Math Is Fun Forum

ke

Member

Registered: 2013-04-08

Posts: 1

1 does not equal .999999.....

The proof submitted:
x=0.999...
10x=9.999...
10x-x=9.999...-0.999...
9x=9
x=1
In the 4th line of the proof we see that from the RHS: 9.999...-0.999... = 9. This is not the case. While it is true that the number of 9s after the decimal point is infinite, it is neither sufficient or trivial to assume this step. For example:
Take an infinitely long integer consisting exclusively of 9s, call this number A. Take second infinitely long integer also consisting exclusively of 9s and call this number B.
A - B = 0
If and only if the number of digits in A and B are the same. Since x has been multiplied by 10, its infinite number of 9s after the decimal point is one less than in x by definition. (sorry to jump in here...but how is one set of INFINITE 9's smaller than another set of INFINITE 9's...by definition, they are the same) It is not sufficient either to say that there are an infinite number of 9s after the decimal place therefore it does not matter, irrespective of whether the repetition is finite or not, the number of digits does need to be the same, and as Mathematicians know, not all infinities are the same. (I laughed. What a joke. Go read some Cantor theorems. infinity +1 is ALWAYS equal to infinity). Therefore the 4th line should say:
That is only true for a finite series.
Say you have two sets, A and B. If A = {0, 1, 2...} and B = {1, 2, 3...) (both infinity), does that mean A has more digits than B simply because it has a zero? No, both sets are equal.
9x = 9 - d
Where d is a infinitesimally small number which is strictly non-zero but limited by zero.
0 is an infinitesimal, and the only one to exist in the real number system
(X=.99999 and X= 1?) how does that work?
You say the number of digits in A and B would not be the same if you multiply x by 10. I'm not entirely sure here, but wouldn't saying that multiplying a number by 10 (or 100, 1000 etc.) moves the decimal place one to the right be a simplified way of expressing the final answer? Doesn't 10 * x = x + x + x + x + x + x + x + x + x + x or x added together 10 times? Therefore, if you add .999... to .999... 10 times, shouldn't you maintain the same number of decimal places? I feel like I'm probably wrong but I'm not seeing where.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,664

Re: 1 does not equal .999999.....

hi ke

Welcome to the forum.

Your new thread brings the number on this topic to 9.  Search on 0.99999 if you want to explore the many comments others have made.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: 1 does not equal .999999.....

Your new thread brings the number on this topic to 9.

Just as long as it is not 8.9999999... That would be intolerable.

That is only true for a finite series.

I like to think of it like this

.99999999... - .9999...

.9999... = .99999999... ( I just took a few measly nines out of those 3 dots) I call it 9 extraction from dots. After all there are gazillions of them in there and they will not be missed. Now it is easy to see

.99999999... - .99999999... = 0. You just have to remember ... - ... = 0. I call it m's law of the dots.

Using this great idea the proof you gave by multiplying by 10 works fine.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: 1 does not equal .999999.....

Have you fully explored whether m's law of the dots can be safely used after 9 extraction from dots?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: 1 does not equal .999999.....

Hi;

Good point, I realize it is not rigorous but if I get a grant from the NSF I will be willing to explore that and a lot more.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,007

Website

Re: 1 does not equal .999999.....

Have you fully explored whether m's law of the dots can be safely used after 9 extraction from dots?

What is this?

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: 1 does not equal .999999.....

Post #3, has the supposed theory.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Mathegocart

Member

Registered: 2012-04-29

Posts: 2,226

Re: 1 does not equal .999999.....

Your new thread brings the number on this topic to 9.

Just as long as it is not 8.9999999... That would be intolerable.

That is only true for a finite series.

I like to think of it like this

.99999999... - .9999...

.9999... = .99999999... ( I just took a few measly nines out of those 3 dots) I call it 9 extraction from dots. After all there are gazillions of them in there and they will not be missed. Now it is easy to see

.99999999... - .99999999... = 0. You just have to remember ... - ... = 0. I call it m's law of the dots.

Using this great idea the proof you gave by multiplying by 10 works fine.

True.

---

The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: 1 does not equal .999999.....

Hi;

I have had a change of heart and no longer think .999999... = 1. I think

.999999... =1.00000000...

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,664

Re: 1 does not equal .999999.....

Much better.   

Those extra zeros make it much more accurate!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: 1 does not equal .999999.....

hi ke

Welcome to the forum.

Your new thread brings the number on this topic to 9.  Search on 0.99999 if you want to explore the many comments others have made.

Bob

I would say 99.999... is the number.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,664

Re: 1 does not equal .999999.....

Well I used sound, thought-out search criteria.  I found 9 (or 8.99999... if you prefer)

I'm talking about threads not posts.  The latter may well be un-countable.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,007

Website

Re: 1 does not equal .999999.....

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: 1 does not equal .999999.....

Well I used sound, thought-out search criteria.  I found 9 (or 8.99999... if you prefer)

I'm talking about threads not posts.  The latter may well be un-countable.

Bob

Which search criteria was it?

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,664

Re: 1 does not equal .999999.....

hi Stefy,

I said this way back in post 2.

Try 0.99999

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: 1 does not equal .999999.....

a thread which does not come up in that search.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

∞

Member

Registered: 2013-08-13

Posts: 5

Re: 1 does not equal .999999.....

personally, I believe 0.999...=1 in the theoretical word, but 0.999...=0.999... in the real world. why? infinity doesn't exist in the real world, only in theory. thus, an infinite number of 9's is impossible in real life. I mean you cant accurately measure pi can you? nopies! so 0.999... is only a theoretical number. this "infinity exists here but not there" causes the whole topic to be whacko. that's also why the value infinity is kinda whacko. you cant fully explain it can you? nopies.....so really, 0.999... can be thought of along side i, pi, infinity, and all other undefined values that make math involving them impossible to fully explain in exact amounts (like integers, etc.)

---

∞ exists as the limit of x/0, where x can be any value that exists. Unfortunately, x/0 doesn't exist in nature, so ∞ doesn't either. That's why its always so elusive and mystifying. Personally? I believe it was 0's brother that never was born into the universe of commonly used math.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: 1 does not equal .999999.....

Hi;

.99999... is just another way to write 1

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,664

Re: 1 does not equal .999999.....

0.999... is only a theoretical number.

So you think some numbers are only theoretical.  So what are the non-theoretical ones?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,007

Website

Re: 1 does not equal .999999.....

lol

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

SteveB

Member

Registered: 2013-03-07

Posts: 595

Re: 1 does not equal .999999.....

In pratice all measurements and engineering calculations are going to have limited accuracy.
However in pure maths... the matter is not quite so simple.
For starters if you have an irrational number like the square root of 2, then if it is rounded off then
it strictly speaking is being given as a rational numbered approximation.
By the same principle is 0.9 recurring the same as one to any finite conventionally rounded approximation,
but if you could have, in theory, literally every 9 listed, then it is not quite the same in pure maths. (???)
I have seen a theorem in a Foundations of Maths book written by a Mathematics professor (Warwick I think)
which states with proof that between any two distinct rationals there is an irrational number,
and between any two distinct irrational numbers there exists a rational number, but they
should not be thought of as alternating along the number line.
The number 0.9 recurring is a rational number because it has a repeating sequence.
So therefore we have to think of it as exactly one, otherwise it has to have a number inbetween it and one.
It certainly converges to one as the number of digits tends to infinity....

Last edited by SteveB (2013-08-16 07:07:27)