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Which one?

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jigokutamago said:
If d = infinitely small number
x = 0.999... = 1 - d
x = 1 - d
10x = 10 - d
10x = 9 + 1 - d
10x = 9 + x
9x = 9
x = 1

This makes sense right? An infinitely small number multiplied by 10 is still an infinitely small number.

Not really I think. If 10d = d then dividing by d gives you 10 = 1. My guess is that the problem takes advantage of this fact.

If 10d = d, then d must equal 0. If d = 0, you can't divide by d. Or, in other words, your d must equal 0. It's the decimal notation that has a problem, nothing else.



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Zkuq said:
jigokutamago said:
If d = infinitely small number
x = 0.999... = 1 - d
x = 1 - d
10x = 10 - d
10x = 9 + 1 - d
10x = 9 + x
9x = 9
x = 1

This makes sense right? An infinitely small number multiplied by 10 is still an infinitely small number.

Not really I think. If 10d = d then dividing by d gives you 10 = 1. My guess is that the problem takes advantage of this fact.

If 10d = d, then d must equal 0. If d = 0, you can't divide by d. Or, in other words, your d must equal 0. It's the decimal notation that has a problem, nothing else.

I see. My thought was that the problem takes advantage of the fact of assuming an infinitely small number to be equal to zero. Then obviously

d = infinitely small number = 0
0.999... = 1 - d = 1 - 0 = 1

but can we assume infinitely small number = 0 ?



When you try to undertand what's happening on this thread but your math knowledge is sub par



                                                                                     

JWeinCom said:
mjk45 said:

my answer to that is  yes a 1/3 and 2/3 = one whole but .333 recuring added to .666 recuring doesn't simply because it ignores the recuring part.

But... 1/3 is equal to .333... and 2/3 equals .666... they're interchangeable. They're different ways to denote the same amount.  

Replace the fractions in the equation with their decimal equivelants.

.333333333 (1/3)+ .666666666 (2/3)= .99999999 (1)

The thing is that you're not really dealing with infinity.  .33333333... is a specific finite amount that is equal to exactly one third. The numbers may go on and on in theory but the quantity they represent is specific amount.  It just so happens that in a base ten system, with the way division works, certain ratios (fractions) can only be represented in decimal form using repeating decimals.  The number itself is not infinite, the language we use to describe it is. It's just a quirk of our number system.  

For example, if we used a different number system, base 3 for example, we wouldn't have the problem of repeating numbers.  .333333333333... (1/3)in base 3 would simply be .1.  And in base 3, .66666666666... (2/3) would simply be .2.  If you added those two numbers, it would come out to a nice even 1.  And if 1/3+2/3=1 in a base 3 system, it has to also equal that in a base 10 system.  

(in base 3 .1+.2=.3  But, in a base 3, the only digits are one and two.  When you get to 3, you carry it over a place, like when you get to ten in base 10.  So, .3 would become 1.  1 in base 3 is the same as 1 in base 10)

Dunno if that helps, but the point is there is nothing inherently repeating about numbers like 1/3 or 2/3.  They don't *have* to be repeating numbers, it just so happens that due to the limitations of working with the particular number system we use, we can't describe these numbers in decimals without them repeating.

Thank you very much , I really appreciate your explainations, I suppose most of my thinking isn't so much about the result , especially since you thoughtfully  explained it to me , more to do with how it has been put forward in some quarter's,  where it is seen in a very black and white way , that being it's a mathmatical fact end of story , while you point out that the reason has as much to do with us dealing with a systems limitation, something that i feel pokoko was alluding to in his replies.



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mjk45 said:
JWeinCom said:

But... 1/3 is equal to .333... and 2/3 equals .666... they're interchangeable. They're different ways to denote the same amount.  

Replace the fractions in the equation with their decimal equivelants.

.333333333 (1/3)+ .666666666 (2/3)= .99999999 (1)

The thing is that you're not really dealing with infinity.  .33333333... is a specific finite amount that is equal to exactly one third. The numbers may go on and on in theory but the quantity they represent is specific amount.  It just so happens that in a base ten system, with the way division works, certain ratios (fractions) can only be represented in decimal form using repeating decimals.  The number itself is not infinite, the language we use to describe it is. It's just a quirk of our number system.  

For example, if we used a different number system, base 3 for example, we wouldn't have the problem of repeating numbers.  .333333333333... (1/3)in base 3 would simply be .1.  And in base 3, .66666666666... (2/3) would simply be .2.  If you added those two numbers, it would come out to a nice even 1.  And if 1/3+2/3=1 in a base 3 system, it has to also equal that in a base 10 system.  

(in base 3 .1+.2=.3  But, in a base 3, the only digits are one and two.  When you get to 3, you carry it over a place, like when you get to ten in base 10.  So, .3 would become 1.  1 in base 3 is the same as 1 in base 10)

Dunno if that helps, but the point is there is nothing inherently repeating about numbers like 1/3 or 2/3.  They don't *have* to be repeating numbers, it just so happens that due to the limitations of working with the particular number system we use, we can't describe these numbers in decimals without them repeating.

Thank you very much , I really appreciate your explainations, I suppose most of my thinking isn't so much about the result , especially since you thoughtfully  explained it to me , more to do with how it has been put forward in some quarter's,  where it is seen in a very black and white way , that being it's a mathmatical fact end of story , while you point out that the reason has as much to do with us dealing with a systems limitation, something that i feel pokoko was alluding to in his replies.

Glad I could help.  When I first heard this I was firmly in the "nuh-uh" camp.  It actually didn't click to me until I started studying postmodernism in college which is largely focussed on the inherent flaws in language.



