# 0.9... = 1

• 164 Replies
• 24085 Views
?

#### Erasmus

• The Elder Ones
• 4242 ##### 0.9... = 1
« Reply #150 on: November 11, 2006, 06:19:35 PM »
Quote from: "Curious"
But does that prove there are more?

It proves that there are more in the sense that, in any way you might try to put them in a one-to-one correspondence -- that is, in any way that you might try to pair them off uniquely -- you will always have some real numbers left unpaired.

Quote
How is infintity less than infinity^2?

It isn't; it isn't really even well-defined to talk about infinity^2.  There are two different infinite "sizes" that we are comparing: one is called "aleph-naught" and is the number of natural numbers; the other is called "c" and is the number of real numbers.  The former is less than the latter.
Why did the chicken cross the Möbius strip? ##### 0.9... = 1
« Reply #151 on: November 12, 2006, 04:58:38 AM »
Quote from: "Curious"
But does that prove there are more?

Well, you need to be clear about what we mean by "more" in the setting of infinite quantities. If you have two infinite sets, A and B, we say that A and B have the same size, (also referred to as "cardinality", and sometimes written A~B) if there is a one to one correspondance between the members of A and the members of B. For example, the set of natural numbers and the set of even natural numbers have the same size, because multiplication by two gives a one-to-one correspondance between the set of all natural numbers and the set of even numbers.

So what erasmus proved is that there is no one-to-one correspondance between the set of real numbers between 0 and 1 and the set of natural numbers, by assuming there was, and then getting a contradiction.

We can say that one set is B larger than another set A if there is no one-to-one correspondance between A and B, but there is a one-to-one correspondance between A and some subset of B. It doesn't take much more work to show that there are actually more real numbers than natural numbers (for example, arctan(pi/2-pi*x) gives a nice one-to-one correspondance between (0,1) and the set of all real numbers, so the set of all real numbers is not in one-to-one correspondance with the set of natural numbers, but contains the set of natural numbers as a subset.)

Quote
How is infintity less than infinity^2?

As erasmus mentioned, infinity is not a number, so you can't take it's square. One thing you could do is ask if an infinite set A is smaller than the set of ordered pairs (a,b) of elements of A. This latter set is often denoted AxA or A^2, by analogy to finite quantities, since there are n^2 ordered pairs of a set containing n elements. It turns out that A and AxA always have the same cardinality if A is an infinite set, so in that sense, you could say that infinity is not less than infinity^2. However, the most important result of the definitions we have been using (which are due to Georg Cantor) is that some infinite sets are larger than others, and in fact there is an infinite class of the sizes of sets.
-David
E pur si muove!

?

#### Curious

• 413 ##### 0.9... = 1
« Reply #152 on: November 12, 2006, 05:57:21 PM »
Quote from: "skeptical_scientist"
[.... However, the most important result of the definitions we have been using (which are due to Georg Cantor) is that some infinite sets are larger than others, and in fact there is an infinite class of the sizes of sets.

I understand the concept, but like the idea of 0.999... = 1, it is more because someone decides it was better to accept the idea, than it is more valid the the alternative.  To my mind, 1 - 0.999... = 1.0 ^ 10^-Infinity ( can't get my keyboard to spin the 8 key  : ) ) is equally valid.

It's all in how you set the conditions.  It's like saying 2 + 2 = 5 (for sufficiently large values of 2) , or the joke about the engineer and the mathematician (the one that goes on about how every second you cover half the distance to your girl friend)

?

#### Curious

• 413 ##### 0.9... = 1
« Reply #153 on: November 12, 2006, 06:00:18 PM »
Quote from: "Erasmus"
Quote from: "Curious"
I'll buy the 0.999... = 1, but not the 0.333... = 1

Ah, hm, sorry 'bout that.... should have been, in all cases, 0.333... = 1/3.

Being human is nothing to be sorry about.

?

#### BOGWarrior89

• 3793
• We are as one. ##### 0.9... = 1
« Reply #154 on: November 12, 2006, 08:41:37 PM »
Quote from: "Erasmus"
I'm just genuinely curious as to what kind of practical application this subject has.

[long, boring, and seemingly useless proof]

Yes, I read it, but I don't understand why you couldn't do it this way:

On interval (0,1), there is/are exactly 1 or 2 (depends on the definition in use for "natural numbers") natural number(s), but there are infinitely many real numbers.

Quote from: "Erasmus"
(this proof is dedicated to BOGWarrior89)

Insult-thrower?  If so, uneffective. ##### 0.9... = 1
« Reply #155 on: November 13, 2006, 01:34:39 AM »
Quote from: "BOGWarrior89"
Quote from: "Erasmus"
I'm just genuinely curious as to what kind of practical application this subject has.

[long, boring, and seemingly useless proof]

Yes, I read it, but I don't understand why you couldn't do it this way:

On interval (0,1), there is/are exactly 1 or 2 (depends on the definition in use for "natural numbers") natural number(s), but there are infinitely many real numbers.

Actually on the interval (0,1) there are no natural numbers, because (0,1) doesn't contain its endpoints, 0 or 1.

