Does 0.9999 repeating 9 equal 1?

[jnhay]

The other easy proof, which has been touched on earlier and is as clear as day, is

If .333 (repeating) is exactly one-third.

Then .999 (repeating) has to be exactly three-thirds.

And three-thirds = one.

credit: Thinking Skins

.3 repeating is not exactly one third. It's just the closest number to 1/3 that we use for simplicity.

[Mark The Homer]

...In addition, maybe it is a limitation with respect to my understanding of infinity, but it is certainly valid to talk about one infinite set being larger than another (two infinite subsets can be combined to make a larger infinite set, and while all are infinite the combination also is larger). I have serious issues thinking about something like that with respect to what I would consider an "entity".

Hmm. I don't follow that. I don't get where two infinite sets being combined make a larger set. Then again, I have never taken an upper level math course higher than Calculus I (math 140, u of Md).

I'll also add with respect to your willingness to change your mind quicker than PokerPacker and others, I'll bet that has something to do w/ respect to the amount of math that you (think you) know vs. them. Understanding that you don't have a firm basis for a belief makes it easy t give up. I also think age plays a role in the willingness to admit that you may have made a decision w/o a firm basis.

I can't not accept reality. Post #244 above is reality. Math doesn't lie.

This is in fact a big deal in science education, especially physics, where a lot of work has been done with respect to first having to "unteach" people what they think they know. The fact that PokerPacker has been recently taught (and probably tested) on one way to think about infinity (with respect to limits) probably increases his diffiuctly in giving that up. Hopefully, his instructor will make an effort to correct that way of thinking by the end of his semester.

Right. I think "infinity" and "limits" are two different things because limits includes the phrase "approaches". Limits discusses what happens to something as it approaches infinity, but it never quite gets there. Infinity is there. Maybe PP is having a problem differentiating between those two ideas.

[Mark The Homer]

hmmm...I have been convinced. I stand corrected. 0.(999) = 1.

Another one has seen the light.

[Mark The Homer]

.3 repeating is not exactly one third. It's just the closest number to 1/3 that we use for simplicity.

You're mistaken.

[jnhay]

You're mistaken.

.9 repeating isn't even a real number then. If infinity means "without bound", then .9 repeating is just a limit meaning that 1 will never be reached.

[Mark The Homer]

.9 repeating isn't even a real number then. If infinity means "without bound", then .9 repeating is just a limit meaning that 1 will never be reached.

You're new to the thread, aren't you.

If you look at some of the proofs, you'll understand it better.

If you don't want to accept that .333 repeating is one third ...

(Dr. Math says it is here http://mathforum.org/library/drmath/view/52310.html)

Then check this proof, which is a little harder but not too hard:

I say they are equal. There's a funner geometric series proof, but I don't have time to post it now, so I'll update later. Here's one though I really like.

x = .9 repeating

10x = 9.9 repeating

10x -x = 9.9 repeating - .9 repeating

9x = 9

x = 1

so 1 = .9 repeating

[Thinking Skins]

Hmm. I don't follow that. I don't get where two infinite sets being combined make a larger set. Then again, I have never taken an upper level math course higher than Calculus I (math 140, u of Md).

Think about this. We have the set of all positive integers:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Call this set Z1.

and we have the set of all negative integers:

{-1, -2, -3, -4, -5, -6, -7, -8, -9, -10, ...}.

Call this set Z2.

These are both infinite sets. And no integer belongs to both sets (since no number is both positive and negative).

But if we look at the union of Z1 and Z2, then this is a set thats infinite and bigger than both Z1 and Z2.

Z1 union Z2 = {..., -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

[jnhay]

You're new to the thread, aren't you.

If you look at some of the proofs, you'll understand it better.

If you don't want to accept that .333 repeating is one third ...

(Dr. Math says it is here http://mathforum.org/library/drmath/view/52310.html)

Then check this proof, which is a little harder but not too hard:

I've read those "proofs" and others, and don't agree with them. You can't say a number is infinite and finite at the same time.

[Mark The Homer]

Think about this. We have the set of all positive integers:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Call this set Z1.

and we have the set of all negative integers:

{-1, -2, -3, -4, -5, -6, -7, -8, -9, -10, ...}.

Call this set Z2.

These are both infinite sets. And no integer belongs to both sets (since no number is both positive and negative).

