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 Why are repeating decimals Rational numbers?
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Szabo
Average Member

USA
8 Posts

Posted - 06/30/2013 :  20:10:22  Show Profile
Please bear with me, the depth of math I've learned is intermediate algebra. That being said, I know that.....
A rational number is any quotient of 2 integers ( 1/4, 2/3, 17/1)
The quotients will be equal to a number that is greater or less than 0 with a terminating or repeating decimal.

My question is why are repeating decimals considered rational if you can't place an exact countable value on them. Also, how do they even equate to the integers being divided. Moreover, how would you be able to determine their place on a number line if the decimal expansion is infinite?
I get that 1/3 of an apple is 3 equal parts of the apple. What I don't understand is why the .3333 repeating wouldn't terminate after a finite number of digits.


Edited by - Szabo on 07/02/2013 00:53:17
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Ultraglide
Advanced Member

Canada
299 Posts

Posted - 07/01/2013 :  13:06:42  Show Profile
We need to sort out a few things. A rational number is the quotient of two integers where the denominator is not zero. The quotient could be any value - less than zero, equal to zero, or greater than zero, i.e. no restrictions. Some rational numbers such as 1/2 have exact decimal values, in this case 0.5. Others are repeating decimals such as 1/3 which is 0.333333... The way to show that it never terminates is to set it up as a division question such as 3 divided into 1.000000... You will notice that there's always a remainder of 1 that can be combined with the next zero in each step. As far as the number line, you can only approximate the location of the number. By the way, you can work back from the repeating decimal to the ratio. Also, you should say "Please bear with me...", we don't need the other 'bare', lol.
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the_hill1962
Advanced Member

USA
1468 Posts

Posted - 07/01/2013 :  13:13:08  Show Profile
To take your last question first, The .333333333... wouldn't terminate because if you multiplied that finite amount of threes by 3, you would get a finite amount of nines. We know that 1/3 multiplied by 3 is 1.
See that .9999999999999999999999999999999999999999999 is not 1.
Sure, for practical purposes, we round .9999999999999999999999999999999999999999 to 1 but truly, you need to go infinite with them. Therefor, 1/3 is an infinite amount of threes.
Now for your first question. A fraction IS an exact countable value. Irrationals are not. To place them on a number line, you just divide the segment up geometrically. Do a search for "how to divide a segment into thirds" or "how to divide a segment into n parts" to see how to do that.
As an interesting note, many beginning math students like decimals better but get confused as to why you can't use a decimal accurately and you need to use fractions. I take you that you understand that. If not, let us know.
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/01/2013 :  16:27:36  Show Profile
quote:
Originally posted by Szabo


My question is why are repeating decimals considered rational



Hi Szabo,

One thing that might help is to become familiar geometric series.
When |r| < 1, 1 + r + r^2 + r^3 + r^4 + r^5 + ... = 1/(1-r)
and a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ... = a/(1-r).

We actually want ar + ar^2 + ar^3 + ar^4 + ... = ar/(1-r).

We can use this to find the representation of a "repeating decimal" as the ratio of two integers. See if you can follow the steps in this example. Then try it yourself with some others repeating decimals, the process will be the same.

Example

0.45823232323232323232323232323232323232323232323...

= 0.458 + 0.000232323232323232323232323232323...

= 458/1000 + (1/1000)(0.2323232323232323232323232323...)

= 458/1000 + (1/1000)( 23(1/100) + 23(1/100)^2 + 23(1/100)^3 + 23(1/100)^4 + ... )

= 458/1000 + (1/1000)( 23(1/100) / (1-(1/100)) )

= 458/1000 + (1/1000)(23/99) <--- (we usually skip the above 2 steps and go directly to this)

= 9073/19800
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Szabo
Average Member

USA
8 Posts

Posted - 07/02/2013 :  00:39:53  Show Profile
First off, http://www.mathopenref.com/constdividesegment.html <--- that might be one of the coolest things I have ever seen. I am getting a geometry text book right now. Thank you.

Thank you for the explanations; they gave me a lot of clarity.

