# 3 Proofs for .9999 = 1

Grizzlycomet said:
I spoke to my math teacher about this issue today, and he was quite firm in his belief that $$0.\bar{9}$$ does not equal 1. He didn't provide any proof though.
The public should understand education. And it would do no harm if teachers and professors understood it, too.
-Hutchins

The public should understand education. And it would do no harm if teachers and professors understood it, too.
-Hutchins
Indeed. I am planning to make a small document with various proofs that 0.999... does equal 1. Any links anyone could provide would be most helpful.

Grizzlycomet said:
I spoke to my math teacher about this issue today, and he was quite firm in his belief that $$0.\bar{9}$$ does not equal 1. He didn't provide any proof though.

It would be a good idea to first ask teacher what endless repeating decimal expressions actually mean, inasmuch as addition is a finite (originally binary) operation between numbers.

quartodeciman said:
It would be a good idea to first ask teacher what endless repeating decimal expressions actually mean, inasmuch as addition is a finite (originally binary) operation between numbers.
Further discussion with my teacher would indeed be a good idea. However, my school year ends this week so there's not exactly a lot of time for this.

I was posting on the math thread before it was closed. .99999 can only be 1 if indeed it was generated by the division of three equal parts of a whole. It would preclude you knowing that the .999999 was generated by this act and then could be equated to one. Other wise no one has a right to make .999999 = 1 because it is not so and non relative at this point. So it could have been generated by three parts of a whole or it was a selected number. If it is just a selected number it is not equal to 1.

ahrkron
Staff Emeritus
Gold Member
Tenyears, as I told you in the math thread, the fact that 0.999...=1.00... is not a matter of convention, "acceptance", or authority. It is a logic inevitability from the definitions of real numbers. The issue is not controversial at all among professional mathematicians, and is based on solidly established branches of math (in particular, Real Analysis).

quick random q
where do you get all your mathematical symbols from on the keyboard... thanks

K_

europium said:
quick random q
where do you get all your mathematical symbols from on the keyboard... thanks

K_
The matematical symbols is created using a code in the forum knows as Latex. It's usage is described in This Thread

Njorl
I am firmly in the .999...=1.0 camp, where 1.0 equals the real number one. Is it allowable to say that the real number one is equal to the integer number one? Is this like mixing apples and oranges (one apple does not equal one orange)?

I feel that the integer one does equal the real one, but I don't know of a rigorous logical foundation for it. Is there one? Is it just defined to be so? Am I wrong?

Njorl

honestrosewater
Gold Member
Njorl said:
I am firmly in the .999...=1.0 camp, where 1.0 equals the real number one. Is it allowable to say that the real number one is equal to the integer number one? Is this like mixing apples and oranges (one apple does not equal one orange)?

I feel that the integer one does equal the real one, but I don't know of a rigorous logical foundation for it. Is there one? Is it just defined to be so? Am I wrong?

Njorl

I have only seen two contructions of R (Dedekind cuts & Cauchy sequences) and both construct R from Q. And Q is defined by Z and Z by N. So I would think that is rigorous logical foundation, but what do I know? :uhh:

Happy thoughts
Rachel

I don't care what proof you come up with.

1 - I take a geometric object a cicle or a square and divide it into three parts and then add the decimal values the number is .999999 but is 1 relative to the object as a whole. This is correct for it is the totality of the object.

2 - I take the number .9999.... out of the blue with no reference to a geometric representation of an object this is not 1. If I take a geometric reference to the universe, divide it into three equadistant rays starting from a central point and extending into infinity running a wall of ray for the length of extension so you have three defined parts, make that a decimal value of .333.... added together, then I will equate .99999..... to one with repsect to the universe.

If there is no geometric reference, and it is just a number, it is not equal to 1.

NateTG
Homework Helper
Really, this seems like a funky philosophy of math question but, there is an obvious subset of $$\Re$$ that is isomorphic to the integers with the appropriate operations. As long as you think of it more as an instatiation of the integer 1 rather than as the only integer 1, you should be fine.

I have only seen two contructions of R (Dedekind cuts & Cauchy sequences) and both construct R from Q. And Q is defined by Z and Z by N. So I would think that is rigorous logical foundation, but what do I know?

You can also construct the real numbers from things like the set of all countable sequences of zeroes and ones that do not end in reapeating 1's, or something, but a construction like that has a PITA factor while Cauchy sequences and Dedekind cuts can readily be shown to have the desired properties. I bring this up because it allows for $$0.\bar{1}_2$$ <binary notation>, not to be a real number.

I expect that, the (mistaken) notion that $$0.\bar{9}$$ and $$1$$ are distinct is a result of the mistaken assumption that decimal representations are unique.

honestrosewater
Gold Member
TENYEARS said:
I don't care what proof you come up with.

I think you are giving ahkron too much credit.
If ahkron has seen further it is by standing on the shoulders of Giants.

