# 0.9999 = 1, true or false and why?

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#### Anaxagoras

##### Banned
y= 0.9999999
10y=9.9999999
9y=10y - y
9y=9.9999999 - 0.9999999
9y=9
y=1
1=0.9999999

Where did I go wrong?

#### Leonsito

##### Member
Anaxagoras said:
y= 0.9999999
10y=9.9999999
9y=10y - y
9y=9.9999999 - 0.9999999
9y=9
y=1
1=0.9999999

Where did I go wrong?

There, I think.

Edit: Wait, 10y is wrong, it's 9.999999, giving 9y = 8.9999991.

10y

#### flsh

##### Banned
You have one 9 too many on that 9.999..
Though 0.9999... (to infinity) does equal 1.

#### StopMakingSense

##### Member
Anaxagoras said:
y= 0.9999999
10y=9.9999999
9y=10y - y
9y=9.9999999 - 0.9999999
9y=9
y=1
1=0.9999999

Where did I go wrong?

0.9999999 * 10 is not 9.9999999

#### Ferrio

##### Banned
Anaxagoras said:
Where did I go wrong?

You suck at multiplying.

flsh said:
Though 0.9999... (to infinity) does equal 1.

edit: bah I'm probably wrong.

#### StuBurns

##### Banned
flsh said:
You have one 9 too many on that 9.999..
Though 0.9999... (to infinity) does equal 1.
This.

Although the OP said it strangely, I ususally see it like this:

1/3=0.333*
0.333*x3=0.999* so...

1=0.999*

#### BananaBomb

##### Member
:lol :lol :lol

edit: to the doubters, it can be proved dozens of different ways that .999* = 1

nerds

#### Parl

##### Member
StopMakingSense said:
0.9999999 * 10 is not 9.9999999
The notation is just wrong. But if these figures tend to infinity, then my understanding is that 0.99999... = 1

#### Earthstrike

##### Member
0.9999... is exactly equal to one.
You went wrong nowhere. This is just a consequence of the definition of a repeating decimal as setup by mathematics. A lot of people tend to have a knee jerk reaction to reject for some reason.

1/3 = .333...
2/3 = .6666...

There's several easy ways to rationalize this.
The first is to consider the difference between 0.999... and 1.
You would have 0.000... with a hypothetical 1 as the inifitieth digit. Consequently, you would have a number that is infinitely small. If you have a number that's infinitely small, then it must be the smallest number (absolute value of course). a la zero.

Suffice it to say though, the math will show through.

#### RevoDS

##### Junior Member
StuBurns said:
This.

Although the OP said it strangely, I ususally see it like this:

1/3=0.333*
0.333*x0.333*=0.999* so...

1=0.999*

I believe you meant to say 0.333* x 3 = 0.999*. Because 0.333* x 0.333* = 0.110889.

As for the OP's question, it all depends on the point of view. If you see things technically, then no, 0.999999 does not = 1 because no matter what you do with it, how many 9s there are, there is still an infinitely small difference which keeps shrinking as you add 9s but never completely goes away. In practical use however, 0.999999 = 1.

The fraction argument is a fallacy, because the conversion of 1/3 to decimals relies on a small amount of rounding which introduces a margin of error big enough to make up for the actual difference between both numbers.

#### tokkun

##### Member

Here's another one:

1/9 = .1111111111111111...
2/9 = .2222222222222222...
3/9 = .3333333333333333...
4/9 = .4444444444444444...
...
8/9 = .8888888888888888...
9/9 = 1

#### Mike M

##### Nick N
Ferrio said:
You suck at multiplying.

flsh said:
Though 0.9999... (to infinity) does equal 1.

No it doesn't.

1/3 = .333333... to infinity
3 X 1/3 = .999999... to infinity

#### Kipz

##### massive bear, tiny salmon
Because .9999... is an imaginary number that is infinitely close to 1 and therefore for all intents and purposes is 1?

#### Coolio McAwesome

##### Member
72/9 = 8.0
73/9 = 8.111111
74/9 = 8.222222
75/9 = 8.333333
76/9 = 8.444444
77/9 = 8.555555
78/9 = 8.666666
79/9 = 8.777777
80/9 = 8.888888

Following this same pattern, the next answer should naturally be 8.999999. However:

81/ 9 = 9.0

#### StuBurns

##### Banned
RevoDS said:
I believe you meant to say 0.333* x 3 = 0.999*. Because 0.333* x 0.333* = 0.110889.
I did indeed, I edited before you posted, but you caught me. Moronic mistake.

