for each square root. The problem is that asking "what is x/y?" is asking the question "what number z satisfies the equation x=zy?" When we are dividing by 0, we're looking at the equation "x=z0" (or rather, x=0) i.e., "for what z does x=0". If x isn't 0, then no a satisfies the equation; if x is 0, then every possible z satisfies it, and so we need to test every possible value to get any information out of the equation. CRG brought up "rings", and it turns out that in any ring, we have this awful behavior. Quote:
 
July 9th, 2010, 02:46 PM  #9 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Irrational Numbers. Repeating Decimals and Infinity?
Would it be correct (or close to correct) to state that infinity is a property of objects rather than a distinct entity?

July 9th, 2010, 06:06 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Irrational Numbers. Repeating Decimals and Infinity?
Greg, cknapp addressed your questions to me, I think. I agree with everything he wrote regarding that post; if you need clarification just ask. Quote:
 

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decimals, infinity, irrational, numbers, repeating 
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