Is 1 = 0.9999......

[silverpig]

How is it closeminded to be correct?


What if I was to argue out of ignorance that the walls in your room are orange and yellow? I could state over and over that they HAVE to be that colour because all walls MUST be that colour. Even if you posted pictures, sent me paint flakes, whatever, I would still argue. How does that make YOU closeminded?

 

[bleeb]

Originally posted by: silverpig
How is it closeminded to be correct?


What if I was to argue out of ignorance that the walls in your room are orange and yellow? I could state over and over that they HAVE to be that colour because all walls MUST be that colour. Even if you posted pictures, sent me paint flakes, whatever, I would still argue. How does that make YOU closeminded?

please calm down. i apologize if I've offended you.

 

[Kyteland]

Originally posted by: bleeb

please calm down. i apologize if I've offended you.

You have offended me with your claim that 0.99...!=1. Please apologize and retract your claim. 😉

 

[silverpig]

Originally posted by: bleeb


please calm down. i apologize if I've offended you.

I'm not upset. I was just drawing an analogy.

 

[bleeb]

Originally posted by: Kyteland

Originally posted by: bleeb

please calm down. i apologize if I've offended you.

You have offended me with your claim that 0.99...!=1. Please apologize and retract your claim. 😉

If you look at all those proofs, there is someplace in the steps where "error" is introduced. i.e. an approximation is made which cancels the infinite sequence of digits past the decimal. Therefore, I still propose that 0.9999... != 1.

Think about it. 1/3 = 0.3333.... YOU are losing the precision there!

Check all of your proofs.

don't use math.. use your reasoning

 

[silverpig]

What about the ones showing that there can not be any number in between 0.999... and 1? They don't do any mathematical operations on any of the numbers.

 

[bleeb]

Originally posted by: silverpig
What about the ones showing that there can not be any number in between 0.999... and 1? They don't do any mathematical operations on any of the numbers.

Why does there have to be a number between those numbers? If you go infinitely into the decimals... you reach the limit which is 1. this doesn't mean that it is EXACTLY 1. 0.9999... = 1 says it EQUALS exactly 1. And besides, since you are going infinitely towards 1, 0.9999... simply is the point before you reach 1.

 

[Haircut]

don't use math.. use your reasoning

I really don't know how to respond to this one.

If we are demonstrating a mathematical proof then we have to use math. If the result is completely counter-intuitive, but can be proven then we have to accept it as true.

 

[Haircut]

Why does there have to be a number between those numbers?

If we have numbers x and y, where y>x then
y = x + d, where d>0

Now, we let z = d/2, z>0

Now, x < x+z < y

i.e. I can place a number between any two numbers that are different.
From this, the only time I cannot get a number that is between x and y is the case where x = y

 

[DrPizza]

OMG, I run into innumeracy while teaching all the time! Now, while I take a brief rest from fighting innumeracy in the high school, I find that this thread is filled with innumerates (and fortunately with many of you who do have some math skills.)

Since it isn't a democracy, the vote at the beginning of this thread doesn't matter.

.9999999 repeating equals exactly 1.0000000000000 no more, no less. If you disagree, you're a moron. It's been proven plenty of times. It can be proven in several different mathematically sound ways. Accept it.

But, wanting to add to the headaches of those innumerates among you, let me toss out this for you to ponder.

There are as many positive even integers as there are positive integers.
That is:
2,4,6,8,10,...... has an infinite number of elements.
1,2,3,4,5,6..... also has an infinite number of elements.
And, both of these infinities are the same size. There aren't twice as many integers as there are even integers. For every integer, there is a corresponding even integer. (multiply that integer by 2). For every even integer, there is a corresponding integer (divide that integer by 2). So, there is an even pairing between the two sets of elements.

Now, for the fun part. While *those* two infinities are the same size, there ARE different sizes of infinity.
Ponder that for a bit... get an interesting math book to read... and take 2 or 3 tylenol for the headache.

 

[bleeb]

Originally posted by: Haircut

don't use math.. use your reasoning

I really don't know how to respond to this one.

If we are demonstrating a mathematical proof then we have to use math. If the result is completely counter-intuitive, but can be proven then we have to accept it as true.

Sorry, I guess I meant to say that sometimes you can't solely base your belief based on numbers. Instead, (in this case) use reasoning and then use mathematics to prove it.

 

[silverpig]

Originally posted by: bleeb

Originally posted by: silverpig
What about the ones showing that there can not be any number in between 0.999... and 1? They don't do any mathematical operations on any of the numbers.

Why does there have to be a number between those numbers? If you go infinitely into the decimals... you reach the limit which is 1. this doesn't mean that it is EXACTLY 1. 0.9999... = 1 says it EQUALS exactly 1. And besides, since you are going infinitely towards 1, 0.9999... simply is the point before you reach 1.

That's the whole point though. There is no "point before you reach 1".

Let's call this point P and give it a value of x.

I can ALWAYS find a point Q that is between 1 and P by defining Q to be:

(1 - x) / 2

Basically this shows that no matter how close you define P to be to 1, I can always find something closer (in fact, I can find an infinite amount of numbers that are closer).


In english this can be summarized as follows:

"If A and B are not different, they must be the same."

So if there is no difference between A and B (ie, there isn't any number in between them), then A = B.

 

[bleeb]

HOW is it possible to have different sized of infinity???? please explain.

0.9999.... != 1

 

[bleeb]

In english this can be summarized as follows:

"If A and B are not different, they must be the same."

So if there is no difference between A and B (ie, there isn't any number in between them), then A = B.

There is a difference... that difference is infinitely small. hehe

 

[silverpig]

Originally posted by: bleeb
HOW is it possible to have different sized of infinity???? please explain.

