Is Dividing 1 by 3 an Invalid Problem?

[tedima]

Why is it that every week or so we get someone who claims that 1 cannot be divided by 3?

1 can be divided by 3. The result is 1/3. It's not my fault that the decimal expansion for that is infinite repeating.

i said in theory ;p

 

[Drakkith]

There is no end, 1 can not actually be divided by 3 in theory. Infinity is not a number... but yet u divide 1 by 3 and you get an infinite amount of 3's.

Sure you can. You don't get infinity as your answer, you get 0.333... with the 3's repeating forever. It is merely a consequence of our number system that we get a value with an infinite number of digits. For example, if we didn't use decimals at all, but always used fractions, we get an easy 1/3.

 

[Drakkith]

i said in theory ;p

And how does that change anything? Even in theory 1 can be divided by 3. It isn't theory, it is reality.

 

[tedima]

We arent talking about 0.9999. We are talking about 0.9999...
The 3 decimals means that it repeats. Forever. If you started writing out the number, you would NEVER get finished.

I know what your saying. For 0.9999... to become 1 its got to be rounded up to 1. 0.9999... is the closest you can get to 1 but it isn't 1.

 

[tedima]

Sure you can. You don't get infinity as your answer, you get 0.333... with the 3's repeating forever. It is merely a consequence of our number system that we get a value with an infinite number of digits. For example, if we didn't use decimals at all, but always used fractions, we get an easy 1/3.

good point.

 

[Drakkith]

I know what your saying. For 0.9999... to become 1 its got to be rounded up to 1. 0.9999... is the closest you can get to 1 but it isn't 1.

No, that is incorrect. 0.999... IS equal to 1. There is no rounding involved.

 

[tedima]

No, that is incorrect. 0.999... IS equal to 1. There is no rounding involved.

why is it equal to 1?

 

[Hurkyl]

0.9999 is smaller than 1 if i put 99p into my bank account it wouldn't show as £1 on the machine

0.9999 only has a 9 in 4 decimal places. 0.9999... has a 9 in every decimal place.

 

[Hurkyl]

why is it equal to 1?

Because that's the way we defined the real number system, and the use of decimal numerals to represent real numbers.

If we defined things in a way that 0.999... and 1 denoted different numbers, the resulting number system wouldn't be very useful. You're free to try and study such number systems if you like -- but none of them will be the number system you learned about in school.

 

[tedima]

0.9999 only has a 9 in 4 decimal places. 0.9999... has a 9 in every decimal place.

put 0.9999... in ur bank account you will see 0.9999... on the machine

 

[Char. Limit]

put 0.9999... in ur bank account ull se 0.9999... on the machine

Good luck "putting in" a number with an infinite amount of decimals into a bank account, an entity that usually doesn't go beyond the second decimal place.

Fact of the matter is, your "bank account" analogy doesn't apply here.

Also, it's spelled "your".

 

[Hurkyl]

put 0.9999... in ur bank account ull se 0.9999... on the machine

You're right. If I deposit $1, I'll see my balance increase by $1.

 

[tedima]

Because that's the way we defined the real number system, and the use of decimal numerals to represent real numbers.

If we defined things in a way that 0.999... and 1 denoted different numbers, the resulting number system wouldn't be very useful. You're free to try and study such number systems if you like -- but none of them will be the number system you learned about in school.

So it was chosen 0.999... was equal to 1.

 

[tedima]

Good luck "putting in" a number with an infinite amount of decimals into a bank account, an entity that usually doesn't go beyond the second decimal place.

Fact of the matter is, your "bank account" analogy doesn't apply here.

Also, it's spelled "your".

Lol thanks for the correction.

 

[Drakkith]

put 0.9999... in ur bank account ull se 0.9999... on the machine

Irrelevant. Your example has nothing to do with what we are talking about.

 

[Hurkyl]

So it was chosen 0.999... was equal to 1.

Effectively yes.

FYI, historically, we used the real numbers long before we ever had the idea to represent them with decimals. I'm not sure what the original idea for decimals was, but I would imagine the value of a decimal was originally meant to be an infinite sum. And the corresponding infinite sum that computes the value of 0.999... is a geometric series whose sum is 1.

 

[Drakkith]

So it was chosen 0.999... was equal to 1.

It was not chosen that specifically 0.999... should equal 1. It is simply the way our number system is set up that makes it that way.

 

[tedima]

Irrelevant. Your example has nothing to do with what we are talking about.

Yeah it kinda moved off topic

 

[tedima]

Effectively yes.

FYI, historically, we used the real numbers long before we ever had the idea to represent them with decimals. I'm not sure what the original idea for decimals was, but I would imagine the value of a decimal was originally meant to be an infinite sum. And the corresponding infinite sum that computes the value of 0.999... is a geometric series whose sum is 1.

That explains everything. Thanks.

 

[Mensanator]

When you divide 1 by 3, you get .33333... repeating forever of course. My question is whether this operation could ever be considered to end. It looks to me like it's an invalid problem since you could never get a final answer, but simply keeping adding threes to the end of it when you try to solve. Does this make any sense?

No, it makes no sense. You are confusing a number with its representation. Try using base 3.

 

[disregardthat]

It was not chosen that specifically 0.999... should equal 1. It is simply the way our number system is set up that makes it that way.

We decided that 0.9999... = 1 when we decided that 0.999.. makes sense in the particular way it does today.

 

[Oriako]

Just read this:
http://en.wikipedia.org/wiki/0.999...

 

[Caramon]

1 = 0.999...

Proof:

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

:)

 

[Char. Limit]

1 = 0.999...

Proof:

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

:)

fify

 

[HallsofIvy]

No, it wasn't "chosen". 0.9999... means .9+ .09+ .009+ .0009+ .00009+ ...= .9(1+ .01+ .001+ .0001+ .00001+ ...)

That last is a "geometric series" which is taught in any good "precalculus" or "algebra II" class.
a(1+ r+ r^2+ r^3+ r^4+ ...) has sum a/(1- r) as long as |r|< 1.

Here, a= 0.9 and r= .1. a/(1- r)= .9/(1- .1)= .9/.9= 1.

 

[disregardthat]

No, it wasn't "chosen". 0.9999... means .9+ .09+ .009+ .0009+ .00009+ ...= .9(1+ .01+ .001+ .0001+ .00001+ ...)

That last is a "geometric series" which is taught in any good "precalculus" or "algebra II" class.
a(1+ r+ r^2+ r^3+ r^4+ ...) has sum a/(1- r) as long as |r|< 1.

Here, a= 0.9 and r= .1. a/(1- r)= .9/(1- .1)= .9/.9= 1.

The point is that it was chosen to mean exactly that, not that we chose to do our calculations correctly...