Does 0.999... equal 1?

Hm, a friend told me something quite strange the other day, that the number 0.999... continued on for eternity equals to the number, 1.

Reasons

Take the fraction, ⅓ for instance.

⅓= 0.333...the decimal that keeps going on.
⅔= 0.666... the decimal that keeps going on.
3/3= But would it make sense for it to be 0.999... as 0.333 x 3 would equal to 0.999.... and that's 3/3 which equals to 1.

Is there any number between 0.999... and the number 1?

In other fractions, such as... x/9. Let x be the whole number.

Let's take a look at these numbers.

1/9= 0.111...
2/9= 0.222...
etc etc
8/9= 0.888...
So wouldn't it be reasonable for 9/9 = 0.999... which is 1, hence 9/9=1.

I can't get the symbol for the line above the number.


*I now believe that 0.9999999... Does NOT equal 1.*

 

Yeah, At first I thought no, but i remembered it from sooo long ago. Needed to refresh my memory.
http://www.mathsisfun.com/9recurring.html
That's how i learnt it.

 

Well, when you put it terms of fractions yes.

 

Srsly, stop googling. LOL. Eh, that's true too.

@ Billy, yes, in fractions. But fractions are numbers too, and the examples I gave are considered a whole which equals to one.

 

some fractions cannot be properly recreated in decimal form.

This flaw in our number system doesn't mean that things are what they are not.

The only reason you can "prove" .999... = 1 is because of the flaws in the way we do math.

 

Personally I don't think so, since its missing 0.001 to actually make it a whole number. But the theory that Adam has posted is obviously right. So, no argueing. It does equal 1.

 

The difference between them is so infinitely small that we treat it as nil.
That's one way they're equal - there is no difference between them.

But if you reject the infinitely small concept... then calculus makes very little sense.

 

Exactly right. .99 repeating gets infinitely close to 1, but will never actually reach it. Why? Well, at what point will you add an extra 9, and realize, "Hey, it's now equal to 1"? That's right, it will not happen.

 

You don't just subtract X randomly from regular numbers....


And fraction wise, it would seem that .99999....=1 but it really isn't. Simplify things and look at it as if you were in the 3rd grade. .9999 is one number, and 1 is a diffrent number therefore making them unequal to each other

 

As Jansen has said, there are some fractions which are impossible to recreate in decimal form. It's impossible to divide 10 into 3 equal parts using decimals while it is fairly simple to do it in fraction form.

Decimals are based on a power of 10 you could say, therefore it is impossible to ever have 3 equal numbers in decimal form equal to 1.

 

you are over thinking this. .999... will get infinatly closer to 1, but will never reach it just as dividing a number by 2 infinatly will never make it equal to 0.

 

No only 1 equals 1 0.9999 gets really really close but never will acctually be one. Thats like 8th grade math...

 

Now you need to understand that fractions and decimals are nothing alike.

1/4 isn't .4 or anything so 1/3 cant be .3333 etc.

The decimal .99999999999 , can carry on forever it will never increase into 1/1 .

Nothing can change that no matter how many 9's you add in there.

 

How can 1/3 NOT be .333? Seriously. Just seriously.

 

1/3 is not .333 >.> Some fractions cannot be defined by decimals.

 

What's 1 - .9999999...?

something so small we can call it 1/infinity. that's zero.

 

No, it's an infinitely small number, of no actual amount.

 

1/3*3/1=3/3
So...
.333...*3=1

Now, here's where things get nasty.
.333...*3=1
1=1
So...

Does .999...=1?
Hard to tell using this method, because .333...*3!=.999...
I don't know any other ways to present this.

Oh, and there are no flaws in our math system. Some things may sound stupid when you first hear them, but there are no real flaws.

 

Ye, seriously. That's just ridiculous. >_>

Eh, I see. There is that 0.0000000001 in between.

 

Last time I saw this the proof was different, it was like
Let x = .99 recurring
Therefore 10x = 9.99 recurring

so 10x - x = 9.99 recurring - .99 recurring
therefore 9x = 9
x=1

And since the original value of x was set to .99 recurring and it is now changed to 1 that was the proof. but um yeh