0.999...=1 and why people believe it is false.

[Powman3]

The thing about tryto's proof is that she put x = .999... AND x = 1 in the second proof when it should only equal one number period.

 

 

 

The whole point is to try and prove that they are really equal. Although, I'm not buying it.

 

 

 

I know that, and I'm not buying it either.

[fastortoise]

Are people here really rejecting the entire use of limits in this thread? Come on guys.. Without simple calculus, we'd still be stuck in the stone age.

 

Come join our advanced society by reading http://en.wikibooks.org/wiki/Calculus/Limits

 

 

 

 

 

That wiki page explains the last part of this proof, which seems to be the part you people can't grasp. The other stuff is cal2 material, but anyone with a right mind can understand the general gist of it... I only used that complicated stuff in order to set up the last part of the proof, the limit part.

 

[Flyingjj]

The problem is not understanding limits, I do that perfectly fine, thank you very much. (I've taken Calc I, II, and III, as well as DiffEq). Limits say that a function approaches a number, which it may or may not reach. In this case, though .999... may approach 1, it never reaches it. The difference between the two becomes infinitesimally small in a limit, and the limit is therefore 1, but that does not mean .999...=1

 

 

 

However, in any practical mathematical sense, such difference is irrelevant, as shown by the limit.

[l0l0lpur34]

With proofs, you can basically make any number equal anything. At least from what I've seen.

 

no

[____]

Yea.... infinately closer to 1 but never actually equal to 1. The 'proofs' are always fascinating but will always be 0.0...1 short of being equal to 1.

[doomy]

Yea.... infinately closer to 1 but never actually equal to 1. The 'proofs' are always fascinating but will always be 0.0...1 short of being equal to 1.

 

How does this end up 0.999... short of 1?

 

 

 

1/3*3=0.333...*3

 

3/3=0.999...

 

1=0.9999...

 

 

 

Then you can say that 0.00....1 = 0 anyway, couldn't you?

[Jaziek]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

[compfreak847]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

Again, all of you are pretending infinity is a number which can be compared. Why is everyone ignoring my post and simply forming equations out of infinity after it was proven that it is completely incorrect and impossible?

 

 

 

What about this? Infinity plus any number equals infinity. Therefore:

 

 

 

Infinity + 7 = Infinity + 38 = Infinity

 

 

 

A true statement. Now, if we pretend infinity is a regular number and compare it mathmatically, we can subtract infinity from both sides using the addidion postulate:

 

 

 

Infinity - Infinity + 7 = Infinity - Infinity + 38 = Infinity - Infinity

 

 

 

And we end up with 7 = 38 = 0

 

 

 

Now, nowhere in math is there an algebraic expression that starts off correct and turns incorrect by treating both sides equally. It's a basic law of math. That leaves us with two choices:

 

 

 

#1: Algebra itself is incorrect, and our whole way of looking at numbers collapses

 

#2: Infinity is not a number, and cannot be treated as such. Therefore, ANY equation with infinity in it is completely wrong and proves nothing whatsoever.

 

 

 

You decide. If it's #1, we've got a lot more problems then this thread; if it's #2, all the amazing "proofs" on this thread are worthless as they all include infinity.

 

 

 

Sorry, but collage professor > teenagers on TIF :-#

[Kaida23]

The proof is interesting, but I think it's just another way of looking at it. Infinity, essentially a nonreal concept cannot be compared to real numbers on our decimal system; it can easily be said that .999... is infinity increasing but never actually reaching 1. It's the same concept as approaching the event horizon of a black hole; to observers, it goes infinity slowly and will never reach the actual horizon.

 

 

 

Basically, yes, it is increasing - but by an infinitely small amount. Your proof represents another way of looking at it, but it isn't accurate to treat infinity as a number when doing comparisons. Essentially, 1/9 and .999... are numbers that cannot be compared in the standard decimal system, as they represent a concept instead of an actual value.

 

Compfreak is right, it is not equal to 1. The size between them becomes infinitely smaller to a point where it is mathematically insignificant, but it will never become 1.

 

 

 

Infinity cannot be used for proper calculations, as it is not an actual number. It can be used for theoretical calculations, but you cannot simply treat it like a regular number.

[fastortoise]

You decide. If it's #1, we've got a lot more problems then this thread; if it's #2, all the amazing "proofs" on this thread are worthless as they all include infinity.

 

 

 

Sorry, but collage professor > teenagers on TIF :-#

 

If my proof included the infinity you speak of, the limit of the function I set up would be undefined since infinity is not approaching any number, that's how infinity works. However, that's not the case (why can't you see this?). Yes, the number 0.999... is increasing infinitely but it's converging to a number (1). Every part of that proof is proper math, have it peer reviewed by all the apparent geniuses that teach at your school.