What is being explained to me is...
Any rational number with a repeating sequence of decimals is countable because it is known that the sequence repeats. (since the .333 goes on repeating infinity decimal places.) Alternatively,√2 is irrational because the proceeding decimal expansion (sorry if I am using the term decimal expansion incorrectly) dose not correlate to any constant pattern. Therefore, √2 is not countable because its exact position on a number line can not be plotted. Whereas, 1/3 can be plotted due to the predictability of the proceeding decimal.

Subsequently, I understand why, conceptually, it is countable as a decimal. The decimal repeats infinity places how could you yield a defined value, let alone a whole number from a never ending decimal who's values can never equate to 1 unless it is stated that it repeats.
.333333333....+.3333333333....+.333333333.... ≠ .999999999...
Geometrically, it makes perfect sense why 1/3 * 3 = 1 = 1/3 + 1/3 +1/3
Algebraically, it makes perfect sense why x = y(x/y) = xy/y = x.
But in reverse, it makes no sense arithmetically why the sum of 3 1/3s = 1. And .333333333....+.3333333333....+.333333333.... ≠ .99999999999999.
Does the expansion go on to infinity places or will it just go on to a limit that is the smallest possible value for the term. In other words, you can cut an orange exactly into 3 segments until the division of the orange itself will yield the term that would be the sum of 2 or more of the actual components that make up the orange itself.
~or is it just they way our number system works and can be seen as some kind of exception or special case.
--Eh, after typing this whole response out I think I get it conceptually now. lol. Correct me if I'm wrong here, 1/3 of a given value is terminated at the smallest possible part of that value.
[ Given a term that is a sum of values--> a pack of cigarettes is the sum of 20 cigarettes. 1/3 of 20 cigarettes is less than 7 and greater than 6 cigarettes. Although, if the values can be terms of smaller values you can be more exact. 1/3 of 20 cigarettes (which is = to 60 cigarette segments) would be 6.33 cigarettes (only .33 because we are assuming the smallest part of the value is 1/3 of a cigarette). All I can conclude from this it that, without defined values of each term in an equation the definition of any ratio of those terms is limited to the smallest possible division of the initial stated value (when decimals repeat). since eventually you will reach a point of division of the value that you yield something which is a component of the value, or the value becomes a term of an equation with a sum of the new (smallest)value. Finally, the number system that is in use can not represent, numerically, 1/3 (or any repeating decimal) in decimal form. Only through graphing can it be proven accurate that 1/3 times 3 equals 1 and not .999 repeating] Maybe I am just over thinking it T_T

-Ultraglide, this is math forum not an English forum >< thanks btw. for the explanation and ensuring I won't make that mistake again)

Edited by - Szabo on 07/02/2013 00:41:09
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/02/2013 :  03:25:16  Show Profile
Hi Szabo,

To address you statement that "Only through graphing can it be proven accurate that 1/3 times 3 equals 1 and not .999 repeating", let me ask you a few questions.


1) Do you agree that two numbers are equal if and only if their difference is 0? (a=b is equivalent to a-b=0)

2) Do you agree that if two numbers are unequal, then the absolute value of their difference is a number that is greater than zero?
If we use <> to represent the "is not equal to", do you agree that
a <> b is equivalent to |a-b| > 0?

3) What do we mean when we write the three little dots at the end of a string of digits?

4) Do you believe that 0.99999999999... < 1?
If you answered yes, what positive value is less than 1-0.999999... ?
After thinking about this for a little while, reread question 2.

Note that all nonzero rational numbers with terminating decimal expansions also have a nonterminating decimal expansion.
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Szabo
Average Member

USA
8 Posts

Posted - 07/02/2013 :  20:52:24  Show Profile
someguy-

if |a-b|=0 then |a| = |b|

"2) Do you agree that if two numbers are unequal, then the absolute value of their difference is a number that is greater than zero?
If we use <> to represent the "is not equal to", do you agree that
a <> b is equivalent to |a-b| > 0?"

If a ≠ b while |a-b| = x then x>0

--the dots after .9999.... indicate that it is repeating infinitely.
So yes .999...<1

So, what are you trying to tell me here that |.999... - .999...| = 0 ?

What I think I got out of all of this is that simply, there is no symbol to represent 1/3 in a decimal form.