0.9999.... != 1

There are actually different sizes of infinity, called cardinality. Aleph-null is the first size of infinity, corresponding to the number of integers. Aleph-one is the next level. I'll try and come up with an explanation if I can find it somewhere here...

 

[bleeb]

Originally posted by: silverpig

Originally posted by: bleeb
HOW is it possible to have different sized of infinity???? please explain.

0.9999.... != 1

There are actually different sizes of infinity, called cardinality. Aleph-null is the first size of infinity, corresponding to the number of integers. Aleph-one is the next level. I'll try and come up with an explanation if I can find it somewhere here...

So if Aleph-Null is the entire set of infinitely, none repeating integers... then how can you have a higher set?The set not included in the set Aleph-null?

 

[Haircut]

Originally posted by: bleeb

Originally posted by: silverpig

Originally posted by: bleeb
HOW is it possible to have different sized of infinity???? please explain.

0.9999.... != 1

There are actually different sizes of infinity, called cardinality. Aleph-null is the first size of infinity, corresponding to the number of integers. Aleph-one is the next level. I'll try and come up with an explanation if I can find it somewhere here...

So if Aleph-Null is the entire set of infinitely, none repeating integers... then how can you have a higher set?The set not included in the set Aleph-null?

Yep, you can prove this by contradiction.
Assume that the interval [0,1] (all the numbers from 0 to 1 inclusive) is countably infinite (has Aleph-null elements)
Then we could assign an order to the elements in this interval (i.e. they correspond to the number of integers as Silverpig posted)

Each number in this interval would be n1 = 0.a11 a12 a13 a14 a15
where the a11, a12 are digits in the decimal expansion.

However, we can then define a number than differs from nk in the kth decimal place.
i.e our new number would have a11 different to n1

Thus we cannot give a 1 to 1 correspondence with the integers, this set is uncountably infinite which is a bigger infinity than countably infinite.


Reading through this again it isn't explained very well, but I hope you get the gist of it 🙂

 

[bleeb]

i think i remmeber something about this in my combinatorics class.

 

[silverpig]

Let's take 3 playing cards. Since I can't show them on here, we'll let 1 = a face up card, and 0 = a face down card. For 3 cards we have 8 possibilities:

000
100
010
001
110
101
011
111

There are 2^n ways to arrange the cards (where n is the number of cards).

Now, let's try that with an infinte deck of cards:

1000101001...
0110100010...
1000101010...
1001101011...
0001010011...
etc...

(note: I'm doing it in a random order because they would all look the same if I was to order them).

The number of columns indicate the number of cards, and the number of rows indicate the number of ways to arrange the cards. Obviously, there are an infinite number of each. If we have an infinite number of rows, then we must be able to represent EVERY possible combination of face-up and face-down cards right? WRONG.

Let's consider the following arrangement:


0000101001...
0010100010...
1010101010...
1000101011...
0001110011...

Notice that this set is exactly the same as the previous one, except for the bolded numbers. All I have done is gone down the diagonal and flipped every card. This diagonal must represent a way of ordering the cards right? After all, it consists of an infinite number of cards either face-up or face-down.

Now comes the cool part. This new diagonal is NOT a member of our original set. Why? Well, it can't be a copy of the first row, because the first card differs. It can't be a copy of the second row because the second card differs. Further, it can't be a copy of the nth row, because the nth card differs. We have just found a way of arranging the cards in such a way that was not included in our infinite set. We have produced a subset that cannot be in our previous list, despite the previous list being infinite.

This shows that such a set cannot be matched one to one with the integers, and is therefore of a different cardinality.


edit: fixed boldness

 

[bleeb]

sweet....I get it.

 

[Haircut]

/ \
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That's what I was trying to say, but you did it better.

 

[Kyteland]

Originally posted by: silverpig

Originally posted by: bleeb
HOW is it possible to have different sized of infinity???? please explain.

0.9999.... != 1

There are actually different sizes of infinity, called cardinality. Aleph-null is the first size of infinity, corresponding to the number of integers. Aleph-one is the next level. I'll try and come up with an explanation if I can find it somewhere here...

Link

This kind of explains the background of cardinality. The author also brushes over the 0.99...=1 thing

There is a slight oversimplification here: it is possible to write .499999.... as a real number, but that is actually the same real number as .5. This problem can be dealt with, though, so it doesn't spoil the proof.

 

[bleeb]

BUT, I'm still holding on to the belief that 0.9999... != 1. (Until the 1000th post)😉

 

[Kyteland]

Originally posted by: bleeb
BUT, I'm still holding on to the belief that 0.9999... != 1. (Until the 1000th post) 😉

Man, I already "paraphrased" your previous post to prove that you actually think 0.99...=1, regardless of anything else you say. My logic is immuatble. Give it up already 😛

 

[JupiterJones]

Originally posted by: silverpig

Originally posted by: bleeb
HOW is it possible to have different sized of infinity???? please explain.

0.9999.... != 1

There are actually different sizes of infinity, called cardinality. Aleph-null is the first size of infinity, corresponding to the number of integers. Aleph-one is the next level. I'll try and come up with an explanation if I can find it somewhere here...

I've read about this mumbo jumbo being put out by set theorist, but it won't last. Infinity = Infinity. 2 X Infinity = Infinity. Cardinality of the set of even positive integers = Infinity. Cardinality of the set of positive integers = Infinity. Infinity = Infinity.

Karl Friedrich Gauss noted that while one can think of an object like the natural numbers always reaching towards infinite size, thinking about infinity as something that has been reached, as if it were existing in a finished form was contrary to the principles of mathematics. THIS is the error of the set theorist.