Does .999... equal 1?

does it?

>>38

READ THIS AND BE PERSUADED OF IT FOR IT IS THE GOSPEL TRUTH OF MATHEMATICS, AS FAR AS THE REAL NUMBERS ARE CONCERNED.

You wish to describe an infinitesimal number.  Call it 1/infinity, or 0.000...0001, or whatever else.  You wish to maintain that it is less than any other strictly positive real number, but more importantly, you wish to maintain that it is nonzero.  call it x.  Then the infinitesimal x is greater than 0.  You might like to say that it is a real number, or something else.  At any rate, if we show that it's merely equal to zero, we discard your claim.  Suppose I write 1/(2*infinity).  you would rightly claim that this is equal to your 1/infinity, while attempting to maintain that neither is equal to zero.  But what is (1/(2*infinity))?  It is just (1/2)(1/infinity).  And there's only one real value I know of that's half of itself, and that's zero.  This is not rigorous, but it is appropriate to your level of contemplation of the problem.  It is also legitimate, since we obeyed regular arithmetic throughout.

THERE IS NO SUCH THING AS A REAL NUMBER WHICH IS LESS THAN EVERY POSITIVE REAL, YET GREATER THAN ZERO.

IN EXACTLY THE SAME SENSE THAT THERE IS NO INFINITELY LARGE POSITIVE REAL NUMBER, SO TOO IS THERE NO INFINITELY SMALL POSITIVE REAL NUMBER.

Let's beat this into your head some more, with respect to the original question.

x = 0.9999... 
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

While not RIGOROUS, this proof works, since it's not like the fallacious 1=2 proof where one divides by zero (zing).  All that it does is to show that the thing we mean by .999... and by 1 have the same numerical value.  1/3 + 1/3 + 1/3 = .333... +.333... +.333... = .999... =1.  Precisely 1.  Nothing other than 1. 

INFINITESIMALS ARE RIGOROUSLY BANISHED FROM THE REAL NUMBER SYSTEM.

Some history!  Newton and Leibniz used infinitesimal quantities and came up with appropriate results.  The rigorous difficulty arose in the fact that they divided by a quantity which was pretended to be nonzero, and later discarded like it was zero.  It took the foundations of limits and functions much later to give calculus a firm(er) base.  Have a mathworld link while we're at it:  http://mathworld.wolfram.com/Infinitesimal.html
Infinitesimal: "A quantity which yields 0 after some limiting process"

'Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson.'-Wikipedia.  Moreover, they were necessary imaginitive tools in the initial development of the calculus.  The point of which you will now be convinced is this:  .999... = 1 and your infinitesimal turns out to be zero when you try to do anything with it as a number in and of itself.  If you want to comment on an non-archimedian field, you are free to read up on it and do so.  But your object has no place in the real number system.  .999... is in every possible sense, 1, and that's the truth.  The object you are referring to has no reality in the only number system on which you are prepared to comment on it.

Here ends the sermon.

>>38

You have no idea what you're talking about.  You're asserting with neither proof nor understanding.  And you're being willfully ignorant out of pure stubborness.  Mathematics at this basic, fundamental level never take into account "realistically" versus "theoretically".  Realistically and theoretically, .9... is identically equal to 1.  Just as .3... is not an approximation, but rather is exactly 1/3, .9... is not an approximation, but rather exactly 3/3.

>>38
Ever heard of Cauchy?

>>42
I agree, alternatively you can look at 0.9.... conceptually. However this brings up another dilemma.

1/3 is modelled as 0.3.... using the decimal system, but what if the number is not concerning 1/3. What properties would this number have?

What about numbers which cannot be simplified such as pi and phi?

>>44

Huh?

First of all, 1/3 isn't "modelled" as 0.3..., 1/3 *is* 0.3...: they're two different names for exactly the same thing.  I assume you understand that, though, and simply made a poor choice of words. 

I have no idea what you're talking about wrt "what if the number [what number?] is not concerning 1/3?"