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JWeinCom said:
mjk45 said:

my answer to that is  yes a 1/3 and 2/3 = one whole but .333 recuring added to .666 recuring doesn't simply because it ignores the recuring part.

But... 1/3 is equal to .333... and 2/3 equals .666... they're interchangeable. They're different ways to denote the same amount.  

Replace the fractions in the equation with their decimal equivelants.

.333333333 (1/3)+ .666666666 (2/3)= .99999999 (1)

The thing is that you're not really dealing with infinity.  .33333333... is a specific finite amount that is equal to exactly one third. The numbers may go on and on in theory but the quantity they represent is specific amount.  It just so happens that in a base ten system, with the way division works, certain ratios (fractions) can only be represented in decimal form using repeating decimals.  The number itself is not infinite, the language we use to describe it is. It's just a quirk of our number system.  

For example, if we used a different number system, base 3 for example, we wouldn't have the problem of repeating numbers.  .333333333333... (1/3)in base 3 would simply be .1.  And in base 3, .66666666666... (2/3) would simply be .2.  If you added those two numbers, it would come out to a nice even 1.  And if 1/3+2/3=1 in a base 3 system, it has to also equal that in a base 10 system.  

(in base 3 .1+.2=.3  But, in a base 3, the only digits are one and two.  When you get to 3, you carry it over a place, like when you get to ten in base 10.  So, .3 would become 1.  1 in base 3 is the same as 1 in base 10)

Dunno if that helps, but the point is there is nothing inherently repeating about numbers like 1/3 or 2/3.  They don't *have* to be repeating numbers, it just so happens that due to the limitations of working with the particular number system we use, we can't describe these numbers in decimals without them repeating.

It's not just a flaw with base 10 though. Base 3 also has fractions that can only be represented by infinitely recurring decimals. The only thing that changes when you change the base is what fractions.

If there is a system that has a flaw, then it's the decimal system itself. However I like to think of it as just another way of writing the same thing. 2/2 (As a fraction), 10^0, 1, 0.999.... All just different ways to write the value (not the number) 1. The recurring 9 notation works for all finite sequences of numbers as well (ie 0.25=0.24999....)

It's like a synonym in language. It's just that people are a bit more stuck up when it comes to math.

It may be a flaw in the system, but the flaw isn't that this should be incorrect, because it is correct. The flaw is that notation like this is ever used/ever has to be used. However when you denounce something as "simply the result of a flaw in the system", people are often quick to think "aha, so it's not actually correct", so you have to be careful when proclaiming things as the result of a flaw in the system.



As an aside how many people here know calculus or some elementary real anaylsis ?



Teeqoz said:
JWeinCom said:

But... 1/3 is equal to .333... and 2/3 equals .666... they're interchangeable. They're different ways to denote the same amount.  

Replace the fractions in the equation with their decimal equivelants.

.333333333 (1/3)+ .666666666 (2/3)= .99999999 (1)

The thing is that you're not really dealing with infinity.  .33333333... is a specific finite amount that is equal to exactly one third. The numbers may go on and on in theory but the quantity they represent is specific amount.  It just so happens that in a base ten system, with the way division works, certain ratios (fractions) can only be represented in decimal form using repeating decimals.  The number itself is not infinite, the language we use to describe it is. It's just a quirk of our number system.  

For example, if we used a different number system, base 3 for example, we wouldn't have the problem of repeating numbers.  .333333333333... (1/3)in base 3 would simply be .1.  And in base 3, .66666666666... (2/3) would simply be .2.  If you added those two numbers, it would come out to a nice even 1.  And if 1/3+2/3=1 in a base 3 system, it has to also equal that in a base 10 system.  

(in base 3 .1+.2=.3  But, in a base 3, the only digits are one and two.  When you get to 3, you carry it over a place, like when you get to ten in base 10.  So, .3 would become 1.  1 in base 3 is the same as 1 in base 10)

Dunno if that helps, but the point is there is nothing inherently repeating about numbers like 1/3 or 2/3.  They don't *have* to be repeating numbers, it just so happens that due to the limitations of working with the particular number system we use, we can't describe these numbers in decimals without them repeating.

It's not just a flaw with base 10 though. Base 3 also has fractions that can only be represented by infinitely recurring decimals. The only thing that changes when you change the base is what fractions.

If there is a system that has a flaw, then it's the decimal system itself. However I like to think of it as just another way of writing the same thing. 2/2 (As a fraction), 10^0, 1, 0.999.... All just different ways to write the value (not the number) 1. The recurring 9 notation works for all finite sequences of numbers as well (ie 0.25=0.24999....)

It's like a synonym in language. It's just that people are a bit more stuck up when it comes to math.

It may be a flaw in the system, but the flaw isn't that this should be incorrect, because it is correct. The flaw is that notation like this is ever used/ever has to be used. However when you denounce something as "simply the result of a flaw in the system", people are often quick to think "aha, so it's not actually correct", so you have to be careful when proclaiming things as the result of a flaw in the system.

I don't believe I said that the system is flawed.  I said it is limited which I'll stand by.  That's not to say other systems are not limited (I chose to use base 3 instead of base 2 because 1/3 and 2/3 still repeat there), but this particular example is one that exists in base 10 and not in other systems.



Yup it sounds weird at first but it makes totally sense. What the OP is showing is just undeniable from a mathematical perspective.



For those saying this really doesn't have a good application/is useless. I found that the concept of a convergent infinite series clicked when this was explained to me when I first learned it in high school, and then again when I learned it in my Calc II class.  So at the very least it has a good use to provide intuition when one learns about series, which are very applicable and useful in the application of higher mathematics (I do series expansions quite often in my physics studies.) When I tutor this particular topic I always start with this series, because it really motivates the students I am tutoring.