# 1 Divided by 3

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

Staff Emeritus
2022 Award
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.

Staff Emeritus
2022 Award
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
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.

Staff Emeritus
2022 Award
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?

Staff Emeritus
Gold Member
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.

Staff Emeritus
Gold Member
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

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Gold Member
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".

Staff Emeritus
Gold Member
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.

Staff Emeritus
2022 Award
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.

Staff Emeritus
Gold Member
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.

Staff Emeritus
2022 Award
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.

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.

Caramon
1 = 0.999...

Proof:

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

:)

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Gold Member
1 = 0.999...

Proof:

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

:)

fify

Homework Helper
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.