---> 1/4 = .25---> .25 * 4 = 1
1/8 = .125---> .125 * 8 = 1
1/3 = .333(repeating)-----> .333(repeating) * 3 ≠ 1
.333(repeating) * 3 = .999(repeating)
Because .3(3) = .9
.3+.3+.3 = .9
.33(3) = .99
.33+.33+.33 = .99
.333(3) = .999
.333+.333+.333 = .999
So by this logic if you have any expansion of .33 that terminates or repeats the sum of that value .33(repeating or terminating) should always equal .99(repeating or terminating)
Why shouldn't that apply for repeating decimals?
Well, it should but it can't other wise the number system we are using would be inconsistent.
In conclusion, 1/3---> How many times does 3 go in to one? .333(repeating). Because .3 * 1.111(repeating ∞ decimal places)≠ .333(repeating). .333(repeating) = 1/3 .333repeating ≠ .33(repeating)
Basically there is no number that equals 1/3 so we just use the TERM .333 repeating for the VALUE 1/3.

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Ultraglide
Advanced Member

Canada
299 Posts

Posted - 07/02/2013 :  21:49:02  Show Profile
One last comment, 0.999999.... is not less than 1, it is equal to 1.
Two numbers are different if there is a number between them. If there is no other number between them, they are equal. I challenge you to find a number between 0.9999999.... and 1.
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Szabo
Average Member

USA
8 Posts

Posted - 07/02/2013 :  23:14:45  Show Profile
Ultraglide
you are 100% correct! .999(repeating)< 1
BUT when the number .333(repeating) is yielded from 1/3; saying its a number isn't really accurate. .333(repeating) is 1/3 because 1/3 * is 1 and not .999(repeating)
Think of the decimal .333repeating, times that by three logically you will yield .999 repeating.
What I am saying is that when you REPRESENT 1/3 as a decimal .333(repeating) isn't really 1/3 of 1 its just the closest thing there is. So 1/3 ≈ .333(repeating) really .333(repeating) and 1/3 are two DIFFERENT VALUES. Its confusing, 1/3 = .333(repeating) is like saying 1/3 = x
our decimal system fails to provide symbols (numbers) for the value of any rational expression that has a repeating pattern.

Edited by - Szabo on 07/03/2013 00:40:56
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royhaas
Moderator

USA
3059 Posts

Posted - 07/03/2013 :  08:43:27  Show Profile
The convention that we use to represent rational numbers by repeating decimals is to place a bar over the repeating group. Thus, bar(.3)=.33333... = 1/3 bar(.142857)=.142857142857... = 1/7, etc. Note that a function "bar" can be defined which implies that there is a way to represent rational numbers as repeating decimals, in other words, as the sum of a convergent geometric series.
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Szabo
Average Member

USA
8 Posts

Posted - 07/03/2013 :  15:47:28  Show Profile
royhaas-

"The convention that we use to represent rational numbers by repeating decimals is to place a bar over the repeating group."

Yes! But the repeating decimals that are REPRESENTING rational numbers really don't have any gravity as numbers because, the sum of parts should equal the whole and it never will with repeating decimals.
1/3 = 1/3
1/3 + 1/3 = 2/3
1/3 + 1/3 + 1/3 = 3/3
^^^^^Duh^^^^^^

1/3 = .333(repeating)
1/3 + 1/3 = .666(repeating)
1/3 + 1/3 + 1/3 = 1? <---- see where did that extra .0(repeating)1 come from?
It came from a lack of symbols to represent all rational numbers in decimal form.
(can you tell me what 1/3 of 1 dollar is? .33 cents? so whats 3 thirds of a dollar? .99 cents.... hey I want my penny back!)

[--slightly off topic-- every number after one is just a symbol to represent a sum of ones. 1 is a presence of a value 0 is an absence of value and -1 is a deficit of value.]

This whole topic seemed to spark some interest (lol at least on math goodies). I will type up a clear and detailed essay and would be happy to debate the issue on a new thread once posted.

-thanks for all the responses they really did help me understand this


Edited by - Szabo on 07/03/2013 17:23:33
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/03/2013 :  18:11:26  Show Profile
Hi Szabo,

What do you mean when you write .0(repeating)1? It is worth your time to think this through.


Anyway, you seem ok with calculations like

0.3333333333333333...
+.3333333333333333...
=.6666666666666666...

What are your thoughts on the following calculations.
Are they valid?


10*0.9999999999999...
=9.999999999999999...


