The Language of Mathematics 1

[CajunCenturion]

Multiplying by 10 increases by 1 order of magnitude, not 1 order of magnitude plus .000...1.

That is correct. There is no addition of .000...1. But remember that 0.9999... is repeating infinitely before the multiplication and it is repeating just the same after the multiplication. The multiplication by 10 has no effect on the repeating non-terminating decimal portion of the value.
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You can have infinites within infinities. You an infinty +1, etc. You can have infinity x 2, which is an infinite series followed by another infinite series.

Yes, you can have infinities within infitinities. They are know as degrees of infinity. However, infinity, infinity + 1, and infinity * 2, are the same infinity, or to be more precise, all these infinities are of the same degree, or those sets of numbers have the same cardinality. If a 1-to-1 mapping can be established between two infinities, then the two infinities have the same cardinality. The cardinalilty is know by the Hebrew Letter Aleph ([ℵ])

The smallest infinity is the set of integers, or counting numbers, and has cardinality of [ℵ]₀ (Aleph-Null). The set of integers, odd integers, even integers, integers + 1, and integers * 2 all have the same cardinality because a 1-to-1 mapping can be established with the integers. The values on either side of the mapping do not have to be, and usually aren't the same, but as long as the 1-to-1 mapping exists, the size of the infinity is the same.

There are just as many even integers as there as integers. The mapping is E_n = I_n * 2 or conversely I_n = E_n [÷] 2
[tt]
Integer Even Integer
1 2
2 4
3 6
4 8[/tt]
For every integer there is exactly one corresponding even integer, and vice versa; for every even integer, there is exactly one corresponding integer. The values of corresponding entries are not equal, in this mapping, the even value is twice that of the integer value, but both sets still have the same number of terms. There is no integer that does not have corresponding even integer, nor is there an even integer that does not have a corresponding integer. That means that a 1-to-1 mapping exists, therefore the numbers of integers is same as the number of even integers, or mathematically speaking, both sets have the same cardinality, which is [ℵ]₀. When dealing with the cardinality of infinities, it is not the values of the terms that matter, it is the number of terms that matter.

Now, can we establish a 1-to-1 mapping between the integers and the real numbers. Let's start with the integer 1 which maps to the real number 1. What is the next highest real number, i.e., the real number that will map to the integer 2? It is unknown. No matter what answer is submitted, there is a real number halfway between that value and 1. Therefore, you cannot map the real numbers to the integers and since the real numbers is clearly the larger set, the real numbers have a higher degree of infinity than the integers. The set of real numbers has a cardinality of [ℵ]₁ (Aleph-One).

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[tgreer]

Can you give an example of two infinities that don't have the same cardinality?



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[tgreer]

Or did you? Re-reading, it seems that the set of real numbers is "infinitely" bigger than the set of integers.

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[CajunCenturion]

Yes, that is correct. The set of real numbers is infinitely larger than the infinity for the set of integers. One more, the set of all strings is an infinity larger than the set of real numbers. The simplified proof is along the following lines. Take any given string and pick an arbitrary midpoint and call that 0. On just one string, every possible distance, positive and negative, can be marked with a real number. You can cut any interval in half, and correspondingly so, the associated real number. That means there is a 1-to-1 mapping between the real numbers and the points on a single string. However, there are an infinite number of possible strings, each with its own individual mapping to the real numbers. That means for every real number, there is not one, but an infinite number of corresponding string points -- one one point per string, but an infinite number of strings, therefore an infinite number of single points. Since that is a 1-to-[∞] mapping, the set of strings must be infinately larger than the set of real numbers, thus it must have a higher cardinalty: [ℵ]₂ (Aleph-Two)

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[tgreer]

I understand all of that. I'm still struggling with the concept of

= 1.

I'm stuck on the visualization of a set of finite values, and every single one of those finite values is less than 1. I realize, intellectually, that we're not talking about any given finite value, but rather the totality we call

.

I'm used to the concept of any number, such as "1", as the "embodiment" of a particular set of infinite real numbers. I have trouble with the reverse, namely, that a particular infinity is equal to a whole number.

I have a highly kinesthetic way of thinking. When I program, I truly see (I'm not using an analogy here), translucent arrows, perhaps forming a loop, with blocks, spikes, and other shapes, each representing a condition or flag.

Coworkers have described me as making certain hand or arm gestures, which they've come to recognize as a "loop" or an "if/else".

That's why I wrote, in the "UP" thread, that words are vectors or chemical compounds. I see them as in motion at all times. Even when, as Esquared wrote, they are "instantiated", I don't see them sitting still, confined to a specific meaning. They are very nearly vibrating with all of their other meanings and connotations. The very act of supplying a context subtly alters all future meanings. He wrote that if two differently spelled words can't mean the same thing, then neither can two "samely" spelled words. I'm sure that was meant as a counter-argument, but I think it's actually so.

What I'm getting to, is that thinking of

as a process, rather than a static value, comes close to reconciling the issue for me.

is the iterative process which BUILDS the value "1".

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[CajunCenturion]

==> What I'm getting to, is that thinking of as a process, rather than a static value, comes close to reconciling the issue for me. is the iterative process which BUILDS the value "1".

