# A shorter proof to 0.999 = 1

Here is Blizzard's proof that 0.999... = 1

http://www.blizzard.com/press/040401.shtml [Broken]

My friend however, recently mentioned that:

1/3 = 0.333...

1.3 * 3 = 0.333... * 3

1 = 0.999... also works.

Has she discovered the shortest proof? Or is there something wrong here?

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This is perfectly right. I don't know if there's a "shortest proof", but your friend's certainly is short.

I'm somehow a newbie here, but I used to visit Tom's Hardware Guide's forums, and there's an interesting topic with... various "opinions" on the subject on it. Check it out http://www.community.tomshardware.com/forum/showflat.m?Cat=&Board=other_poll&Number=167099&page=0&view=collapsed&sb=5 [Broken]

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BenGoodchild
There is, ofcourse,

1/9 = 0.111....
multiply be nine
1 = 0.999.....

Integral
Staff Emeritus
Gold Member
juef said:
This is perfectly right. I don't know if there's a "shortest proof", but your friend's certainly is short.

I'm somehow a newbie here, but I used to visit Tom's Hardware Guide's forums, and there's an interesting topic with... various "opinions" on the subject on it. Check it out http://www.community.tomshardware.com/forum/showflat.m?Cat=&Board=other_poll&Number=167099&page=0&view=collapsed&sb=5 [Broken]
I made a few contribution to that thread. SilverPig started that thread to see if the response was significantly different from a similar thread in the HiTech Forum at anantech, Which I also participated in.

It was very disappointing that the majority of members of Anantech and Toms felt that it was not true.

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matt grime
Homework Helper
That isn't a proof. Why is arithmetic deinfed on infinitely long decimals? In short, you're confusing real numbers with their representations as decimals. The fact that they are equal is immediate from the definition of the real numbers, not that anyone who thinks they're different even knows what the real numbers are.

BenGoodchild
Matt - don't become worried - it isn't meant to be a proof - it is just a cunning trick used by high school mathematics teachers to trick their students and to make them think.

No one is really saying that 0.9 recurring equals 1

Regards

Ben

matt grime
Homework Helper
Oh great; the cranks come out of the wood work. Can we have an instant ban for anyone who, despite the explanation to the contrary being in the thread, asserts that they are not equal?

Apologies if that's a typo and you clarifyting no one *denies* that they are equalivalent as representations of real numbers, or if | misunderstand and you are trying to differentiate between representations of real numbers and the numbers themselves, but I doubt that is your intention.

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Integral
Staff Emeritus
Gold Member
I consider this sort of algebraic manipulation more of a demonstration then a proof. It is a valid demonstration of a mathematical fact, but not a proof.

Zurtex
Homework Helper
BenGoodchild said:
Matt - don't become worried - it isn't meant to be a proof - it is just a cunning trick used by high school mathematics teachers to trick their students and to make them think.

No one is really saying that 0.9 recurring equals 1

Regards

Ben
I'm saying 0.9 recurring equals 1 and so will any mathematical approch on it.

arildno
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Gold Member
Dearly Missed
Now, I don't know if I qualify as one of Zurtex' approch's (sounds like some lumbering, prehistoric animal to me), but I agree with his view as well.

I consider this sort of algebraic manipulation more of a demonstration then a proof. It is a valid demonstration of a mathematical fact, but not a proof.

I'm saying 0.9 recurring equals 1 and so will any mathematical approch on it.

I think BENGOODCHILD was making the point that the value of 0.999... as a number is not 1.

It is true that 0.999... comes from the formula for the series 9/10 + ;9/10^2 + 9/10^3,

therefore the limit of the series is infact 0.999... and therefore one but the value is different.

Unless we want to start the whole debate on infinity and what happens at the last 9 etc - well there i no last 9 becasue 0.99...is non-equatable. Okay?!

-M

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arildno
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Yes it it is, maverick (goodchild?). Learn about how real numbers are defined as equivalence classes on the set of (increasing, bounded) sequences of rationals.

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matt grime
Homework Helper
Who would have to talk about infinity and the last 9? You, Maverick? Only those who don't understand mathematics would cite that. Indeed there is no reason to invoke infinity at all, indeed the appearance of any infinty is only a short hand fomr something to do with finite things and we need not ever mention it. Now, as I'm apparently not in a charitable mood, can the cranks go away?

All refutations of this fact arise from not understanding maths - the definitions are straight forward, though hard to visualize perhaps, but in the completion of Q (ie R) those are the same number. Fin. Just as 1/2 and 2/4 are the same rational number.

HallsofIvy
Homework Helper
I think BenGoodchild was making a joke.

I hope maverickmathematics was also (look at the user name!).

BenGoodchild
we're studying at Trinity College Cambridge- look at the notes mav posted about number theory, so yes I'm messing you guys around.

and maverick is a long time friend of mine so there you go

regards,

Ben

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arildno
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Gold Member
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Level of jocular funniness:
Harrumph, heh-heh

matt grime
Homework Helper
maverickmathematics said:
I think BENGOODCHILD was making the point that the value of 0.999... as a number is not 1.

It is true that 0.999... comes from the formula for the series 9/10 + ;9/10^2 + 9/10^3,

therefore the limit of the series is infact 0.999... and therefore one but the value is different.

Unless we want to start the whole debate on infinity and what happens at the last 9 etc - well there i no last 9 becasue 0.99...is non-equatable. Okay?!

-M

Funny? Nurse, my sides have split.

BenGoodchild
you guys are just plain boring - you'll get all worked up if I tell you that 2+2=5 and start crying...

Zurtex
Homework Helper
BenGoodchild said:
we're studying at Trinity College Cambridge- look at the notes mav posted about number theory, so yes I'm messing you guys around.

and maverick is a long time friend of mine so there you go

regards,

Ben
I'm studying number theory at UMIST and I'm sure half the people in my class wouldn't know that 0.999... = 1 :grumpy:

arildno
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Gold Member
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And, from BenGoodchild's second post, I don't think he knows either.
This damage control action he's undertaken afterwards is unconvincing.

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BenGoodchild
Give me a couple of hours and I'll bring back evidence!

kreil
Gold Member
I'm thoroughly convinced that 0.9r=1. I have been reading the posts on the hardware forum and here all day during study halls, and when my calculus class finally came I asked my teacher what she thought. Her reply was that it did, but one of my friends wasn't convinced. I argued with him about it on the way to physics. Knowing that my physics teacher is educated in math, I said we would ask him. This is how the conversation went:

"Mr. H, does point 9 repeating equal 1?"
"No"
"Yes it does, we have been arguing about it. Number theory says it does"
"Number theory was created by a bunch of mathematicians that don't know anything about the real world, thats why it is called number theory and not number really"

I walked away, angered and surprised at his ignorance. I never got a chance to show him a proof, which I am hoping will convince him.

Why is this fact so hard for people to accept?

BenGoodchild
Because to most people it appears as though you are saying somthing equivalent to

0.99999999999999999999999 = 1

They do not understand the actualy CONCEPT of number and to a certain extent it appears as though you are saying that the little bit that one expects should be inbetween the 0.9r and 1 does not exist - people cannot accept it.

For them 0.999999999999999 + 0.000000000000001 = 1

It is the idea of a repeating decimal people cannot fathom

how do we know that 1/3=0.33333........ by this, we need to look at the geometric series to prove such.

If 1/3 cannot be defined as 0.333...

Then wouldn't that imply that it is illegal to even write the equation 0.999... = 1?