#### BananaBomb

##### Member
Kipz said:
Because .9999... is an imaginary number that is infinitely close to 1 and therefore for all intents and purposes is 1?

no it's a real number that is equal to one.

#### tokkun

##### Member
Kipz said:
Because .9999... is an imaginary number that is infinitely close to 1 and therefore for all intents and purposes is 1?

The term 'imaginary number' means something in math.

.999.... is not an imaginary number.

#### Defcon

##### Banned
Seriously, people.

#### bachikarn

##### Member
RevoDS said:
As for the OP's question, it all depends on the point of view. If you see things technically, then no, 0.999999 does not = 1. In practical use however, 0.999999 = 1.

This is wrong (no offense). .9999... = 1 in theory and in "practice."

The easiest way to think about it is that numbers (real numbers specifically for this problem) are abstract quantities. We represent them with "numerals". So when we write "5," we are writing down a representation of the abstract quantity that is the real number 5. It turns out because of how our number system was set up, the numeral ".999...." and the numeral "1" represent the same real number. Hence, we say .9999... = 1.

Edit: PS, there is no fallacy in the fraction argument.

#### SmokeMaxX

##### Member
Haven't seen one of these threads on here in a while. They occurred so often in a short period of time and then suddenly stopped. I thought they were bannable at some point in time.

Anyway, 0.9999 repeating is 1. Numbers are weird, but that's just how it is. You can rationalize it in many ways or you can just acknowledge it as truth. The latter is easier. While there are many ways to prove that 0.99999... = 1, there is no way to confirm the opposite.

If you had a hat with an infinite amount of slips of paper consisting of an infinite amount of numbers, the probability of pulling any random number is zero (1/infinity = 0). In fact, the probability of pulling any real subset of numbers is zero (i.e. 10000/infinity = 0). However, if you actually reach down in there and pull out a number, the probability obviously isn't zero because you just pulled out a number. Why is that?

I had my Cal professor explain the answer to this one for me, but I totally forgot what he said haha.

#### Cruceh

##### Banned
It's 1. Learned this in HS math. Something about limits or wahtever. Forgot everything already.

#### Evlar

##### Banned
0.9999999 != 1 (does not equal)

0.9999... = 1 (the ellipsis means the previous series continues perpetually)

#### Earthstrike

##### Member
RevoDS said:
I believe you meant to say 0.333* x 3 = 0.999*. Because 0.333* x 0.333* = 0.110889.

As for the OP's question, it all depends on the point of view. If you see things technically, then no, 0.999999 does not = 1 because no matter what you do with it, how many 9s there are, there is still an infinitely small difference which keeps shrinking as you add 9s but never completely goes away. In practical use however, 0.999999 = 1.

The fraction argument is a fallacy, because the conversion of 1/3 to decimals relies on a small amount of rounding which introduces a margin of error big enough to make up for the actual difference between both numbers.

There's not rounding in the fractional representation of 1/3. Also, answers to mathematical questions do not depend on "points of view".

Plain and simple mathematics is the evaluation of the set of consequences and relations amongst a given set of definitions. By the definitions in our number system 0.99999 repeating is exactly equal to one.

You're argument contains a fallacy however because you say "there is still an inifinitely small difference that never goes away". Actually it HAS to go away by definition.

Proof: Lets say there is some infinitely small difference that is greater than zero. Divide that by 10. It is now smaller, meaning it wasn't an infinitely small difference. The only infinitely small difference is zero.

#### Mr. Spinnington

##### Banned
sorry, math nerds

#### Door2Dawn

##### Banned
Math is the devil.

#### Tntnnbltn

##### Member
RevoDS said:
As for the OP's question, it all depends on the point of view. If you see things technically, then no, 0.999999 does not = 1 because no matter what you do with it, how many 9s there are, there is still an infinitely small difference which keeps shrinking as you add 9s but never completely goes away. In practical use however, 0.999999 = 1.
No, it doens't depend on your point of view. They are different representations of the same number. Technically, practically, and in every way.

