Does 0.9999 repeating 9 equal 1?

[PeterMP]

There is no end to an unending string. You're thinking finitely.

But is it really any different than saying:

infinity + infinity = 2*infinity

[Spec138]

But is it really any different than saying:

infinity + infinity = 2*infinity

Infinity + Infinity = Infinity

It's the highest you can go.

Just like the speed of light, 100 mph + c = c.

Just to add 2Infinity implies you can quantitate infinity, which you cant.

[PeterMP]

Infinity + Infinity = Infinity

It's the highest you can go.

Just like the speed of light, 100 mph + c = c.

Just to add 2Infinity implies you can quantitate infinity, which you cant.

But that isn't true or I couldn't say that:

(set of real numbers between 0 and 1) > (set of all natural numbers)

Both are infinite, but one is more infinite.

[Spec138]

But that isn't true or I couldn't say that:

(set of real numbers between 0 and 1) > (set of all natural numbers)

Both are infinite, but one is more infinite.

How is one more infinite?

[Thinking Skins]

I just love this thread!!!!

I really love math and I love seeing people actually discussing it.

[Slacky McSlackAss]

i just love this thread!!!!

I really love math and i love seeing people actually discussing it.

more math less politics!!!!

[Larry]

Those number equal the same amount. .25 isn't equal to .255.

But your argument wasn't that .25 was different from .255. Your argument was that 1 was not equal to .9 repeating, because they looked different.

1/4 and 4/16 look different, too. But they're the same number.

EDIT: if .9 repeating equals 1, then .9 repeating doesn't exist. It's really just an imaginary number. People shouldn't use it if they're just going to say that it equals one.

Your reference to imaginary numbers (an actual mathematical term, which mathematicians and engineers actually do use in calculations), reminds me of a column I'd read written by Isaac Asimov.

(Not making fun or arguing. Just relating a funny story.)

He said that when he was in college, a philosophy professor was illustrating a point, in which he stated that the world consists of people who deal with reality, and people who deal with abstractions of reality. He listed several groups of people, including, in the first group, "engineers", and in the second, "mathematicians".

Isaac Asimov (math major) rose to object, claiming that mathematicians did, in fact, deal with reality, not with abstractions.

The instructor threw him a piece of chalk, and told the student to hand him the square root of negative one pieces of chalk.

Asimov tossed the chalk back to the Professor, and asked him to give Asimov one half of a piece of chalk. The Professor broke the chalk into two pieces, and tossed one of them back.

Asimov observed that this wasn't one half of a piece, it was one piece. And that even if you were to claim that "one piece" of chalk was a standard unit of mass or length, how could the professor prove that he held one half of a piece, and not .49 or .52 piece?

Asimov then announced that he was incapable of discussing the mathematical basis of "imaginary numbers", (and their correspondence to "real world" phenomena, like AC current), with a person who, by his own standards, was unqualified to discuss fractions.

(He got a D in the class.)

[PeterMP]

How is one more infinite?

It must contain more numbers because it is possible to include all of the natural numbers in all possible combinations.

http://en.wikipedia.org/wiki/Infinite_set

"If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite."

So if I have an infinite set, which I'll call x and subset it into a number of subsets (y1....yi) one of the subsets must have an infinite number of members so let's say y1 has an infinite number of members. However, y1 is a subset of of x so:

the number of things in x > the number of things in y1

Yet both of them conatain an infinite number of things.

[Larry]

Infinity + Infinity = Infinity

It's the highest you can go.

Just like the speed of light, 100 mph + c = c.

Just to add 2Infinity implies you can quantitate infinity, which you cant.

Although I understand that, supposedly, infinity squared is not equal to infinity.

(Freely admitting that my eyes glazed over at that point, and it all ran out my other ear.)

[Spec138]

But your argument wasn't that .25 was different from .255. Your argument was that 1 was not equal to .9 repeating, because they looked different.

1/4 and 4/16 look different, too. But they're the same number.

Your reference to imaginary numbers (an actual mathematical term, which mathematicians and engineers actually do use in calculations), reminds me of a column I'd read written by Isaac Asimov.

(Not making fun or arguing. Just relating a funny story.)

He said that when he was in college, a philosophy professor was illustrating a point, in which he stated that the world consists of people who deal with reality, and people who deal with abstractions of reality. He listed several groups of people, including, in the first group, "engineers", and in the second, "mathematicians".

Isaac Asimov (math major) rose to object, claiming that mathematicians did, in fact, deal with reality, not with abstractions.

The instructor threw him a piece of chalk, and told the student to hand him the square root of negative one pieces of chalk.

Asimov tossed the chalk back to the Professor, and asked him to give Asimov one half of a piece of chalk. The Professor broke the chalk into two pieces, and tossed one of them back.

Asimov observed that this wasn't one half of a piece, it was one piece. And that even if you were to claim that "one piece" of chalk was a standard unit of mass or length, how could the professor prove that he held one half of a piece, and not .49 or .52 piece?

Asimov then announced that he was incapable of discussing the mathematical basis of "imaginary numbers", (and their correspondence to "real world" phenomena, like AC current), with a person who, by his own standards, was unqualified to discuss fractions.

(He got a D in the class.)

Brilliant.

Also just to add, I've .999... related to pi in that they're both unnecessary.

[PokerPacker]

Which has nothing to do with the question.

The proof presented is a valid mathematical proof. It is 100%, absolutely, without question, proven.

All of these "well, no matter how many 9's you have" thought exercises are nothing more than changing the question.