 

 

 

And collage is spelled college, professor :roll:

[Faux]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

99.9999 / 10 = 9.99999 so this whole thing is wrong.

[Utopianflame]

Jaziek as you have a finite number of decimals the positions change when you divide by 10, as such your wrong.

 

 

 

Compfreak, infinity can not be used in equations as a number and that has been the case since GH Hardy. However infinity is perfect correct used in the context of a limit of a function as a variable tends to infinity.

 

 

 

 

 

 

While algebraic demonstrations of 0.9...=1 are invaraibly sloppy Fastortoise's proof is perfectly correct as are a number of analysis (which is what you would call the foundations of calculus really) based methods.

[Soma2035]

Rather than responding to any responses to my original post, I thought I would simply post a clearer, simpler logical proof as to why 0.999... is not 1.

 

 

 

I think everyone has heard of this theoretical situation.

 

 

 

A pie is baked and left on the counter. Each person who sees it takes half of what is left. Will the pie ever be taken?

 

 

 

This is a logic question that is asked fairly frequently, that most of you have heard of. However, this question is bound by one major constraint: how small can pie get? Simply put, pie can not be infinitely small. Eventually, you'll have a crumb. If you go farther, you'll have a speck of dust. If you go as far as to bring out micro technology, you could have an atom. Then what? You break the atom in half? Eventually, you'll have to stop.

 

 

 

However, let's rephrase this question to remove the size constraint.

 

 

 

You have a number. You repeatedly take away 90% of what remains. Will the number ever be entirely gone?

 

 

 

What you have taken away can be represented as f(x) = 1 - 0.1^x

 

What you have remaining can be represented as g(x) = 0.1^x

 

 

 

The size boundary is no longer an obstruction.

 

 

 

f(1) = 0.9

 

g(1) = 0.1

 

f(2) = 0.99

 

g(2) = 0.01

 

f(3) = 0.999

 

g(3) = 0.001

 

 

 

I think this is a proof as clear as day. The formula is an accurate and direct mathematical representation of the scenario, and the rules of the scenario represented clearly state that you must always take only 90%. Simply put, there must always be something remaining.

 

 

 

How many digits are in 0.9 repeating? Because this is how many times you have taken away 90% of the remaining value. Infinity is not a number, and thus, is only theoretical. 0.9 repeating does not actually exist.

 

 

 

This is why all your proofs work: you fail to account for the fact that 0.9 repeating does not exist as a real number. The limit of 0.9 repeating is certainly one. Look at the equations once more.

 

 

 

f(x) = 1 - 0.1^x

 

g(x) = 0.1^x

 

 

 

Look at the second equation, which is also the latter part of the first. Even if x is 10,000,000, or 100,000,000,000,000,000,000,000, or a number even bigger, g(x) is not 0. g(x) can never be 0. Let's translate it to a fractional problem:

 

 

 

g(x) = 1 / (10^x)

 

 

 

Guess what? The one on top is still a one.

 

 

 

HOWEVER

 

 

 

The value below, as x approaches infinity, also approaches infinity. Thus, the limit of g(x) as x approaches infinity is 0.

 

 

 

And therefore, the limit of f(x) as x approaches infinity, is 1.

 

 

 

And the limit of f(x), as x approaches infinity, can also be written as .9 repeating, as essentially, you are just adding more 9s.

 

 

 

.9 repeating is a limit. It is not a number. Saying .9 repeating = 1 is inaccurate, as 1 is a defined, real number whilst .9 repeating is not. No number with an infinite number of digits is "real". 1/3 is real, but in decimal form, it is a limit, as 1/3 can not accurately be represented in decimal form. You can either approximate, or treat it as a limit. Not pretend it's a real number with an infinite amount of digits, because nothing can have an infinite amount of digits.

[Soma2035]

Soma, your terminology is wrong as is your conclusion.

 

 

 

0.9... isnt an approximation of anything, it is a representation. The elipsis simply means that the preceding characters are repeated forever.

 

 

 

In particular 0.9... is a representation of the sum , (-1)^2 and -e^(i*pi), they are all exactly the same number. Just because they look different doesnt mean that they are.

 

 

 

Also Pi, and e are numbers, transcendant numbers, yes, but numbers nethertheless.

 

 

 

Taken straight from wikipedia:

 

 

 

 

 

 

Pi is not a number. Pi is a limit. As x approaches infinity, the value of the function approaches Pi. Pi is never actually reached, because infinity is never actually reached. The same is true of all irrational or transcendental numbers. In a mathematical sense, they are not numbers in the same way 5, 10, or 2.5 are.

