My Math Forum  

Go Back   My Math Forum > Welcome > New Users

New Users Post up here and introduce yourself!


Reply
 
LinkBack Thread Tools Display Modes
July 7th, 2010, 02:12 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 non-fractional number meaning there are no 2 (err conventional? real?) numbers that can be used as a fraction to express Pi.

Pi is a non-repeating, non-fractional 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?
MattJ81 is offline  
 
July 7th, 2010, 04:27 PM   #2
Global Moderator
 
CRGreathouse's Avatar
 
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:
Originally Posted by MattJ81
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 non-fractional number meaning there are no 2 (err conventional? real?) numbers that can be used as a fraction to express Pi.
"Integers" is what you're looking for: there are no two integers such that their ratio is equal to pi. (And pi isn't magical. )

Quote:
Originally Posted by MattJ81
Pi is a non-repeating, non-fractional decimal that goes on, I suppose for all intents and purposes, to Infinity.
Non-repeating: yes. Irrational (what you call "non-fractional"): yes. Goes on to Infinity... not mathematically well-defined. What I think you mean is that it has no finite decimal representation. But it has other representations that are finite: the symbol , for example, or the string "100" in base . So whether the representation is finite or not has everything to do with how we choose to represent it.

Quote:
Originally Posted by MattJ81
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?
I have no idea what you mean by this.

Quote:
Originally Posted by MattJ81
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.
I think the expression you're looking for is per se.

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:
Originally Posted by MattJ81
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.
0.999... = 1 but 1/3 ? 0.34, I agree.

Quote:
Originally Posted by MattJ81
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 don't know what you mean. 1/3 = 1/3 = 1/3, so there's no problem.

Quote:
Originally Posted by MattJ81
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.
Yep, perfectly fine.

Quote:
Originally Posted by MattJ81
So if I attempted to cut this meter rule, the 3 sections would have to get infinitely bigger but still never reach 34 centimetres.


What in the world are you talking about?

Quote:
Originally Posted by MattJ81
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 makes sense to me that some things aren't allowed. You can divide 3 by 5 but not 3 by "cherry pie" or "humanity". It happens that in any ring (a special mathematical object) it doesn't make sense to divide by 0. There are other mathematical objects (e.g. the projectively extended real numbers) where dividing by 0 is (usually) allowed. It all depends on exactly what you're talking about.

Quote:
Originally Posted by MattJ81
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.
Actually infinite numbers are really easy to work with. I don't know why people think they're difficult to grasp or imagine.

Quote:
Originally Posted by MattJ81
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.
You seem to have trouble imagining something that can't be expressed as a finite sum of powers of 10. I don't know why this is such a mental block for you. What's wrong with pi, or e, or phi? Why are they so unnatural to you?

From my perspective, numbers like 0.23 or 210423.7 are aberrant, "weird" numbers. Almost all real numbers are non-terminating 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:
Originally Posted by MattJ81
1/3 of something appears to be infinitely large but still smaller than 0.34?
Don't confuse the size of the number with the size of its representation. "The number of nations CRGreathouse has personally overthrown" is just a fancy, long way of writing 0 -- but that representation is certainly longer than the representation "10^1000" of a much larger number.
CRGreathouse is offline  
July 7th, 2010, 10:21 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:
Originally Posted by CRGreathouse
"Integers" is what you're looking for: there are no two integers such that their ratio is equal to pi. (And pi isn't magical. )
Integers thank you.. my understanding of a non-fractional decimal I suppose is limited to having little knowledge of the subject, and assuming that if the ratio of no two integers can be equal to pi, then pi becomes weird. Yes Pi is not magical I shouldn't have referred to it as such but it is interesting?

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:
Originally Posted by CRGreathouse
Non-repeating: yes. Irrational (what you call "non-fractional"): yes. Goes on to Infinity... not mathematically well-defined. What I think you mean is that it has no finite decimal representation. But it has other representations that are finite: the symbol , for example, or the string "100" in base . So whether the representation is finite or not has everything to do with how we choose to represent it.
I am not a mathematician so, "not mathematically well-defined" means I don't know what I'm talking about but yes, pi has no finite decimal representation. I have no knowledge of base but is infinity not a word we choose to represent something that has no end? So we represent pi by because we cannot represent it with a terminating decimal using base 10? So I suppose this has to be a misunderstanding of math on my part regarding base 10 such that you cannot represent pi accurately with it, so pi being a non-terminating decimal is neither here nor there?