Now, if you cut a giant into 3 parts... only kidding, in good fun

Okay, now I can't help myself ;)

"If I have not seen as far as others, it is because giants were standing on my shoulders." -- HalAbelson

"In the sciences, we are now uniquely privileged to sit side by side with the giants on whose shoulders we stand." -- GeraldHolton?

"If I have not seen as far as others, it is because I was standing in the footprints of giants"

"If I have seen farther than others, it is because I was standing on a really big heap of midgets." -- EricDrexler (Nice for those of us who believe the inspiration of giants isn't the only engine of progress.)

"If I have seen further than others, it is because I was surrounded by dwarves." -- attributed to MurrayGellMann?, possibly maliciously.

"I cannot see very far, because my eyes are full of midgets."

:rofl:

honestrosewater
Gold Member
NateTG said:
You can also construct the real numbers from things like the set of all countable sequences of zeroes and ones that do not end in reapeating 1's, or something, but a construction like that has a PITA factor while Cauchy sequences and Dedekind cuts can readily be shown to have the desired properties. I bring this up because it allows for $$0.\bar{1}_2$$ <binary notation>, not to be a real number.

Great, now I'm confused too :yuck: Is there a quick way to explain how that construction proceeds? Oh, countable is a clue methinks. No, the set is countable? Or the sequences are countable? Yeah, What is this PITA factor you speak of? (I get the PITA, but what is it?)

Hurkyl
Staff Emeritus
Gold Member
Basically, you define all the arithmetic operations by the method of elementary school arithmetic, but the trick is that you have to perform addition from left to right. The "PITA" factor is in cases where 'ambiguous'
whether you should have a carry or a borrow when doing an operation.

E.G. when adding 0.1100... and 0.011000..., you can "look ahead" to see that the second place to the right of the decimal point generates a carry, and it's propagated through the next place, so you can set the one's digit to be a 1, and then so on.

However, when adding 0.101010... and 0.010101..., there's nothing to say that there should or should not be a carry. Thus, you make a definition; you either say that in this situation you will always consider there to be a carry, never consider there to be a carry, or define both options as being equal.

honestrosewater
Gold Member
Hurkyl said:
However, when adding 0.101010... and 0.010101..., there's nothing to say that there should or should not be a carry. Thus, you make a definition; you either say that in this situation you will always consider there to be a carry, never consider there to be a carry, or define both options as being equal.

1) carry-> 1.000...
2) no carry-> 0.111...
3) equal-> 1.000...=0.111...

Yes? No? So why can .111... not be real? I am still missing something; hopefully I will see clearer after I some shuteye.

Happy thoughts
Rachel

EDIT- Oh duh- if you decide to carry, you cannot ever get the "noncarry" number, and vice versa.

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My understanding of infinte numbers is that you can not treat them like any other number. Like what 'tenyear" was talking about.

Infinite numbers are just a mathimatical idea and dont realy exist in the real universe as we know it. I thought that is why "limits" were created anyway.

infinity, as I've been thinking about it, is whithin every real number as a result of mathamatical processes. e.g, divide to infinity, add to infinity, subtract to infinity, .....etc. Even whithin 0, there is infinity.

Infinity is the true nature of all things from greatly massive to micro. "Limits", are just easier ways of looking at things and are nothing more than a generalization of reality.

mikesvenson said:
infinity, as I've been thinking about it, is whithin every real number as a result of mathamatical processes. e.g, divide to infinity, add to infinity, subtract to infinity, .....etc. Even whithin 0, there is infinity.

every real number has an infinite number of digits but no real number has an infinite magnitude. e.g. ...000000001.0000000...

it has an infinite divisible magnatude

1/3 is a non-mathematical representation of a potential infinity: 0.333...etc. There is no such thing as an actual infinity. I refer to the Hilbert's hotel paradox. Potential infinities exist in mathematics because it is a theoretical tool.

Let me put it this way:
does 0.9 = 1? No
does 0.99 = 1? No
does 0.999999999999999999 = 1? No

In each case we are getting closer to 1, but will we ever get there? No.

0.9~ is not 1.
If you can think of space as infinately large,
then just imagine the concept of infinately small.
Distance in space can always be cut in half. And in half again.
Our mind just has a hard time comprehending it...

Integral
Staff Emeritus
Gold Member
There is no such thing as an actual infinity.

Mathematically speaking there is and it is carefully defined. Maybe you should take a few minutes to read some of the posts in this thread.
0.9~ is not 1.
Certainly is! I guess your concept of the universe does not apply to the real number line. Perhaps you should modify your concept of the universe.

Hurkyl
Staff Emeritus
Gold Member
Let me put it this way:
does 0.9 = 1? No
does 0.99 = 1? No
does 0.999999999999999999 = 1? No

Does 0.9 = 0.9~? No.
Does 0.99 = 0.9~? No.
Does 0.999999999999999999 = 0.9~? No.

In each case, we are getting closer to 0.9~ but will we ever get there? No.

Why do you think your observation has any bearing on whether 0.9~ = 1?