 

 

 

Limits can be numbers. Not all limits are numbers. Any limit that can not be represented without the use of infinity is not a number, and the "..." afterwards is just that, a representation of infinity.

[Utopianflame]

Your classifications of what you consider to be a number are wrong.

 

 

 

Number sets starting with the smallest and moving from there, each set is a subset of the one that follows it.

 

 

 

N - The set of natural numbers

 

 

 

1,2,3,4,....

 

 

 

Z - The set of integers

 

 

 

...-3,-2,-1,0,1,2,3...

 

 

 

Q - The set of the rational numbers - that is any number that can be expressed as an element of Z divided by an element of N.

 

 

 

R - The set of reals, all non-complex numbers (this includes irrational numbers).

 

 

 

C - The set of complex numbers , any number that can be expressed as a+ib where a and b are elements of R.

 

 

 

Just because something isnt nice doesnt make it not a number. Pi is still a number, just because you cant write it down exactly on a sheet of paper doesnt make it non existant. Just because a number is the limit of a function doesnt mean it doesnt exist.

[Faux]

[hide=]Are people here really rejecting the entire use of limits in this thread? Come on guys.. Without simple calculus, we'd still be stuck in the stone age.

 

Come join our advanced society by reading http://en.wikibooks.org/wiki/Calculus/Limits

 

 

 

 

 

That wiki page explains the last part of this proof, which seems to be the part you people can't grasp. The other stuff is cal2 material, but anyone with a right mind can understand the general gist of it... I only used that complicated stuff in order to set up the last part of the proof, the limit part.

 

[/hide]

 

 

 

Wanna explain to me where you got the [s - (1/100)S]?

[Laura]

And collage is spelled college, professor :roll:

Well if we are going to be scrutinizing over grammatical errors, sentences generally end with a form of punctuation.

[Joes_So_Cool]

And collage is spelled college, professor :roll:

Well if we are going to be scrutinizing over grammatical errors, sentences generally end with a form of punctuation.

 

 

 

 

 

Headshot, professor.

[fastortoise]

[hide=]Are people here really rejecting the entire use of limits in this thread? Come on guys.. Without simple calculus, we'd still be stuck in the stone age.

 

Come join our advanced society by reading http://en.wikibooks.org/wiki/Calculus/Limits

 

 

 

 

 

That wiki page explains the last part of this proof, which seems to be the part you people can't grasp. The other stuff is cal2 material, but anyone with a right mind can understand the general gist of it... I only used that complicated stuff in order to set up the last part of the proof, the limit part.

 

[/hide]

 

 

 

Wanna explain to me where you got the [s - (1/100)S]?

 

Look at the two lines above it. The bottom is subtracted from the top, so the terms "99/100^2 + 99/100^3 + 99/100^4 + ... + 99/100^n" are all deleted, which gives the result you see.

 

 

 

And collage is spelled college, professor :roll:

Well if we are going to be scrutinizing over grammatical errors, sentences generally end with a form of punctuation.

 

There's a difference between backing up an entire argument on "collage professor > teenager" and not ending a sentence with a point. Especially when the former was wrong :thumbdown:

[Powman3]

With proofs, you can basically make any number equal anything. At least from what I've seen.

 

no

 

 

 

Would you like to prove me wrong, oh great sage?

 

 

 

edit: And fastor, there's a difference between math and grammar. People can be experts in one thing but not the other.

[Kiriyama]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

WRONG!

 

 

 

x= 9.9999, therefore 10x = 99.999

[Powman3]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

WRONG!

 

 

 

x= 9.9999, therefore 10x = 99.999

 

 

 

BOOM, headshot!

[Rebdragon]

This argument is the language arts equivalent of arguing the spelling of Djibouti.

 

 

 

I don't give a flying s**t if that silent 'd' looks off to you, YOURE WRONG.

[Jaziek]

Simplest proof of this I know is

 

 

 

10x = 99.9999

 

x = 99.9999 / 10

 

x = 9.9999

 

 

 

10x - x = 9x

 

 

 

therefore

 

 

 

9x = 99.9999 - 9.9999

 

= 90

 

 

 

9x = 90

 

 

 

therefore

 

 

 

x = 90/9

 

 

 

x = 10

 

 

 

10 = 9.9999

 

 

 

99.9999 / 10 = 9.99999 so this whole thing is wrong.

 

 

 

what? thats what I wrote.

 

 

 

and sworddude as well. I fail to see your problem. you're just repeating whats in my post.

[Rebdragon]

Jaziek, ignore all the [puncture]s nitpicking the fact that you don't have an ellipse after the nines.

 

 

 

Christ, I thought everyone here was supposed to be a nerd.