Does 0.9999999... = 1?

[Fluffy]

Edit: To avoid confusion please note that I am not talking about "0." with a large number of 9's following it but an infinite amount of 9's following it.

Out of curiosity and because of a poll on another forum with the same question, do you believe that 0.9 recurring (that is to say 0.9 with an infinite number of 9s after 0.9) is exactly equal to 1?

The result on the other forum surprised me so I wish to see if it was an anomaly.

The two most common approaches for proving 0.9^=1 ("^" indicates an infinitely recurring decimal)

1/3 = 0.3^
3 X 1/3 = 3 X 0.3^
3/3 = 0.9^

3/3 = 1

Therefore 0.9^ = 1

x = 0.9^ (Stating the definition of x)
100x = 99.9^ (Multiplying both sides by 100)
100x - x = 99.9^ - 0.9^ (Subtracting x from both sides)
99x = 99 (Simplifying the previous step)
x = 1 (Dividing both sides by 99)

Therefore, 1 = 0.9^ (Inserting the value of x back into the original definition)

Infinite Convergent Geometric Series: a/(1-r) = a + ar + ar*2 + ar*3...

Let a = 0.9 and r = 0.1.

Therefore, first term = 0.9, second term = 0.09, third term = 0.009. Extended into infinity and totalled, the series equals 0.9^.

Therefore a/(1-r) = 0.9^

Input a and r into left hand side
0.9/(1 - 0.1) = 0.9^
1 = 0.9^

 

[Djamila]

It might as well be, but no - I don't believe it would ever be exactly equal to one. 0.99999 ... to infinity means exactly that.

 

[nutshell]

No.

If it did, then the algebra text book would say 0.999999.... = 1

It seems insignificant, but if we take a trip to the far reaches of space to meet some aliens, we'll miss them by a huge margin.

 

[nutshell]

Yes. My dad studies mathematical theory for fun. No, seriously. And he explained it to me like this.
What is 1 divided by 9? .111111 repeating
What is 2 divided by 9? .222222 repeating
What is 5 divided by 9? .555555 repeating?
What is 9 divided by 9? Well, logic says a number divided by itself is 1, but the previous examples show that anything divided by 9 is a repeated number. Therefore, the two answers are one and the same. 9 divided by 9 is .999999 repeating, or in other words, 1.

Well, I don't think I'd want your dad planning my trip to the far reaches of space then.

 

[Ðanisty]

Yes. My dad studies mathematical theory for fun. No, seriously. And he explained it to me like this.
What is 1 divided by 9? .111111 repeating
What is 2 divided by 9? .222222 repeating
What is 5 divided by 9? .555555 repeating?
What is 9 divided by 9? Well, logic says a number divided by itself is 1, but the previous examples show that anything divided by 9 is a repeated number. Therefore, the two answers are one and the same. 9 divided by 9 is .999999 repeating, or in other words, 1.

What is 18 divided by 9? 2
What is 27 divided by 9? 3
What is 36 divided by 9? 4
Anything divided by 9 is not a repeated number.

 

[ch'ang]

Yes. My dad studies mathematical theory for fun. No, seriously. And he explained it to me like this.
What is 1 divided by 9? .111111 repeating
What is 2 divided by 9? .222222 repeating
What is 5 divided by 9? .555555 repeating?
What is 9 divided by 9? Well, logic says a number divided by itself is 1, but the previous examples show that anything divided by 9 is a repeated number. Therefore, the two answers are one and the same. 9 divided by 9 is .999999 repeating, or in other words, 1.

Patterns prove nothing they merely suggest somthing, 9 divided into 9 equal parts is 1 not .99999......

Okay, explain this to me then. You take one object, just one, and divide it into three, equal sections. In reality, you know the three are equal, but on paper, 1 divided by 3 is .3 repeating. If you add .3 reapeating plus .3 repeating plus .3 repeating, you'll get .9 repeating, not 1. Yet, in reality, when you add all three of the pieces back together, you'll get a whole again.

This problem only arises when you use decimals, which are normaly considered an estimation.

 

[MaddLlama]

No. 1=1. End of story.

If some advertisement said "this product is 99.9% effective against germs", would we say that it's failure rate is 0? No, we would say it's failure rate it .01. Which is still a number, a very small one, but still a real number.

 

[Tiberius]

Have you guys even looked at the proofs in the first post?

Have a look at the Wiki article as well. Specifically this part of the article, explaining why many people find it hard to comprehend that one number can be written in two different ways.

0.999 repeating IS equal to one.

 

[Ody]

In all practical purposes yes, in concrete fact, no.

 

[nutshell]

For those who think 0.9999.... is 1, do you think 1.999999999..... is 2?

 

[Ody]

For those who think 0.9999.... is 1, do you think 1.999999999..... is 2?

If we are doing equations for basic level problems.. I don't see the major problem this would create...

 

[YmirGF]

In all practical purposes yes, in concrete fact, no.

I tend to sit with Alan/Ody and Jmoum on this one. For all intents and purposes, in the real world, .99999999999999999999 etc... DOES = 1 because you cannot split an object into an infinite amount of parts. It is simply not possible. In theoretical terms, we have another story. In theoretical tems, an infinite number can be multiplied by an infinite number simply by ascribing variables. It has been a very long time since I have given it much thought but there is a difference between "pure" mathmatics and applied mathmatics.... although I could just be showing my ignorance. The last time I checked, the above holds true.

 

[SoyLeche]

Well, I don't think I'd want your dad planning my trip to the far reaches of space then.

You may have forgotten the "infinite" part of the discussion. If we were ever to stop the 9's, yeah, we'd miss them by a lot. As long as the 9's keep going, we're going to hit them dead on.

 

[Kungfuzed]

What if you multiplied 0.9999999.... by itself an infinite number of times? Would it turn out to be 0 or 1?

 

[jonny]

I'm trying to dig back into Calculus, but doesn't .999999 approach 1?

 

[SoyLeche]

I'm trying to dig back into Calculus, but doesn't .999999 approach 1?

Yeah, it does. You'll never find anything closer to 1 (except, of course, 1)

 

[Mike182]

the difference between 0.99999999^ and 1 is 0.000000000^

 

[Ryan2065]

Hrm... I voted no before I read the exact question, I thought this was a rounding question.

So... To non math people... Why do you care? Then again why would math people care? I don't know of any mathematicians who use decimals instead of fractions... Its one of those fun facts that hardly ever is used in the real world or in math.

 

[sahra-t]

Oh dearie me, the result of the poll is shocking!!

My favourite proof is the second one in the OP. If people aren't going to accept flawless, logical algebra, there's really nowhere else we can go with this. *shrugs*

 

[Willamena]

I say no, they are not equal, simply because 0.999... means something different than 1. It literally means "a number approaching 1 that can never quite get there."