Is Dividing 1 by 3 an Invalid Problem?

  • Context: High School 
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Discussion Overview

The discussion revolves around the mathematical operation of dividing 1 by 3, specifically questioning whether this operation can be considered valid given that the result is a repeating decimal (0.33333...). Participants explore the implications of this representation and the nature of division itself.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that dividing 1 by 3 results in an infinite repeating decimal, which raises the question of whether the operation can ever be considered to end.
  • Others argue that the notation 0.333... represents a completed process and is equivalent to 1/3, thus implying that the division is valid and finished.
  • A participant proposes examining the division in a base 3 number system, where 1/3 equals 0.1, suggesting that different number systems can represent rational numbers differently.
  • There is a discussion about the nature of division itself, with some participants clarifying that division is an inverse operation to multiplication and questioning what it means for division to "end."
  • Some participants highlight the distinction between the representation of numbers and the actual numbers themselves, emphasizing that 0.333... is a representation of 1/3 rather than a process that continues indefinitely.
  • Concerns are raised about division by zero, with participants noting that it is fundamentally different from dividing by a non-zero number and cannot be defined meaningfully in standard arithmetic.

Areas of Agreement / Disagreement

Participants express differing views on whether the division operation can be considered valid or if it is inherently flawed due to the nature of the repeating decimal. There is no consensus on the implications of this discussion, as multiple competing perspectives remain.

Contextual Notes

The discussion touches on the limitations of representing certain rational numbers in decimal form and the implications of infinite series and limits in mathematics. Participants also note that the understanding of division and number representation can vary based on the mathematical framework being used.

  • #31
Tedima, there is no end to an infinitely long number.
The number 0.999... REPRESENTS a value that is equal to the number 1.
 
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  • #32
Drakkith said:
Tedima, there is no end to an infinitely long number.
The number 0.999... REPRESENTS a value that is equal to the number 1.

0.9999 is smaller than 1 if i put 99p into my bank account it wouldn't show as £1 on the machine
 
  • #33
Mark44 said:
Where exactly is the end of an infinite number of 0's?

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.
 
  • #34
tedima said:
0.9999 is smaller than 1 if i put 99p into my bank account it wouldn't show as £1 on the machine

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.
 
  • #35
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.
 
  • #36
Char. Limit said:
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
 
  • #37
tedima said:
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.
 
  • #38
tedima said:
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.
 
  • #39
Drakkith said:
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.
 
  • #40
Drakkith said:
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.
 
  • #41
tedima said:
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.
 
  • #42
Drakkith said:
No, that is incorrect. 0.999... IS equal to 1. There is no rounding involved.

why is it equal to 1?
 
  • #43
tedima said:
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.
 
  • #44
tedima said:
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.
 
  • #45
Hurkyl said:
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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  • #46
tedima said:
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".
 
  • #47
tedima said:
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.
 
  • #48
Hurkyl said:
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.
 
  • #49
Char. Limit said:
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.
 
  • #50
tedima said:
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.
 
  • #51
tedima said:
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.
 
  • #52
tedima said:
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.
 
  • #53
Drakkith said:
Irrelevant. Your example has nothing to do with what we are talking about.

Yeah it kinda moved off topic
 
  • #54
Hurkyl said:
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.
 
  • #55
Drakkith said:
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.
 
  • #56
Drakkith said:
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.
 
  • #58
1 = 0.999...

Proof:

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

:)
 
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  • #59
caramon said:
1 = 0.999...

Proof:

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

:)

fify
 
  • #60
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.
 

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