"4/4" looks different to "1", but people easily accept that these are different representations of an identical number. The exact same deal applies to 0.999... = 1

#### RevoDS

##### Junior Member
Earthstrike said:
There's not rounding in the fractional representation of 1/3. Also, answers to mathematical questions do not depend on "points of view".

Plain and simple mathematics is the evaluation of the set of consequences and relations amongst a given set of definitions. By the definitions in our number system 0.99999 repeating is exactly equal to one.

You're argument contains a fallacy however because you say "there is still an inifinitely small difference that never goes away". Actually it HAS to go away by definition.

Proof: Lets say there is some infinitely small difference that is greater than zero. Divide that by 10. It is now smaller, meaning it wasn't an infinitely small difference. The only infinitely small difference is zero.

By "infinitely small", I meant that it was equal to 1x10^-n, where n is the amount of 9s after the 0. As the 0.9999.... grows ad infinitum towards 1 without ever actually reaching it, the difference between 1 and that number shrinks at the same rate.

There IS rounding, for the very same reason there is a 0.0000....1 difference between 0.9999....9 and 1

In practice, we simply assume 0.999999 = 1 for convenience and because the 0.000001 difference is mathematically insignificant. But it is still there no matter what, thus 0.999999 and 1 are not the same number.

Assuming 0.999... is 1 is the very same thing as assuming 99 is 100, only on a smaller scale.

99/100 =9.9/10
9.9/10=0.99/1

#### Evlar

##### Banned
RevoDS said:
By "infinitely small", I meant that it was equal to 1x10^-n, where n is the amount of 9s after the 0. As the 0.9999.... grows ad infinitum towards 1 without ever actually reaching it, the difference between 1 and that number shrinks at the same rate.

There IS rounding, for the very same reason there is a 0.0000....1 difference between 0.9999....9 and 1

In practice, we simply assume 0.999999 = 1 for convenience and because the 0.000001 difference is mathematically insignificant. But it is still there no matter what, thus 0.999999 and 1 are not the same number.
You are correct that 0.9999...9 does not equal 1. However, we aren't talking about 0.9999...9.

Or, to put it another way,

0.9999...9 != 1
0.9999...9 != 0.9999...
RevoDS said:
Assuming 0.999... is 1 is the very same thing as assuming 99 is 100, only on a smaller scale.

99/100 =9.9/10
9.9/10=0.99/1
Specious reasoning. You are offering one pair in a convergence series. The correct observation is that the difference between the 0.9...9 series and 1 approaches 0 as the series is extended.

#### Enkidu

##### Member
RevoDS said:
There IS rounding, for the very same reason there is a 0.0000....1 difference between 0.9999....9 and 1
This magical 0.000...1 number doesn't exist. You can't have an infinite amount of 0's and then add a 1 at the end, because if you can add a 1 at the end then you obviously could have added more 0's instead hence it wasn't an infinite amount. Therefore, as you say, the difference between 0.999... and 1 is 0.000... which is simply another way of writing 0.

#### Tntnnbltn

##### Member
RevoDS said:
By "infinitely small", I meant that it was equal to 1x10^-n, where n is the amount of 9s after the 0. As the 0.9999.... grows ad infinitum towards 1 without ever actually reaching it, the difference between 1 and that number shrinks at the same rate.

There IS rounding, for the very same reason there is a 0.0000....1 difference between 0.9999....9 and 1

In practice, we simply assume 0.999999 = 1 for convenience and because the 0.000001 difference is mathematically insignificant. But it is still there no matter what, thus 0.999999 and 1 are not the same number.
But there is no 0.000000.........1 difference (for the proper 1-0.999...= argument). No matter how far you go, that tailing one doesn't exist. And because there is no difference between the numbers, they are the same.

The entire mathematic world agrees that they are equal to each other, and not just due to "mathematical insignificance" like you claim.

Assuming 0.999... is 1 is the very same thing as assuming 99 is 100, only on a smaller scale.

99/100 =9.9/10
9.9/10=0.99/1
No it doesn't, because the second it stops being an infinite series it stops being equal.

#### ratcliffja

##### Member
tokkun said:

Here's another one:

1/9 = .1111111111111111...
2/9 = .2222222222222222...
3/9 = .3333333333333333...
4/9 = .4444444444444444...
...
8/9 = .8888888888888888...
9/9 = 1

This is the one I always use. It seems weird, but it's actually true. The entire concept of repeating decimals is to express a ratio or fraction in decimal form. Repeating numbers can always be expressed as a number out of a series of 9s. For instance, .123123123... = 123/999.