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by MattJ81
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?
I have no idea what you mean by this.
Quote:
Originally Posted by CRGreathouse
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.
Rationalizing? something that I don't understand. If I have a set diameter that I say is exactly 1 inch, and I draw a circle, the circumference of that circle would be exactly pi (1 of anything x pi?), but I cannot measure it(how can I measure 1 inch). If I can measure, accurate to the equipment I am using, around the outside of my pencil line, and then again around the inside of my pencil line I will have two numbers close to pi, but NOT pi. If I draw a thinner line and measure the same, I will again have two numbers that are close, closer than before, but are still NOT pi. In order for me to accurately physically measure pi the thickness of my line would have to be.....0? but is 0 not nothing?. I don't believe this is a mental block as much as it is not having enough information.

Quote:
Originally Posted by CRGreathouse
0.999... = 1 but 1/3 ? 0.34, I agree.
Using the logic of 0.999... is 9/10 + 9/100 + 9/1000 etc, I would never reach 1, the same as 3/10+3/100+3/1000 etc would never reach 0.34, although, for example trying to divide something close to 3/3, 2.99999.../3 with as many places as my calculator can handle , I end up with ....9967, so what equals 0.999.... this is obviously me not knowing enough about what I am talking about isn't it? or is it that my calculator cannot handle infinty represented by base 10? I'm still asking you're opinion not arguing, I cannot.

Quote:
Originally Posted by CRGreathouse
I don't know what you mean. 1/3 = 1/3 = 1/3, so there's no problem.
Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by MattJ81
So if I attempted to cut this meter rule, the 3 sections would have to get infinitely bigger but still never reach 34 centimetres.


What in the world are you talking about?
Sure 1/3 + 1/3 + 1/3 = 3/3 = 1, so is this a problem again that I have with base 10, or a problem understanding how you can physically cut something into a decimal that has no end? So 1/3 of something will never be 17/50 of something but 1/3 does go on forever, although me saying "1/3" of a metre ruler(yard stick?, its not a yard) gets infinitely larger is just plain stupid, the amount you add on gets infintely smaller(is that the right way to say it? I know its not). So although 3/10+3/100 etc will keep growing, the fraction you add on is getting smaller. So, what is 17/50-(something infinitely small?) Once again I feel I must reiterate I am not arguing, just seeking some direction.

Quote:
Originally Posted by CRGreathouse
It makes sense to me that some things aren't allowed. You can divide 3 by 5 but not 3 by "cherry pie" or "humanity". It happens that in any ring (a special mathematical object) it doesn't make sense to divide by 0. There are other mathematical objects (e.g. the projectively extended real numbers) where dividing by 0 is (usually) allowed. It all depends on exactly what you're talking about.
That makes sense, I don't have any other knowledge about division by 0 other than it doesn't make sense, and it makes sense to me why it doesn't make sense(heh), within the realm that it is being used as you said. This raises a somewhat perhaps entirely stupid question, if our existence is all that it is and nothing can be created nor destroyed(this probably is a misunderstanding of basic phyics on my part by not paying attention), then although I(being everything) can say I have 0 of something, surely I must have any number of something else.

Quote:
Originally Posted by CRGreathouse
Actually infinite numbers are really easy to work with. I don't know why people think they're difficult to grasp or imagine.
Easy to imagine, but hard to understand because I cannot express them as a number within my understanding, what do I need to know to make that easy?

Quote:
Originally Posted by CRGreathouse
You seem to have trouble imagining something that can't be expressed as a finite sum of powers of 10. I don't know why this is such a mental block for you. What's wrong with pi, or e, or phi? Why are they so unnatural to you?

From my perspective, numbers like 0.23 or 210423.7 are aberrant, "weird" numbers. Almost all real numbers are non-terminating 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.