Actually, that's a good way to look at it. Remember, in the "UP" thread, I referred to it a infinite convergent series with a finite sum.

0.999... can be shown as

9/10 + 9/100 + 9/1000 + 9/10000 + 9/100000 + ...
[tt]
sum = 0;
for pwr = 1; pwr < [&infin;]; pwr++)
sum += 9/(10 ^ pwr);
[/tt]
Granted, the compiler might have some trouble with [&infin;], and I'm using ^ instead of the pow() function, and it is an infinite loop, but in the end,
[tt]sum = 1[/tt]

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[RodKnowlton]

it is an infinite loop, but in the end,
sum = 1

Either it is infinite, or it has an end.

Your explanation of Aleph-One cardinality says it best: You can always name a number half way (or in this case 9/10s the way) between it and one.

For any n digits of .999..., you are precisely 10^-n away from one, but you only add 9x10^-n-1. There are always an infinite number of nameable real numbers between the sum and one.

sum = 1 - 10^-[&infin;]

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[CajunCenturion]

==&gt; For any n digits of .999...
.999... doesn't have n digits, it is a non-terminating repeating decimal. It is infinite. It has [8] digits. Mathematically, convergant infinite series do have finite sums. In the above post, I was trying to help tgeer understand that by manufacturing a &quot;loop&quot; to help show the summation is an infinite process, but nevertheless, with a finite sum. Mathematically, that is shown as

[&amp;Sigma;]^{[&amp;infin;]} 1 [&amp;divide;] 10ⁿ
[&amp;nbsp;]ⁿ⁼¹

==&gt; You can always name a number half way (or in this case 9/10s the way) between it and one.
There can only be an infinite number of real numbers between the sum and one if and only if you terminate the series, and I'm not, nor ever have I, terminted the series.

For any n digits of 0.999... is a terminating decimal. It is a finite value. It is less than one.

0.999... is a repeating decimal. Is it a convergant infinite series, and it's value (the sum of the series) is 1.

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[tgreer]

It's strange to say, but I don't actually have a problem with "it goes on forever, but in the end...". It all depends on the point of observation, or how you choose to look at it.

Consider how computers produce a circle. A circle is a final shape, completely bounded. Yet it is built from short line segments, starting with a square. The process is infinite, however, each device decides how far to go before calling it quits and saying "this approximation of a circle is as close as I can achieve".

The infinite loop builds the circle. When it's "done", you have a circle. From the point of view of an observer standing outside of time, they could say that "that algorithm, when it finishes, will produce a circle."

That's how I understand the statement:

= 1.

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[Chacalinc]

do you know? for me, the statement 0.9999... = 1 is the justification for the limitation of the human brain against the infinity of the universe.

Can anyone just imagine the infinity?

Cheers.

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[tgreer]

No one has proven that the human brain has any limits, nor that the universe is infinite. In fact, the universe is probably more comprehensible than the brain.



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[Zathras]

I may be too simple minded, but this convinces me:

if x = .999999999......

then 10x = 9.99999999......

now 10x - x = 9x
and 9.99999.... - .999999.... = 9 (exactly)

so 9x = 9

Divide both sides by 9 and you have

x = 1

thus .99999999999...... = 1 Q.E.D.

 

[craigsboyd]

I feel this only shows that subtraction and division are not suited for infinite numbers. Just as division cannot readily handle a divisor of 0.

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[strongm]

The Zathras illustration is the one I used when this came up in the early days of the Squaring the Circle forum, and seems convincing enough to me ... Just because the result seems counterintuitive does not mean that the maxims of mathematice are wrong (statistics, for example, often provides counterintuitive results)

 

[CajunCenturion]

This topic originated in the 'UP' thread and two additional 'proofs' were presented. For continunity, the other two are the following:

[tt]0.111… = 1/9
0.222… = 2/9
0.333… = 3/9
0.444… = 4/9
0.555… = 5/9
0.666… = 6/9
0.777… = 7/9
0.888… = 8/9
0.999… = 9/9 = 1[/tt]

and

[tt] 1/3 = 0.333…
+ 1/3 = 0.333…
+ 1/3 = 0.333…
================
1 = 0.999…[/tt]

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[strongm]

Ah - sorry, I'd given up on the "UP" thread some time ago. Didn't realise it had got onto infinite series and the meaning of numbers ...

 

[BobRodes]

Perhaps the brain is the universe, and the universe the brain.

 

[CajunCenturion]

In another thread, more appropriate to this one

Plus cut the mathematics tricks using 9 Given a choice between 1 whole cake and .999999........ of the cake, I am not leaving any crumbs I will take the 1 whole cake.

I on the other hand will take the .999... because that means there will always be more cake coming.

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[tgreer]

The mathematical proof I found most convincing, was to do

1.0 / 1

Of course the easy answer is "1", and that's valid.

However, 1.0 / 1 also equals .9 remainder .1. That's valid, because .9 + .1 = 1.

But if you continue the long division, instead of leaving a remainder:

     0.999...
    ______
  1| 1.0
      .9
     ------
      .10
      .09
     ------
      .010
      .009

And so on, it never ends. And we've already shown that this isn't a trick, it's valid division.

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[CajunCenturion]

I hope that will cause the nightmares to subside.

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