But if we look at the union of Z1 and Z2, then this is a set thats infinite and bigger than both Z1 and Z2.

Z1 union Z2 = {..., -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Okay. I guess I'm confusing "set" for "value."

You're right.

[Mark The Homer]

I've read those "proofs" and others, and don't agree with them. You can't say a number is infinite and finite at the same time.

.333 repeating is finite. The expression of it in decimal (base 10) is infinite, but the value of it is finite.

[Thinking Skins]

I've read those "proofs" and others, and don't agree with them. You can't say a number is infinite and finite at the same time.

are you saying you don't accept the definition of a rational number?

The number is finite. Its 1/3.

But it is also the sum of an infinite number of numbers:

.3 + .03 + .003 + .0003 + ....

This is a series that converges to 1/3, hence the statement .333333(repeating) equals 1/3. Thats the shorthand for saying that the sum of the series

\sum_{k = 1 to infinity} 3*10^(-k) = 1/3, which we know to be true.

[jrockster21]

I've read those "proofs" and others, and don't agree with them. You can't say a number is infinite and finite at the same time.

The difference is - .(9) has an infinite number of 9's, but its VALUE is finite.

[jnhay]

.333 repeating is finite. The expression of it in decimal (base 10) is infinite, but the value of it is finite.

So you're saying I'm right.

See how annoying that is? Where does .3 repeating end if it's finite?

[Thinking Skins]

See how annoying that is? Where does .3 repeating end if it's finite?

It never 'ends', but what happens is that no matter how many 3's you add to the end of it, all you do is get closer and closer to 1/3. Thats why any finite number of 3's will not be equal to 1/3, but when you make the 3's repeat an infinite number of times, then it converges to 1/3.

[Mark The Homer]

So you're saying I'm right.

See how annoying that is? Where does .3 repeating end if it's finite?

The expression is annoying, sure. That's because we're expressing a third in a base ten system. Expressing a half in a base ten system is easy: .5

But that doesn't change the fact that its value is a finite number.

Edit: See, now you got me writing like Larry

[jrockster21]

where does .3 repeating end if it's finite?

1/3.

[jnhay]

It never 'ends', but what happens is that no matter how many 3's you add to the end of it, all you do is get closer and closer to 1/3. Thats why any finite number of 3's will not be equal to 1/3, but when you make the 3's repeat an infinite number of times, then it converges to 1/3.

The expression is annoying, sure. That's because we're expressing a third in a base ten system. Expressing a half in a base ten system is easy: .5

But that doesn't change the fact that its value is a finite number.

Edit: See, now you got me writing like Larry

I think this is a situation I can choose to be ignorant of, and not experience any repercussions. This debate is without bounds.

BTW, if I wanted to say that a value approaches one but never actually reaches it (as in a function), what number would I use?

1/3.

I :applause: that answer even though it's wrong. I'm the green one>>:saber:

[Mark The Homer]

I think this is a situation I can choose to be ignorant of, and not experience any repercussions. This debate is without bounds.

Except that it's not a debate. It's about trying to hammer a fact into your head.

BTW, if I wanted to say that a value approaches one but never actually reaches it (as in a function), what number would I use?

You wouldn't use a number. You'd have to use a limit of a function.

[jnhay]

Except that it's not a debate. It's about trying to hammer a "fact" into your head.

That's better.

[Mark The Homer]

That's better.

Okay. Find a fault with this:

I say they are equal. There's a funner geometric series proof, but I don't have time to post it now, so I'll update later. Here's one though I really like.

x = .9 repeating

10x = 9.9 repeating

10x -x = 9.9 repeating - .9 repeating

9x = 9

x = 1

so 1 = .9 repeating

You won't be able to. Which means you have to accept it.

[jnhay]

Okay. Find a fault with this:

You won't be able to. Which means you have to accept it.

I accept that algebra isn't perfect.

[Mark The Homer]

I accept that algebra isn't perfect.

lol

I give up.

[jnhay]

lol

I give up.

I warned you didn't I?

[Mark The Homer]

I admit defeat.

You win.

[PokerPacker]

http://www.cut-the-knot.org/arithmetic/999999.shtml

Check out the part in bold.

Its like you're not accepting proofs as proof.

yes, I know what convergence is. As a matter of fact, I just had an exam that covered convergence and guess what? It deals in limits!