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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[craigsboyd]

AngelB,

"I don't wish to untwist your head again, but if you had trouble understanding why 0.999999... = 1, don't you have the same problem with 1/3 = 0.3333333333...?"

I would have, yes. But, I can visualize something being divided into 3 parts, and those 3 parts constituting the whole. This allowed me to at least understand the concept of 0.999999... = 1, whereas before I was completely bewildered by it.

I will grant you that logically, I should be just as uncomfortable with 0.333333... = 1/3 as I was with 0.999999... = 1, however since my brain has absolutely no problem equating 0.333333... with 1/3, it allows me some new found level of comfort with 0.999999... = 1. That simple example was all it took.

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[AngelB]

Fair enough mate

Gez



Let's think the unthinkable, let's do the undoable, let's prepare to grapple with the ineffable itself, and see if we may not eff it after all

 

[strongm]

>Carl Sagan once noted

He was simplifying things ...

 

[Trevoke]

QED is stands for "quod erat demonstrandum" which is Latin for 'thus it has been demonstrated'. It is conventionally used to signify the end of a proof.

Actually, it stands for "What was to be demonstrated", thus signifying that if you have poor memory, looking at the beginning of the proof should tell you that we've now come to the end.

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[GwydionM]

Don't confuse a third with the representation of a third. Calculator displays have to round, so 0.333333 become maybe 0.33. 0.999999 becomed 1.00.

I got this accidentally, years ago, writing an on-line calculator before such things were standard. The trick is to have work-fields with more decimal places than the display.

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