Don't confuse the size of the number with the size of its representation. "The number of nations CRGreathouse has personally overthrown" is just a fancy, long way of writing 0 -- but that representation is certainly longer than the representation "10^1000" of a much larger number.
It is not a mental block, nothing is wrong with pi, it is undebatable but it seems unnatural because I cannot represent it physically and I cannot accuaretly determine anything with it in the context of my understanding because i have to approximate it.

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?
MattJ81 is offline  
July 8th, 2010, 06:52 AM   #4
Global Moderator
 
CRGreathouse's Avatar
 
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:
Originally Posted by MattJ81
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?
The mistake is your implicit assumption that physical reality is limited to "displaying" objects of lengths that can be represented as terminating decimals.

(Also, assuming that (mathematically perfect) circles exist in the real world....)

Quote:
Originally Posted by MattJ81
I am not a mathematician so, "not mathematically well-defined" means I don't know what I'm talking about
Suppose I tell you that x is the number such that x times x is 4. x, in this case, is not well-defined: there are two choices for x (-2 and 2). In your case you said, "Infinity" and that's similarly not well-defined. Do you mean aleph_0, beth_3, omega, epsilon_0? Or perhaps the from the extended reals? Or the projective ?

Quote:
Originally Posted by MattJ81
I have no knowledge of base but is infinity not a word we choose to represent something that has no end? So we represent pi by because we cannot represent it with a terminating decimal using base 10?
We represent pi by for convenience. We could also write it as

or as any number of different things.

Similarly, I can write 1 as "1" or or .

Quote:
Originally Posted by MattJ81
So I suppose this has to be a misunderstanding of math on my part regarding base 10 such that you cannot represent pi accurately with it, so pi being a non-terminating decimal is neither here nor there?
Right!

It's very usual to calculate exactly with non-terminating decimals. Sometimes the final result is even an integer!


Quote:
Originally Posted by MattJ81
Rationalizing? something that I don't understand. If I have a set diameter that I say is exactly 1 inch, and I draw a circle, the circumference of that circle would be exactly pi (1 of anything x pi?), but I cannot measure it(how can I measure 1 inch). If I can measure, accurate to the equipment I am using, around the outside of my pencil line, and then again around the inside of my pencil line I will have two numbers close to pi, but NOT pi.
You can't draw a line that's exactly 1 inch long, so it's not surprising that you can't draw a circle with circumference exactly pi inches long either.

Quote:
Originally Posted by MattJ81
Using the logic of 0.999... is 9/10 + 9/100 + 9/1000 etc, I would never reach 1
You're saying that if you add up a finite number of terms, that is 9/10 + 9/100 + ... + 9/10^n, you won't get one. This is true, but has nothing to do with 0.999... = 1, since that has infinitely many terms. Of course if you leave some off you won't get the same thing!

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:
Originally Posted by MattJ81
the same as 3/10+3/100+3/1000 etc would never reach 0.34
This is different, since 0.333... is not the same as 0.34. The difference between them is precisely 1/150.

Quote:
Originally Posted by MattJ81
although, for example trying to divide something close to 3/3, 2.99999.../3 with as many places as my calculator can handle , I end up with ....9967, so what equals 0.999.... this is obviously me not knowing enough about what I am talking about isn't it? or is it that my calculator cannot handle infinty represented by base 10?
Right, you're catching on. Your calculator can handle only a few decimal places; it gives an approximation rather than a true answer. That could be accomplished by keeping infinitely many decimal places (but that seems hard...) or calculating with rational or symbolic arithmetic.

Quote:
Originally Posted by MattJ81
Sure 1/3 + 1/3 + 1/3 = 3/3 = 1, so is this a problem again that I have with base 10, or a problem understanding how you can physically cut something into a decimal that has no end?
You have a funny idea I'd like to dissuade you of, that the real world can exactly represent any length that can be written as a terminating decimal, while it cannot represent any other number. Quantum mechanics' Planck length (and, for that matter, the uncertainty principle) means that you can't represent any length *exactly*. If we ignore that and go to the world of classical mechanics, you can represent any positive real length.

Quote:
Originally Posted by MattJ81
So although 3/10+3/100 etc will keep growing, the fraction you add on is getting smaller. So, what is 17/50-(something infinitely small?)
0.333... isn't growing, it's a fixed number. You're imagining a process of adding 3/10, then adding 3/100, and so on; but the number is what you get at the end of this process -- it's not the process itself. Numbers don't "go" anywhere.

