# .99999 repeating = 1? Whats your opinion?

Do you think that .9 repeating equals one? What is your theory. I’ll give you mine tommorrow.

Lots of people had theories in this thread (most recently posted to on 8/21/01). It seems the consensus of opinion is that .99999 does, in fact, equal 1.

I really appreciate your consideration in avoiding stepping on my penis - Spiny Norman
Jeg elsker dig, Thomas

This isn’t a matter of opinion. .999… does equal 1 in the real number system.

There’s nothing to opine. By definition 0.999… equals one.

Nooooooo! Mods, kill it before it breeds.

It’s not a definition, just a consequence of the axiom of completeness and the definition of decimal expansions. This may seem like a minor point, but definitions are supposed to be non-creative (i.e., you can’t prove anything with a definition that you can’t prove without it), and a definition like that could be creative.

Yes. Look at it a different way, in fractions. 1/9 = .11111… 9 x 1/9 = 1. .1111… x 9 = .99999…

Therefore, .99999… = 1.

Sound good?

No. .9999999… does not equal 1. Say you can extend .9999… out to k decimal places. I can then pick the number k + 1 (after all, the set integers is infinite). Multiply .99999… by k + 1 and you will get a different answer from 1 * (k+1). If you assume that .99999… = 1 than (k + 1) * .9999999… must = (k + 1) * 1.

Furthermore, the set of real numbers, to which .9999… and 1 both belong, is a type of mathematical structure called a field. One of a field’s properties is that it has a multiplicative identity, a number y such that y * x = x for all x’s belonging to the field. In the case of the real numbers the multiplicative identity is 1, and by the definition of a field, the multiplicative identity is UNIQUE. Therefore, .99999… cannot be 1 as that would mean the real numbers would have two multiplicative identities which is impossible by definition.

A series is not a sum of an infinite number of terms. A series is the limit of the sequence of partial sums. .999… and 1.000… are both shorthand for two series with the same value. I refer the reader to Walter Rudin’s Principles of Mathematical Analysis.

The multiplicative identity element of a field is unique, but it need not have a unique representation. .999… and 1.000… are two different representations for the multiplicative identity element of the real numbers.

In fact, as the identity of any group G can be written as g*g[sup]-1[/sup] for any g[sym]Î[/sym]G, the identity never has a unique representation unless |G| = 1. The non-zero elements of a field form a group under multiplication, so the multiplicative identity of a field never has a unique representation.

If you want to argue that .999… is not equal to 1.000…, you need to argue against the property of completeness. Specifically, you need to show that the set of Dedekind cuts of rational numbers is not complete. You can’t, of course, but it might be fun to try.

[list]I’d say that; ______
The value 0.999999 is equal to the number one minus the reciprocal of infinity.

Someone please demonstrate that this is not the case.

gry

cornflakes - lol.

(for those who don’t know, there are some threads that crop up again and again on these boards - .9999… = 1 is one of them, and ‘gry’ refers to another which I don’t dare mention for fear it sparks off yet another debate)

EagleEye - that .999… = 1 is not a matter of opinion - AFAIK, it is accepted by the mathematical community in the same way that sin[sup]2[/sup]x + cos[sup]2[/sup]x = 1 .
Here is a formal proof for .9 recurring = 1. I found it in Google by searching for “.9999 = 1” - amusingly, you can find various different proofs by changing the number of 9s in your search. It’s almost a proof by “reductio ad nauseum”

Infinity is not a discrete mathematical value. Hence such terms as infinity minus one, infinity plus one, and other products and sums of a similar form are indeterminate. Infinity really means, “I don’t know how much, but it seems like a whole lot.” When you find a use for the term where that definition is nonsensical, you have made an error in your use of the term.

## Tris

“Here Kitty, Kitty, Kitty.” ~ Erwin Schrodinger ~

I realise this seems puzzling, but it’s not a matter of opinion.
‘Who is the greatest athlete of all time?’ is a matter of opinion.

As previous posters have said, mathematics is founded on axioms, which give rise to all sorts of consequences. There is no opinion involved.
By using the phrase repeating (forever) you have introduced infinity. Infinity, as Triskadecamus showed, is tricky!

Does this help?

1 - .9 recurring = 0.0 recurring = defined as 0.