And what does any of this have to do with irrational numbers?

1/3 is not 0.3... 

There are real numbers and rational numbers.  Rational numbers are embedded in real numbers.     

1/3 is rational, meaning "the ratio of 1 to 3."

It is represented as 0.3... when in decimal form.  Decimals are also called decimal fractions.  Decimals are base 10, and are a ratio based on a power of 10. 

1/3 is not based on a power of ten. 

0.3... can only be infinitely "close" to 1/3.    

GOD DAMN YOU ALL ARE FUCKING STUPID
WHY DOES THIS THREAD REAPPEAR EVERY FEW DAYS

.999~ = 1 EXACTLY

PROOF:

let x = .999~ (where ~ indicates repeating forever)
multiply each side by 10
10x = 9.999~
subtract x from each side
9x = 9 (because x = .999~, so we subtracted .999~ from 9.999~)
divide both sides by 9
x = 1
and we started by saying x = .999~, so...

.999~ = 1

NOW SHUT THE FUCK UP ABOUT IT

How would subtracting .999~ from 10 equal 9 and not 9.111~  ?

>>48
Math.
.9 repeating ~= .999 (because you and >>47 obviously like threes)
10 - .999 =  9.001
9.001 != 9.111

>>48
gah retard. 10-0.99... = 9.0000000....1

the one is never reached because there are infinite zeros

lol oops. i didn't read >>49

It still isn't = 1

Math is based on belief.  And faith.  Always remember that.  In God We Trust Math.

The proof isn't correct though.

This thread is threadstopped. You can't reply anymore.

0.0...1 + 0.999... = ...

0.0...1 is not a real number if you're implying some sort of "transfinite" number, and therefore doesn't play into the discussion at all.

I am going to kill you all.

Wow, third time I've been told that today.

>>58
I'm still taking notes.

1 = definately 0.999...
when you're in a store and you order 0.999...pound of meat and the  sailsman says "might it be a little bit more" you say "NO IT HAS TO BE 0.9999... OR I'LL CUT OFF YOUR BALLS" because you are an obsessive little bastard and you gotta have 0.999... AND NOT 1.

now does this story sound rite to you? i didnt think so...

HEY FUCKASS THAT'S NOT A VERY GOOD ANALOGY: YOU CAN'T SAY "HEY I WANT PI POUNDS OF MEAT" OR "I WANT THE SQUARE ROOT OF TEN POUNDS OF MEAT" EITHER

JESUS FUCKING CHRIST YOU COCKOOS

>>61

orly? an obsessive little bastard wouldnt think of that now would he?

It's SCIIIIEEEEEENNNNNCNCCCCCE!

.9999 is infinitely close to 1

.9999 is .0001 away from 1.  .9 repeating, .9..., is equal to 1.

0,999... =  1 / 1,000...1

>>52
>>53
>>64
>>65
>>66

You are all wrong.  Read the god damn proofs written earlier in the thread.  It's a mathematical proof using only algebra.  Infinities don't enter in to it.

.999... = 1

Game over.

>>67

I'm >>65.  Read my own god damn statement.  I said that .9... is equal to 1.  I've been saying it since I was >>6.

>>68

D'oh.  I fail at reading.

>>67 noob @ math

0.999... and 1 are two different names we use for the same concept

1/3 = 0.333...
1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333...
1/3 + 1/3 + 1/3 = 3/3 = 1
0.333... + 0.333... + 0.333... = 0.999...

ergo 1 = 0.999...

That's not a valid proof

>>73

which is not a problem, being as how this thread is filled with them.

Answer to the original post: yes

Yeah, I'm just sayin' that it ain't a valid proof, is all.

Yeah, and I'm just saying it's not a problem, thats all.

>>73>>75>>76 yes it is, you are retarded

>>77

I'm a math student =(

>>78
MATHS STUDENTS SHOULD SAY THEY ARE MATHS STUDENTS NOT MATH STUDENTS

>>79

I'M NOT PART OF THE COMMONWEALTH KTNX