9.999999999999999...
-.999999999999999...
____________________
9.000000000000000... = 9
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Szabo
Average Member

USA
8 Posts

Posted - 07/03/2013 :  20:15:25  Show Profile
quote:
Originally posted by someguy

Hi Szabo,

What do you mean when you write .0(repeating)1? It is worth your time to think this through.


Anyway, you seem ok with calculations like

0.3333333333333333...
+.3333333333333333...
=.6666666666666666...

What are your thoughts on the following calculations.
Are they valid?


10*0.9999999999999...
=9.999999999999999...


9.999999999999999...
-.999999999999999...
____________________
9.000000000000000... = 9



So in algebraic terms....
10(a)-a = 10a-a
In your example the term "a" is .333(repeating)
Since "a" is a repeating decimal you can't add it to any other term because you can't determine the numeric value.
10a-a when a=.9999
10(.9999)=9.999
9.999-.9999=8.9991

What I think was intended from that example:
10(.9999)=9.999
9.999-.999=9
It is the same arithmetic it should apply to any countable value shouldn't it?

the .0(repeating)1 is saying what you would have to add to get the true sum , 1.( from 1/3(3) ).
I want to go back to the dollar example. Here it is in a word problem.

Lee gives Lisa 1/3 of one dollar every hour. How much will Lisa make in 2 hours? How much will Lisa make in 3 hours? How much will Lisa make in 5 hours?
->1 hours = .33 cents
->2 hours = .66 cents
->3 hours = .99 cents
If they are rounded
->1 hours = .32 cents
->2 hours = .64 cents
->3 hours = $1.28
She would have to get paid every 3 hours to make 1 dollar, opposed to getting paid every hour where should would make .99 cents; in three hours.---> x = hours worked y = pay per hour a = xy
(x)(y)= a
x=2 y = .5
2(.5)=1 Because 2(.5)= .5 + .5 =1
x=3 y=1/3 of a dollar
3(.33) = .99 because 3(.33) = .33 + .33 + .33 = .99
This will only work through addition if you only use fractions to do the addition.
we have a term that is very very close to the value of 1/3 but we have nothing to represent it in decimal form it would be more accurate to just label it as 1/3 since 1/3 times 3/1 = 1 the rational number is the closest thing we have to a signal expression for the value 1/3. And, .333(repeating) the closest value we can write in a decimal form for the actual value.
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/03/2013 :  21:40:46  Show Profile
So you sometimes seem to accept the use of nonterminating decimal expansions and sometimes you don't.

It is good to question just what such an expression means and also to question the use of our standard algorithms for decimal arithmetic that we are taught in elementary school since they are meant to be used with finite strings of digits. Wouldn't it be nice if there was some way that we could still take advantage of them when dealing nonterminating decimal expansions .

We somehow have to make the leap to dealing with the infinite.

Lets continue your idea of working with finite strings of 3's and 9's.

0.3 + 0.3 + 0.3 = 0.9
0.33 + 0.33 + 0.33 = 0.99
0.333 + 0.333 + 0.333 = 0.999
0.3333 + 0.3333 + 0.3333 = 0.9999
0.33333 + 0.33333 + 0.33333 = 0.99999

As long as we only have a finite number of digits, our standard algorithms for addition work and we get the right answer. The problem is that none of these are equivalent to
0.333... + 0.333... + 0.333... = 0.999... .
Can we use what we know about terminating decimal expansions to make meaningful statements about nonterminating decimal expansions?


Before we answer this question, we need to ask just what those three little dots mean.
What does 0.333333333333... actually mean?
What does 0.999999999999.... mean?
Do these expressions represent actual values?
How should we interpret them?

You suggest earlier that 0.999999... = 1 - 0.000...001. I am assuming based on your argument at the time that when you said 0.0(repeating)1 that you meant place a 1 after an infinite number of 0's. Can we do this? Does 0.0(repeating)1 represent a valid number? (Did you think about this?)

If you think about what a decimal representation means, you will be led to the fact that
0.9999999999... means 9/10 + 9/100 + 9/1000 + 9/10000 + 9/100000 + ... .

Can we sum an infinite number of terms? Do such sums make sense?
What are your thoughts so far?