Quote:
Originally Posted by MattJ81
Easy to imagine, but hard to understand because I cannot express them as a number within my understanding, what do I need to know to make that easy?
Maybe I'll tech you some time to calculate with ordinals. You'd quickly learn that simple skill, and the mystery will drop off. The trouble is that nonmathematicians talk about something called "infinity", and there is no such thing in math -- just many collections of objects described as "infinite". So omega is infinite, omega + 1 is infinite, omega times ten plus two is infinite, etc. But there are unrelated "infinities" like beth_1, the cardinality of the continuum (as well as related ordinals like epsilon_0).
CRGreathouse is offline  
July 8th, 2010, 03:01 PM   #5
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,885
Thanks: 1088

Math Focus: Elementary mathematics and beyond
Re: Irrational Numbers. Repeating Decimals and Infinity?

Quote:
Originally Posted by CRGreathouse
The trouble is that nonmathematicians talk about something called "infinity", and there is no such thing in math . . .
I'm not sure I agree with you there, CRG. For example,



greg1313 is offline  
July 8th, 2010, 07:20 PM   #6
Global Moderator
 
CRGreathouse's Avatar
 
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:
Originally Posted by greg1313
I'm not sure I agree with you there, CRG. For example,

I call that the "calculus infinity". Generally, calculus is taught without explicit reference to the infinite: I was taught the notation ("limit as x approaches a") and then the entirely separate notation ("limit as x increases without bound") which was similar only notationally. Similarly for limits equal to infinity.

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.
CRGreathouse is offline  
July 8th, 2010, 07:24 PM   #7
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,885
Thanks: 1088

Math Focus: Elementary mathematics and beyond
Re: Irrational Numbers. Repeating Decimals and Infinity?

Quote:
Originally Posted by CRGreathouse
fake infinities
Why are they "fake"?

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?
greg1313 is offline  
July 9th, 2010, 02:31 PM   #8
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: Irrational Numbers. Repeating Decimals and Infinity?

Quote:
Originally Posted by greg1313
Quote:
Originally Posted by CRGreathouse
fake infinities
Why are they "fake"?
Because they don't actually refer to a precise object. Saying "as x approaches infinity" does not mean "x approaches some number". It means x grows without bound. Using the extended real line (with a "real" infinity... forgive the pun), then "as x approaches infinity" does mean "as x approaches some number", namely, "infinity" is a number in the extended real line (but it is not in the standard real line.)

Quote:
Also, what about the concepts of infinity introduced by Cantor? Aren't they "real"?
CRG's point about "fake infinities" is that "a fake infinity" is a notational convention-- it doesn't refer to a specific mathematical object.

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:
Originally Posted by MattJ81
but is infinity not a word we choose to represent something that has no end?
I think you've figured this out, but... saying "pi is infinite" is ambiguous, and "[insert number] is infinite" is normally a statement used to say that a number is larger than all integers, which pi certainly is not (4 is larger than pi). The correct thing to say would be that "pi has an infinite decimal expansion."

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:
Rationalizing? something that I don't understand. If I have a set diameter that I say is exactly 1 inch, and I draw a circle, the circumference of that circle would be exactly pi (1 of anything x pi?), but I cannot measure it(how can I measure 1 inch).

snip

In order for me to accurately physically measure pi the thickness of my line would have to be.....0? but is 0 not nothing?. I don't believe this is a mental block as much as it is not having enough information.
Let's forget about the real world for a bit, and pretend we can draw an infinitely thin line, we can measure everything exactly, and we have calculators which can display an infinite number of digits...

Quote:
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?
The standpoint of the mathematical community on such things is called mathematical analysis.
For having very little formal training in math (as you say), you've come surprisingly close to some of the key ideas.