Anyways, that shows there are more real numbers on the interval (0,1) then there are natural numbers on the interval (0,1). What Erasmus showed was there are more real numbers between 0 and 1 than there are natural numbers of any size. Both quantities are infinite, but one is larger than the other.
-David
E pur si muove!

?

#### BOGWarrior89

• 3793
• We are as one. ##### 0.9... = 1
« Reply #156 on: November 13, 2006, 08:15:34 AM »
Quote from: "skeptical_scientist"
Quote from: "BOGWarrior89"
Quote from: "Erasmus"
I'm just genuinely curious as to what kind of practical application this subject has.

[long, boring, and seemingly useless proof]

Yes, I read it, but I don't understand why you couldn't do it this way:

On interval (0,1), there is/are exactly 1 or 2 (depends on the definition in use for "natural numbers") natural number(s), but there are infinitely many real numbers.

Actually on the interval (0,1) there are no natural numbers, because (0,1) doesn't contain its endpoints, 0 or 1.

Anyways, that shows there are more real numbers on the interval (0,1) then there are natural numbers on the interval (0,1). What Erasmus showed was there are more real numbers between 0 and 1 than there are natural numbers of any size. Both quantities are infinite, but one is larger than the other.

Couldn't you have also turned them into functions and used L'Hospital's Rule?

On a side note, I knew what he was doing.  Oh, AND I agree with it. ##### 0.9... = 1
« Reply #157 on: November 13, 2006, 10:23:33 AM »
Quote from: "BOGWarrior89"
Couldn't you have also turned them into functions and used L'Hospital's Rule?

On a side note, I knew what he was doing.  Oh, AND I agree with it.

I have no idea what you mean by "them", but l'Hopital's rule does not apply in any way, as it talks about the limits of sequences, and this is about sizes of infinite sets. Remember that l'Hopital's rule doesn't actually let you take the ratio of infinite quantities, but merely lets you compute the limit of a ratio, where both the numerator and denominator tend towards infinity. Here we are actually directly comparing the size of infinite quantities.
-David
E pur si muove!

?

#### BOGWarrior89

• 3793
• We are as one. ##### 0.9... = 1
« Reply #158 on: November 13, 2006, 11:03:17 AM »
Quote from: "skeptical_scientist"
Quote from: "BOGWarrior89"
Couldn't you have also turned them into functions and used L'Hospital's Rule?

On a side note, I knew what he was doing.  Oh, AND I agree with it.

I have no idea what you mean by "them", but l'Hopital's rule does not apply in any way, as it talks about the limits of sequences, and this is about sizes of infinite sets. Remember that l'Hopital's rule doesn't actually let you take the ratio of infinite quantities, but merely lets you compute the limit of a ratio, where both the numerator and denominator tend towards infinity. Here we are actually directly comparing the size of infinite quantities.

Damn.  Nevermind.

?

#### BOGWarrior89

• 3793
• We are as one. ##### 0.9... = 1
« Reply #159 on: November 13, 2006, 11:05:29 AM »
This method only seems to work for 0.9999... variances (9.999..., 99.999..., etc.).  I tried it with others, and this is what I got:

12.121212... = 400/33
3.3333... = 30/9

But, whenever I did this method with the aforementioned 0.999... variances, I always got some power of ten.

?

#### Erasmus

• The Elder Ones
• 4242 ##### 0.9... = 1
« Reply #160 on: November 13, 2006, 12:29:06 PM »
Quote from: "Curious"
I understand the concept, but like the idea of 0.999... = 1, it is more because someone decides it was better to accept the idea, than it is more valid the the alternative.

Hmm... I don't think that's the case.  Try to remember that any "decimal number", by which I mean a string of decimal digits, optionally followed by a decimal point followed by an optionally-infinite string of digits, means something quite particular: it refers to an infinite sum as I have discussed above.  In some cases of infinite sums, the sum has no value, but in the case of decimal numbers, they always have an exact value.  In some cases, the exact value of the sum happens to be the ratio of two integers.  In the specific case of the string "0.999..." where we take "..." to represent an infinite string of 9s, the exact value of the sum (computing by conventional means -- in other words, not simply picked for convenience) is the integer 1.

It's a consequence of existing mathematics, not somebody's decision that it was "just better that way".  It's a consequence in the sense that if it didn't take you any time to add numbers together, you could actually add up all the terms (9/10, 9/100, 9/1000, etc) in the infinite sum, and you would get the number 1.

Quote
To my mind, 1 - 0.999... = 1.0 ^ 10^-Infinity ( can't get my keyboard to spin the 8 key  : ) ) is equally valid.

It's understandable that to your mind they might be equally valid, but that's because you are not insisting that every symbol you use has a well-specified meaning, and that that meaning combines in a well-specified way with the meanings of other symbols in context.  In other words you have not specified what "Infinity" means, nor what it means for Infinity to be the exponent of a number in an expression.

Some less important comments on that statement: 1.0^(anything) is 1.0.    Your equation is, in fact, false, since 1 - 0.999.... = 0, but the stuff on the right-hand-side, if we assume it to have a value at all, is probably going to have the value 1.

