Proof that 0.999r=1

Alright I've always been baffled by this...

0.999(recurring)=1

Here's some proof I've collected.

Algebra Proof
In this proof, X will equal 0.999. When a number in decimal notation is multiplied by 10, the digits in that number will not change. Instead, the decimal separator (.) moves one place to the right. So 10*0.999=9.999.

X=0.999
10X=9.999
9X=9.999-0.999=9
X=1
0.999=1

The above proof using algebra already is proof that 0.999 can resemble 1.

Infinite series and sequences
This is something one of my class mates at collage showed me.

A sequence (x0, x1, x2, …) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:


thanks to wikipedia for this image, it was hard finding the right symbols

The last step — that is often justified by the axiom that the real numbers have the Archimedean property. The 1895 Arithmetic for Schools textbook says,

Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1.


If anyone knows anymore proof please share them on the thread.

 

Theres a tiny difference but .999 repeating doesn't equal one.

 

The limit defintion, which I don't completely understand, makes a lot more sense mathematically than the digit manipulation. First method is pretty sketchy to me

 

Could you please go further in your explanation of method 1? I don't think you can use that situation on what you're trying to prove, or the relations between the x's that you've used.
I do not know the limitation method you have used, but the difference between 0.999 and 1 is 0.001. The only time 0.999 will equal 1 is when a question in a text will ask the student to round 0.999, or any other variation, to the nearest tenth, hundredth, or thousandth.

 

Yes, it does. They represent the exact same thing.

I remember a teacher explaining it a few years ago, I'll try to show you here.

1/3 = .333... right?
and 1/3 + 1/3 + 1/3 = 1
since 1/3 = .333...
it stands that .333... + .333... + .333... = 1
if you simplify the left half of that equation
you get .999... = 1


There are many more ways to demonstrate this though, that's just one of the simplest to understand imo.

 

He just proved that 0.99999999 is equal to 1, fool.

 

On a calculator it'll round it up. Your point? .333+.333+.333=.999

.333 repeating + .333 repeating + .333 repeating on a calculator rounds up to 1 on a calculator since its so small. its called programming, there's always a flaw in it. the (long) type cant go forever.. it has to end somewhere.

you cant cross the equal sign when proving something either.

 

Yall just confused the hell out of me.

 

We're not talking about calculators and rounding
We're talking about mathematical proof that 0.9 recurring is 1

 

When it's 0.333..., it means 1/3, not just typing 3 into your TI-84 fifty thousand times.

 

He was relating the fact that calculators round off 0.333 and turn it into a fraction, and was responding to DropKickMurphys 'proof' that 1/3 multiplied by three is the same as 0.333 multiplied by three.

 

^ That's the best way to put it.

 

He's explaining that 0.999999 (RECURRING) = 1.
the 9999 in 0.99999 goes on forever.

Yes 0.999999999999r does equal 1 exactly. There isn't a small difference. it's the same.

 

I'm pretty amazed that I'll be the first to say this, but, who the fuck cares?

 

I tried to read it, it looks like .999 to me. haha
Suppose you proved it nice, i failed maths unfortunately... should have tried harder, calculator did it all for me.. well guess it didnt since i failed it :L

Limits look familiar though, graduated last year Nice work haha

 

/Sigh

He made a point to mention that he is using 0.999 recurring,

As for the math, I can't see any problem with it, except that rather than proving 0.999=1, it probably proves recurring numbers are inaccurately portrayed in current mathematics (or maybe that infinity is, in fact a number, so moving the decimal point to the left actually changes the number of 9s after the decimal point from infinity to infinity minus one. I'm not going to try it, but maybe that would make x=0.999). Anyways, the point is that working with things that we don't fully understand, such as infinity, can't deliver conclusive evidence, mostly that which disputes the very core of math. Next thing you know, someone will prove right angles are not congruent by using string theory.

 

X=0.999
10X=9.999

This isn't true... 10X should equal 9.99

 

Ok, what i don't understand is how you get 1 to equal .999r

 

0.999... means that it is 0.9999999999999999999999999999999999999999999999999999999999999999999999999999 etc
so when you multiply by 10, you are just shifting the decimal space

 

x = .999
10x = 9.99
9x = 8.991
x = .999
.999 = .999