Because no matter how many 9's you have, it won't be an infinite number of 9's.

And the problem specified an infinite number.

and I can prove something else:

-1/1=1/-1

sqrt(-1/1)=sqrt(1/-1)

sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

i/1=1/i

i*i/1=i*1/i

i^2/1=i/i

-1=1 ?

[Thinking Skins]

Let x = 0.99999999 (repeating) and z = 1.

Suppose x < z.

Then there exists a y such that x + y = z, where y < 1. Then y^2 is less than 1, which implies that y^2 < y.

So x < x + y^2 < x + y = z. This implies that if x != z, then there exists a number strictly between x and z.

How does this number look? Well since its greater than x, it must have some digit that's not a 9. If this digit is less than 9 (i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8), then x is greater than this number because x has an infinite number of nines i.e

.99999999 is greater than

.99999998

But the last digit cannot be a nine since that would imply that the x + y^2 = x.

So since the nonzero digit of x + y^2 cannot be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, this nonzero digit cannot exist.

Hence one of our assumptions must be wrong. But the only assumption we made was that x < z. So this must be false, and we must have that x >= z. But we can eaily see that x > z is false, so x = z must be true.

[Mark The Homer]

Let x = 0.99999999 (repeating) and z = 1.

Suppose x < z.

Then there exists a y such that x + y = z, where y < 1. Then y^2 is less than 1, which implies that y^2 < y.

So x < x + y^2 < x + y = z. This implies that if x != z, then there exists a number strictly between x and z.

How does this number look? Well since its greater than x, it must have some digit that's not a 9. If this digit is less than 9 (i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8), then x is greater than this number because x has an infinite number of nines i.e

.99999999 is greater than

.99999998

But the last digit cannot be a nine since that would imply that the x + y^2 = x.

So since the nonzero digit of x + y^2 cannot be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, this nonzero digit cannot exist.

Hence one of our assumptions must be wrong. But the only assumption we made was that x < z. So this must be false, and we must have that x >= z. But we can eaily see that x > z is false, so x = z must be true.

I followed about half of this and then I got tired. lol

Did you copy that out of a book or just write it out? lol

[Mark The Homer]

Hey Velocet did you get your package?

[Larry]

Hey Velocet did you get your package?

What's his package got to do with imaginary numbers?

[mjah]

and I can prove something else:

-1/1=1/-1

sqrt(-1/1)=sqrt(1/-1)

sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

i/1=1/i

i*i/1=i*1/i

i^2/1=i/i

-1=1 ?

The third line doesn't follow from the second. Naughty boy!

[PokerPacker]

The third line doesn't follow from the second. Naughty boy!

sqrt(a/b)=sqrt(a)/sqrt(

[Mark The Homer]

What's his package got to do with imaginary numbers?

It's actually not a package. It's .999999999 (repeating) package.

[PokerPacker]

It's actually not a package. It's .999999999 (repeating) package.

so he'll have to wait for all of eternity to get the whole thing?

[PeterMP]

so he'll have to wait for all of eternity to get the whole thing?

No, because it is already there. The infinite set of 9s aren't appearing. They are all there.

[GibbsFactor]

.9 repeated can't be infinite for it can not be smaller then plank time. So therefore, it can never reach that infinite.1 value.

:twitch:

[Thinking Skins]

How is one more infinite?

Lets say we have a set of numbers {1, 2, 3}. The power set of that set is the set of all subsets of that set. So the power set of {1, 2, 3} consists of the elements:

{}, {1}, {2}, {3}, {1, 2}, {1,3}, {2, 3}. {1, 2, 3}.

Notice that the original set had 3 elements and the power set has 8 elements. This is not just by accident, but for any set of size n, its power set will be of size 2^n.

And whats more is that for any n, we know that the size of the power set of that set is strictly bigger than the size of that set.

So suppose we had an infinite set, say the set of all positive numbers. How big would the power set of that be? Well from what I just said, its gonna be bigger than the size of the original set. So its gonna be bigger than infinity. So there's a 'bigger' infinity than infinity. And there's an even bigger infinity than this bigger infinity, and an even bigger one and so on and so forth.

[GibbsFactor]

Lets say we have a set of numbers {1, 2, 3}. The power set of that set is the set of all subsets of that set. So the power set of {1, 2, 3} consists of the elements:

{}, {1}, {2}, {3}, {1, 2}, {1,3}, {2, 3}. {1, 2, 3}.

Notice that the original set had 3 elements and the power set has 8 elements. This is not just by accident, but for any set of size n, its power set will be of size 2^n.

And whats more is that for any n, we know that the size of the power set of that set is strictly bigger than the size of that set.

So suppose we had an infinite set, say the set of all positive numbers. How big would the power set of that be? Well from what I just said, its gonna be bigger than the size of the original set. So its gonna be bigger than infinity. So there's a 'bigger' infinity than infinity. And there's an even bigger infinity than this bigger infinity, and an even bigger one and so on and so forth.

Dude, it's infinity.

:whoknows:

[Thinking Skins]

I followed about half of this and then I got tired. lol

Did you copy that out of a book or just write it out? lol

I wrote one out earlier, but I wasn't quite satisfied with it. So I came back to the thread and wrote this one out.

I'm trying to think of simple ways to prove it that aren't saying the same thing over and over again.

I think the 1/3 = .3333333, 2/3 = .66666666, 3/3 = .99999999 = 1 argument is the best one. Its simple and gets to the point.

Same with the 1/11 = .11111111 argument.

[Stew]

:::wrapping duct tape around head to prevent brain matter explosion:::