Quote
It's all in how you set the conditions.  It's like saying 2 + 2 = 5 (for sufficiently large values of 2) , or the joke about the engineer and the mathematician (the one that goes on about how every second you cover half the distance to your girl friend)

Well, you could pick other values for the symbol "2" such that the sentence "2 + 2 = 5" is true, but it would probably make other sentences very false, such as "2^2 = 4".  It all comes back to having formal, consistent definitions for all our symbols, as well as a "calculus" for combing those symbols into larger expressions and getting consistent meanings for those expressions.  So yes, you can pick any definition of the symbols that you want, but some will be better than others (some will be inconsistent; some will be consistent but very boring).  The set of definitions and calculus that modern mathematicians use is a rather good one -- it's powerful enough to talk about how to build bridges so that they don't collapse, and it seems so far to be consistent -- and one of its consequences is that 0.999... = 1.
Why did the chicken cross the Möbius strip?

?

#### Erasmus

• The Elder Ones
• 4242 ##### 0.9... = 1
« Reply #161 on: November 13, 2006, 12:32:19 PM »
Quote from: "BOGWarrior89"
This method only seems to work for 0.9999... variances (9.999..., 99.999..., etc.).  I tried it with others, and this is what I got:

12.121212... = 400/33
3.3333... = 30/9

But, whenever I did this method with the aforementioned 0.999... variances, I always got some power of ten.

What method are you talking about, exactly?  If you mean the "multiply by a power of ten and subtract the original" method, then:

x = 0.999...
(10^n)x - x = 99..9.999.... - 0.999... = 99..9   (where .. is a string of n-3 9s)
99..9x = 99..9
0.999... = x = 1.

No matter what power of ten you chose to multiply 0.999... by (in other words, no matter what "variance" you pick in your lingo), you should always arrive at 0.999... = 1.
Why did the chicken cross the Möbius strip?

?

#### Curious

• 413 ##### 0.9... = 1
« Reply #162 on: November 14, 2006, 09:32:25 AM »
Quote from: "Erasmus"
Quote from: "Curious"
I understand the concept, but like the idea of 0.999... = 1, it is more because someone decides it was better to accept the idea, than it is more valid the the alternative.

Hmm... I don't think that's the case.  T...Some less important comments on that statement: 1.0^(anything) is 1.0.    Your equation is, in fact, false, since 1 - 0.999.... = 0, but the stuff on the right-hand-side, if we assume it to have a value at all, is probably going to have the value 1.[\quote]

My mistake. I mistyped, by "1 - 0.999... = 1.0 ^ 10^-Infinity " I meant "1 - 0.999... = 1.0 * 10^-Infinity"

I know, how can an infinite string of 0's be followed by anything? But how can you have an infinite string of anything?

Quote
Quote
It's all in how you set the conditions.  It's like saying 2 + 2 = 5 (for sufficiently large values of 2) , or the joke about the engineer and the mathematician (the one that goes on about how every second you cover half the distance to your girl friend)

Well, you could pick other values for the symbol "2" such that the sentence "2 + 2 = 5" is true, but it would probably make other sentences very false, such as "2^2 = 4".  It all comes back to having formal, consistent definitions for all our symbols, as well as a "calculus" for combing those symbols into larger expressions and getting consistent meanings for those expressions.  So yes, you can pick any definition of the symbols that you want, but some will be better than others (some will be inconsistent; some will be consistent but very boring).  The set of definitions and calculus that modern mathematicians use is a rather good one -- it's powerful enough to talk about how to build bridges so that they don't collapse, and it seems so far to be consistent -- and one of its consequences is that 0.999... = 1.

The 2 +2 = 5 is a standard, at least for us computer geeks.  Since for "display purposes" you normally have some kind of rounding, but for calculations you may be working with higher levels of precision, you often have to explain to users why the see something similar to 2 +2 = 5, where the more precise equations might be 2.4908 + 2.34653

?

#### Erasmus

• The Elder Ones
• 4242 ##### 0.9... = 1
« Reply #163 on: November 14, 2006, 03:37:15 PM »
Quote from: "Curious"
The 2 +2 = 5 is a standard, at least for us computer geeks.

I'm familiar with the joke; in fact, I own the t-shirt on which it is printed.  That said, it is a joke and by no means an accepted-as-true statement among experts in the field of computing -- it isn't anything that resembles "a standard".
Why did the chicken cross the Möbius strip?

?

#### Curious

• 413 ##### 0.9... = 1
« Reply #164 on: November 14, 2006, 08:41:57 PM »
Quote from: "Erasmus"
Quote from: "Curious"
The 2 +2 = 5 is a standard, at least for us computer geeks.

I'm familiar with the joke; in fact, I own the t-shirt on which it is printed.  That said, it is a joke and by no means an accepted-as-true statement among experts in the field of computing -- it isn't anything that resembles "a standard".

The joke was around before I started working in the field, and that was more than 20 years ago.  A having had to explain to people that just because their web report shows whole cents, it does not mean that that is how the computer calculated it, so when their totals don't match what they see I use the line to get their attention.  Don't knock it, it works.