Quote:
If I can measure, accurate to the equipment I am using, around the outside of my pencil line, and then again around the inside of my pencil line I will have two numbers close to pi, but NOT pi. If I draw a thinner line and measure the same, I will again have two numbers that are close, closer than before, but are still NOT pi.
Quote:
Originally Posted by MattJ81
Using the logic of 0.999... is 9/10 + 9/100 + 9/1000 etc, I would never reach 1, the same as 3/10+3/100+3/1000 etc would never reach 0.34, although, for example trying to divide something close to 3/3, 2.99999.../3 with as many places as my calculator can handle , I end up with ....9967, so what equals 0.999.... this is obviously me not knowing enough about what I am talking about isn't it? or is it that my calculator cannot handle infinty represented by base 10? I'm still asking you're opinion not arguing, I cannot.
Quote:
Sure 1/3 + 1/3 + 1/3 = 3/3 = 1, so is this a problem again that I have with base 10
, or a problem understanding how you can physically cut something into a decimal that has no end? So 1/3 of something will never be 17/50 of something but 1/3 does go on forever, although me saying "1/3" of a metre ruler(yard stick?, its not a yard) gets infinitely larger is just plain stupid, the amount you add on gets infintely smaller(is that the right way to say it? I know its not). So although 3/10+3/100 etc will keep growing, the fraction you add on is getting smaller. So, what is 17/50-(something infinitely small?) Once again I feel I must reiterate I am not arguing, just seeking some direction.[/quote]

Quote:
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.
All 3 of these things that you mention have something in common: you are iterating some process, and each new iteration has a "smaller" change from the previous one, than the previous one had from the one before it. Let's try to state this more precisely: Let's call an "infinite" (i.e. endless) list of numbers (e.g. 0.9, 0.99, 0.999, ...) a sequence, and let's call an endless sum of numbers a series. We can think of your "never-ending sum" (what analysts call a series) 0.3+0.03+0.003+... as a sequence (mathematicians call it the "sequence of partial sums")-- the first term is 0.3, the second term is 0.33 (0.3+0.03), the third is 0.333 (0.3+0.03+0.003), and so on. This way, whenever we want to talk about infinite sums, we can just talk about serquences.

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 non-zero distance (any distance, however big or small you choose, just non-zero), 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:
This raises a somewhat perhaps entirely stupid question, if our existence is all that it is and nothing can be created nor destroyed(this probably is a misunderstanding of basic phyics on my part by not paying attention), then although I(being everything) can say I have 0 of something, surely I must have any number of something else.
I'm going with the "entirely stupid" part. You are not being "divided" by objects you contain, nor are objects you own "multiplied" by you.

Anyway, regarding division by 0: the reason it "doesn't make sense" to divide by 0 is even deeper than it not being mathematically well-defined-- the square-root of a number is not well-defined, 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:
Easy to imagine, but hard to understand because I cannot express them as a number within my understanding, what do I need to know to make that easy?
A little bit of set theory, which is unfortunately rarely taught to anyone besides math or computer science students... The basic idea behind infinite arithmetic is that certain types of sets (called cardinals) represent sizes. As it turns out, there are different "sizes" of infinity, and each one of these has a cardinal. There are also certain sets which represent a "position in an ordering" (e.g., 1st, 2nd, 3rd), and these can also be treated as numbers (so there is both cardinal arithmetic and ordinal arithmetic). Both of these types of numbers correspond exactly to (non-negative) integers in the finite case.
cknapp is offline  
July 9th, 2010, 03:46 PM   #9
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,885
Thanks: 1088

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?
greg1313 is offline  
July 9th, 2010, 07:06 PM   #10
Global Moderator
 
CRGreathouse's Avatar
 
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:
Originally Posted by greg1313
Would it be correct (or close to correct) to state that infinity is a property of objects rather than a distinct entity?
Being infinite is a property, yes. That's meaningful, while the concept of "infinity" as such is not.
CRGreathouse is offline  
Reply

  My Math Forum > Welcome > New Users

Tags
decimals, infinity, irrational, numbers, repeating



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
irrational numbers Albert.Teng Algebra 4 February 12th, 2014 05:55 PM
Repeating decimals shunya Elementary Math 6 January 29th, 2014 06:08 AM
Irrational numbers niki500 Number Theory 5 October 7th, 2012 10:10 PM
Fractions and repeating decimals midwestbryan Elementary Math 5 September 20th, 2012 04:58 PM
Irrational Numbers Mighty Mouse Jr Algebra 1 October 16th, 2010 08:46 PM





Copyright © 2018 My Math Forum. All rights reserved.