#### Earthstrike

##### Member
RevoDS said:
Assuming 0.999... is 1 is the very same thing as assuming 99 is 100, only on a smaller scale.

You do not make assumptions in math. We're not assuming 0.999... is equal to one. We have proven it with nothing more than addition, subtraction, division, multiplication, and algebraic substitution.

Math is not felt out or worked out by analogy to thought processes. You have the rules defined within the system and the definitions of the system. With these you can demonstrate the 0.999... = 1, ergo, 0.999... is exactly equal to one.

no imo

#### LCGeek

##### formerly sane
Marko Rodin......... :lol :lol :lol :lol

You can also use 7 and number grouping through division to create sequences of 9 derivatives like 9/99/999 and so forth. I prefer 84 and 12 in general to the usage of nine.

#### Ashes

##### Member
Earthstrike said:
You do not make assumptions in math. We're not assuming 0.999... is equal to one. We have proven it with nothing more than addition, subtraction, division, multiplication, and algebraic substitution.

Math is not felt out or worked out by analogy to thought processes. You have the rules defined within the system and the definitions of the system. With these you can demonstrate the 0.999... = 1, ergo, 0.999... is exactly equal to one.

Excellent argument against a false analogy. Although I would argue against using the word exact. It is 'theoretically equal' within 'our' number system.

#### Zaptruder

##### Banned
Ok. Accepting that 0.9recurring = 1, then how to you represent the number that is smaller than 1 by the smallest margin?

#### ccbfan

##### Member
I think I never got about the proof is that

.3333333333333333333333333~ * 3 = .99999999999999999999999~

Is there any proof that says you can even multiply this sort of number like real numbers.

its similar to trying to say

3! * 3 = 9!

#### BananaBomb

##### Member
Zaptruder said:
Ok. Accepting that 0.9recurring = 1, then how to you represent the number that is smaller than 1 by the smallest margin?

What does the smallest margin even mean? There isn't such a thing. How do you represent the number one less than infinity?

And to the guy above me, unless I am in grievous error, 3! and 9! are both real numbers...

#### Stat Flow

##### He gonna cry in the car
If you had 99 cents and what you were trying to buy was a dollar, you wouldn't be getting shit.

Nah, but most people have already said it, 0.999... = 1, I prefer to go with the 1/3 method of proving it.

#### Evlar

##### Banned
Zaptruder said:
Ok. Accepting that 0.9recurring = 1, then how to you represent the number that is smaller than 1 by the smallest margin?
The "smallest margin" is 0. In the real number system there is no number that is the "smallest possible size but larger than 0". Number systems that include such numbers (called infinitesimals) include hyperreal numbers; however, for analysis purposes these are considered non-standard.

#### Slavik81

##### Member
Earthstrike said:
You do not make assumptions in math.
Wikipedia said:
Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. (source)
There are assumptions made in math, but they are fundamental assumptions.

Zaptruder said:
Ok. Accepting that 0.9recurring = 1, then how to you represent the number that is smaller than 1 by the smallest margin?
I'd guess:
y = 1 - &#949;

I think I never got about the proof is that

.3333333333333333333333333~ * 3 = .99999999999999999999999~

Is there any proof that says you can even multiply this sort of number like real numbers.

its similar to trying to say

3! * 3 = 9!
Code:
`````` 12.8
+ 6.1
-----
18.9

The sum of each column can be taken to find
the total sum.

0.333333...
+ 0.333333...
+ 0.333333...
-------------
0.999999...``````

#### Socreges

##### Banned #### BananaBomb

##### Member
Since I feel like there is more controversy in this thread than is acceptable, let's just seal the issue with a simple (and mathematically sound) proof.

Sum of a geometric series with a = 9 and r = 1/10 is 1. End of story.

#### Tntnnbltn

##### Member
ccbfan said:
I think I never got about the proof is that

.3333333333333333333333333~ * 3 = .99999999999999999999999~

Is there any proof that says you can even multiply this sort of number like real numbers.
There probably is, but I'm not a mathmetician.

If it's just the multiplication bit you don't like, try... Otherwise, there are a whole bunch of the proofs at http://en.wikipedia.org/wiki/0.999...#Proofs don't use decimal representation of a fractions at all.

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