0.99999~ = 1 ?

I've seen this thing come along a couple of times on the internet, and was curious what our local math guys think about this.

It is my understanding that 0.9999~ does indeed equal 1.

I could be wrong though.

So, any thoughts on this?

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My image is of agony, my servants rape the land
Obsequious and arrogant, clandestine and vain
Two thousand years of misery, of torture in my name
Hypocrisy made paramount, paranoia the law
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-Lyrics by Lemmy Kilmister of Motorhead

It is my understanding that 0.9999~ does indeed equal 1.

Yes, indeed it does... unless you happen to be a constructivist, that is!

cavediver writes: Huntard writes: It is my understanding that 0.9999~ does indeed equal 1. Yes, indeed it does... unless you happen to be a constructivist, that is!

Shouldn't there be some sort of conditional involved or something? I don't see how it could be considered accurate to say that 0.99999~ is precisely equivalent to 1.

I appreciate that, for all intents and purposes, they are equivalent. But mathematically they simply are not, right?

Maybe I'm missing something.

Yes, you are correct. 0.9999..... = 1. Equals exactly.
Although it looks confusing, it can be broken apart. It's easy to see what 0.99 means. It's 9/10 + 9/100, which is clearly not 1. However when you see an expression like 0.9999...., this is the infinite sum:
9/10 + 9/100 + 9/1000 + 9/10000 + .....................
So the real question is, what are infinite sums?

I can add more detail if people want, but under the conventional definition given to an infinite sum, then this infinite sum equals one.

Here is a proof:
1. One of the axioms of real numbers: Between any two distinct real numbers there is a third real number not equal to either. (The density property)

2. Adding two numbers and dividing by two, results in the number half way between them. By (1), if they are distinct, this should be a new number.

3. 0.999999..... + 1 = 1.999999999.....

4. (1.99999.....)/(2) = 0.99999....., you can check this with long division.

5. Hence, the number halfway between 1 and 0.9999.... is 0.99999....

6. By (2) this is in contradiction with (1), hence

7. 1 and 0.999.... are not distinct and are the same number.

QED

What this actually means is that there are two different ways to represent the real number 1.

Briterican writes: Shouldn't there be some sort of conditional involved or something? I don't see how it could be considered accurate to say that 0.99999~ is precisely equivalent to 1. I appreciate that, for all intents and purposes, they are equivalent. But mathematically they simply are not, right? Maybe I'm missing something.

I guess you are then, because they are.

If you don't think they are equal, try answering this question: "What is the difference between 1 and 0.9999~." If they are not equal, there should be a difference, no?

Also try this explanation

X = 0.9999~
10X = 9.9999~
10X - X = 9
9X = 9
X = 1

---

I hunt for the truth

I am the one Orgasmatron, the outstretched grasping hand
My image is of agony, my servants rape the land
Obsequious and arrogant, clandestine and vain
Two thousand years of misery, of torture in my name
Hypocrisy made paramount, paranoia the law
My name is called religion, sadistic, sacred whore.
-Lyrics by Lemmy Kilmister of Motorhead

x = 0.99...
10x = 9.99..
9x=9
x=1

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1/3 = 0.333~
2/3 = 0.666~
3/3 = 0.999~

No, mathematically they're precisely equivalent - they are, in fact, different ways of writing the same thing. What's more every terminating decimal has two ways of being written, i.e.

0.5214 = 0.521399999~
7.2 = 7.199999~

And so on.

Yeah I get it now.

My brain doesn't want to cooperate, but thanks to the specific examples given by you guys, I understand now.

Well... I am 99.99999~% sure that I understand anyway

I am intoxicated right now by an psychoactive phenethylamine but i will attempt to respond. English is not my first language so bare with me.

0.9999~ will always equal 1 in calculations. Think of the infinity - hotel analogy.

I removed a lot of babbling from this post, seeing as i am too intoxicated to do a proper reply right now, ill will post more on these following days.

2C-I?

Hi Briterican,

I'm with Catholic Scientist on the simple approach, I just look at it a slightly different way:

(1/3)x3 == 1

1/3 = 0.3 + 0.1/3
    = 0.3 + 0.03 + 0.001/3
    = 0.3 + 0.03 + 0.003 + ... + 3x10^-n + 1x10^-n/3
    = 0.33333333~ and there is always a remainder of 1 to divide by 3 wherever you stop.

Now multiply that by 3

(1/3)x3 = (0.33333333~)x3 = 0.99999999~ == 1

This is really a product of using base 10, not anything magic with the math.

If we used base 9 then 1/3 would be 0.3:

  0.3
3\1.0 
  1.0
    0

However 1/2 would be 0.44444~

  0.444 ...
2\1.000
  0.8
  0.10
  0.08
  0.010
  0.008
  0.001
etc.

My personal favorite repeating decimal fraction is 7ths:

1/7 = 0.142857142857~ (where the underlined portion repeats)
2/7 = 0.285714285714~
3/7 = 0.428571428571~
4/7 = 0.571428571428~
5/7 = 0.714285714285~
6/7 = 0.857142857142~

Now 1/7 + 6/7 = 0.999999~ == 1
and 2/7 + 5/7 = 0.999999~ == 1
and 3/7 + 4/7 = 0.999999~ == 1

... now if we used base 7 ...

Enjoy.

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Son Goku's explanation is the one that deals best with an actual mathematical understanding of what is going on.

When I got on to my degree course, one of the first things we were taught was a new definition of equality. It's not the same as Son Goku's point, but it is very closely related.

That is, two numbers a and b are equal if there is no number, e such that
0 < e <|a - b|

(Which is the same as saying that there is no number between them, if you think about it).

In fact we needed this definition to deal with the sums of infinite sequences. And if we couldn't do that, integration wouldn't work (and we'd need to find a physical solution to Zeno's paradox, too !) So it is important to mathematics that 1 and 0.999... are the same number. If they were different we'd have real problems.

Edited by PaulK, : Corrected after prompting from Dr. Adequate.

Hi RAZD

As always, nice presentation.

RAZD writes: My personal favorite repeating decimal fraction is 7ths

These sorts of things are fascinating. Even the simple graphic representation you provided using a yellow font to show the repeating pattern is fascinating. (My favourite example of this kind of thing is the Mandlebrot Set, which has nothing to do with 1=0.999~ but is an example of visualised mathematics).

Even wiki has a large page regarding .999999~

I wonder what other sort of mathematics concepts I'm underinformed on hehe.

PaulK writes: That is, two numbers a and b are equal if there is no number, e such that |a - b| < e (Which is the same as saying that there is no number between them, if you think about it).

This illuminates the matter even further.

I thank you, and Huntard for the OP as well.