Is 1 > .99999....?

Question

Based on a survey of about 10,000 related questions on this Web site, it appears 49.9999...% of the population believes 1 > .99999..., while 49.9999....% do not. What about the rest of you?

Answer 1

hello zanti.
okay, to me and others may disagree, they are equal.
0.9999.... is a rational number, and if you express it as the ratio of two integers, its value is 1.
There is also an elegant proof by contradiction, where you can assume it is less than 1.
if 0.99999...<1, then there exists x
0.999999....< x < 1
the units digit has to be zero, else it is not less than 1.
the first decimal has to be 9, else it is not greater than 0.9. And so does the second and third and fourth and so on.
EDIT: yes freevariable, I am convinced that zanti knows this and he is only testing us. :).

Answer 2

I say that you don't necessarily come to a contradiction by saying either

1) 1 > 0.99999....
2) 1 = 0.99999....

In other words, by convention, we say that 1 = 0.99999... because we don't contradict ourselves when we assume that, but we don't contradict ourselves either if we say that 1 > 0.99999.... I'm not familiar with any mathematical proofs showing that 1 must = 0.99999... that doesn't already assume certain axioms about limits and infinity. And we know that axioms aren't chiselled in stone.

There, does that make me the minority view between the 49.9999...% that say 1) and the 49.9999...% that say 2)?

Addendum: In answer to popeye's and others' "proofs", let's say that 1> 0.99999... That means there's an x such that 1 > x > 0.99999.... Let's say that the number of terms in 0.99999...is N, some huge number. Then x is 0.999999... to N+1 terms. What's the problem? If you argue that infinity + 1 = infinity, I can then argue that 1 = 0. Oh, I can't do that? Let's see you prove that infinity + 1 = infinity, WITHOUT using some axiom somewhere.

Addendum 2: Benoit, I KNOW about all the "proofs" that 1 = 0.99999.... But we also do have non-Euclidian geometries which are just as valid and self consistent as Euclildian geometry. We have the choice of making an axiom that infinity + 1 = infinity, or that infinity + 1 > infinity. We go with the latter only because it's practical and allows us to express functions as "exactly" the same as their infinite series expansion. Do you know what? This will be my next question.

Addendum 3: Benoit, see the answers being offered to my question. I rest my case.

Answer 3

The percentages joke made me laugh. Seriously, I'm surprised nobody has commented on it yet.

Of course they're equal. Zanti knows, he just wants to know if YOU know. I know a few proofs of it, but somebody already posted a link to some proofs, so I guess that would be redundant.

It's amazing how this question can surface even in non-math forums, and most people (in my experience) have a hard time believing it. Heck I've been trying to convince my father for years, but no matter how many proofs I come up with, he still believes his intuition over the proof.

Answer 4

1 = .9999...
This horse has been beaten to death.

Scythian says:

"Let's say that the number of terms in 0.99999...is N, some huge number."

No. The number of terms is infinite. That's what the dots mean. Some "huge number" N is still going to fall wayyyyyyyy short.

Addendum: Saw your other question, Schythian. Zanti3's answer nailed it: {1,2,3...} and {0,1,2,3...} have the same cardinality.

Fact is, you don't get to choose axioms. when we talk about decimals such as 0.9999..., we are assuming the context of real numbers. And in the set of real numbers, 0.999... and 1 are the same thing.

Answer 5

.9 repeating does in fact equal 1. And the website below explains why, as well as provides some mathematical proofs to back it up.

Answer 6

The mathematical answer is 1 = .9999.......... (assuming an infinite number of 9's). Anyone who wants to see the proof should pick up a Calculus 1 textbook.

Answer 7

1 > .99999....

For example 1/3 can be represented as 0.3333333......
However they are not equal because no matter how many threes there are, it can never be equal to 1/3.

Another example, 0.499999..... and 0.5.
Round them to the nearest whole number.
0.4999999...... goes to 0 as rounding off only depends on the next digit, while 0.5 goes to 1.

Therefore 1>0.99999.....

Answer 8

No, they are exactly equal. I enjoy your humor, as well. I can't wait to see how many people point out the fact that the percentages "don't add to 100."
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EDIT:
I am absolutely shocked at how many people are getting this wrong. If the 9's go on infinitely, there is no difference at all. Especially you, detektibgapo; your response was incredibly stupid. Refer to the 3rd link I sent you guys, as it describes a proof using a completely concete infinite series. I think the problem is that none of you have a concept of what something infinite means. It doesn't mean that there are "a lot of 9's" and then they just stop. They go on forever, you morons.

Answer 9

I think it's actually 50% that do not believe that 1 > .99999..., myself included. Since 50% > 49.99999...%, we must be right.

I think the other 0.0000...% may believe that 1 < .99999...; if I meet one, I'll let you know.

(Before anyone tries to ridicule me, read it carefully.)

Answer 10

There are several answers that are on track.

See the following question that we already discussed the same math issue.