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Szabo
Average Member

USA
8 Posts

Posted - 07/04/2013 :  02:09:08  Show Profile
someguy-
"Can we sum an infinite number of terms? Do such sums make sense?
What are your thoughts so far?"
No we can't sum an infinite number of terms. such things don't make sense because repeating decimals are used as a value to represent rational number but do not equal the value they look like.
you can't get a number with an infinite any thing from a ratio of 2 numbers that terminate because then the ratio wouldn't be real. the same applies to things like √2 we use this term because with our number system there is no way to represent a value that when multiplied by it self = 2 (1 + 1 = 2 but wait 1 times 1 = 1 and 1 times 2 = 2) so instead of throwing the baby out with the bad bath water we simply represent the square root of 2 as √2 a value with its own symbol all together. Its just that repeating decimals are really crappy symbols to represent rational numbers that can't be calculated into an exact decimal.
In response to...
"Before we answer this question, we need to ask just what those three little dots mean.
What does 0.333333333333... actually mean?
What does 0.999999999999.... mean?
Do these expressions represent actual values?
How should we interpret them?"

.333333333... actually means .3repeating forever, never ending the repetition--->a pretend number--->actually means 1/3 when referring to thirds.
.999999999... actually means .9repeating---> a pretend number

The sum of .3repeating and any other repeating decimal should yield a number less than or greater than 1 and with a repeating decimal.

I use the term pretend. Well lets look at definitions before I start throwing words around.
Infinity-(symbol: ∞) refers to something without any limit...In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[3] (http://en.wikipedia.org/wiki/Infinity)
The research was hasty but still, I think it is accurate.

When x = an element in a set of rational numbers
ʃ(x)=x+1. Infinite answers still countable. Also, for every dependent there is only one independent

ʃ(x)=x(.3repeating) Infinite answers, not countable because .3(repeating) isn't a real number the .3 repeats to an infinitesimal decimal expansion; beyond the abilities of observation period.

My thoughts on this topic are, that saying .3(repeating) means .3(repeating) AND 1/3 at the same time is wrong.
Saying .3(repeating) times 3 = 1 would be implying that eventually in an infinite expansion you would reach a singularity. Because it is implying that .9(repeating) = 1. This is wrong because an infinite expansion of .3 would then be .4. The reason that we can state something like .3(repeating) times 2 = .6(repeating)is because we are stating it is a fact that the next decimal place will be a 3 and that the pattern will repeat forever so we can say that since we know it will always be another 3following, the repeating value plus it self equals 2(.x(repeating). this function changes for 3 because our number system fails to represent the value 1/3 as a decimal.

when x = .3

(.xRepeating)+ (.xRepeating) = (.yRepeating)
(.xRepeating)+ (.xRepeating)+ (.xrepeating) = y
(.xRepeating)+ (.xRepeating)+ (.xRepeating)+ (.xRepeating) = y.xRepeating

.3repeating and .3repeating = .6repeating not .7
.3repeating and .3repeating and .3repeating = 1 not .9repeating?

Again, unless someone can prove that .9repeating (or any similar instance)= 1 then is is irrefutable that our number system is flawed in that it fails to have an actual decimal value for every rational number. Only those with terminating decimals represent the true value in decimal form.


Edited by - Szabo on 07/04/2013 02:47:50
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/05/2013 :  06:56:31  Show Profile
Hi Szabo,

Nobody is saying that
1/3 = 0.33333333... = 0.4 = 4/10
or that
1/3 + 1/3 = 0.33333... + 0.33333... = 0.7 = 7/10
except for you.

We are saying that 0.333333... is just another representation for 1/3
and that 0.999999... is just another representation for 1.
Also, 8.99999999999... is just another representation for 9.
Does the fact that decimal representations are not unique trouble you?

You claim that these are not valid representations of numbers and call them pretend numbers. What makes something a valid number?

You have provided no reason to justify your claim that our number system is flawed. You have yet to actually put forth a reason to call it into question.
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royhaas
Moderator

USA
3059 Posts

Posted - 07/08/2013 :  09:14:41  Show Profile
A rational number like 1/3 is the limit point of the sequence
.3,.33,.333,... . Using the notion of a geometric sequence, it is easy to see that any rational number can be represented as the limit of a geometric sequence. The very nature of a geometric sequence is that not all of them terminate. I suggest that those who do not understand this try to understand the nature of a limit.
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