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Old 2018-12-07, 23:58   #100
jvang
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Quote:
Originally Posted by Nick View Post
Suppose xilman always takes back the highest numbered coin that you have.
Then he gives you coins 1 thru 10,
takes back 10 and gives you 11 thru 20,
takes back 20 and gives you 21 thru 30, etc.
At midnight, you still have coins 1 thru 9, 11 thru 19, 21 thru 29, etc. going on forever,
which is infinitely many coins, as you said.

Suppose instead that xilman always takes back the lowest numbered coin that you have.
Then he gives you coins 1 thru 10,
takes back 1 and gives you 11 thru 20,
takes back 2 and gives you 21 thru 30, etc.
At midnight, you don't have any coins left: for any positive integer, we can calculate the
precise time at which he took the coin with that number back from you.

Now suppose instead that he gives you coins 1 thru 10,
takes back coin 8 and gives you 11 thru 20,
takes back coin 9 and gives you 21 thru 30,
takes back coin 10 and gives you 31 thru 40, etc.
At midnight, you have the coins numbered 1 thru 7 and no others.
So it might also be argued that you end up with precisely 7 coins.
And this works not just for 7 but for any positive integer!

This is why we never write ∞-∞ in mathematics: if someone gives you infinitely many objects,
and then takes infinitely many back, how many you are left with depends on which objects he took back.
Oh, cool! I never considered that the order of the coins could matter like that. Reminds me of a couple of problems involving infinity that I pose to a couple of my friends:

If I have an infinite number of $1 bills and you have an infinite number of $100 bills, who has more money?

Are there more decimal numbers between 0 and 1 or between 0 and 100? Alternatively, plot y = x. Which has more points, the line or the rest of the coordinate grid?

I’m pretty sure that I understand why the answer to both is that they are equal (amounts of money/points). But I’m not sure how to explain to my friends, who think that the amounts in both questions are unequal...

Quote:
Originally Posted by Dr Sardonicus View Post
One thing you do indeed have to be careful of is that the rule, valid with positive real numbers a and b,

\sqrt{\frac{a}{b}} \; = \; \frac{\sqrt{a}}{\sqrt{b}}

no longer works with negative real numbers. Ignoring this complication can lead to "wrong square root" fallacies. One of Matin Gardner's Mathematical Games columns had a basic one, starting with

\sqrt{\frac{-1}{1}} \; = \;\sqrt{\frac{1}{-1}}

and then cheerfully ignoring the above warning:

\frac{\sqrt{-1}}{\sqrt{1}} \; = \; \frac{\sqrt{1}}{\sqrt{-1}}

Then, "multiplying up" gives

\sqrt{-1}\times\sqrt{-1} \; = \; \sqrt{1}\times\sqrt{1}

so that

-1 \; = \; 1
I’m a fan of “proofs” based upon false assumptions. The Art of Problem Solving book that I used for algebra and trigonometry had a couple for both topics to illustrate fundamentals. One notable one tricked you into dividing by 0; I didn’t notice until the solutions part pointed it out!
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Old 2018-12-08, 09:30   #101
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Quote:
Originally Posted by jvang View Post
Are there more decimal numbers between 0 and 1 or between 0 and 100? Alternatively, plot y = x. Which has more points, the line or the rest of the coordinate grid?

I’m pretty sure that I understand why the answer to both is that they are equal (amounts of money/points). But I’m not sure how to explain to my friends, who think that the amounts in both questions are unequal...
One way to explain it is that sets A and B have the same number of elements if you can pair off
the elements of A with distinct elements of B, with none left over on either side.
In this case, the function f(x)=100x associates each number between 0 and 1 with a
distinct number between 0 and 100 and none are left over.
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Old 2018-12-08, 15:11   #102
jvang
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We finally went over the chain rule, which means that I was able to describe the quotient rule in terms of it and the product rule. We start off by using the quotient rule to have something to compare to:\[\frac{d}{dx}\dfrac{7x^2-4}{3x^2+2}=\dfrac{14x(3x^2+2)-6x(7x^2-4)}{(3x^2+2)^2}=\dfrac{42x^3+28x-42x^3+24x}{9x^4+12x^2+4}=\dfrac{52x}{9x^4+12x^2+4}\]Setting this up with the other two rules:\[\frac{d}{dx}(7x^2-4)(3x^2+2)^{-1}+\frac{d}{dx}(3x^2+2)^{-1}(7x^2-4)=\dfrac{14x}{3x^2+2}+\dfrac{-1*6x(7x^2-4)}{(3x^2+2)^2}\]Gotta have the same denominators for both terms...\[\dfrac{14x(3x^2+2)}{9x^4+12x^2+4}+\dfrac{-42x^3+24x}{9x^4+12x^2+4}=\dfrac{42x^3+28x-42x^3+24x}{9x^4+12x^2+4}=\dfrac{52x}{9x^4+12x^2+4}\]Cool! I’m not sure how someone came up with the quotient rule, which less complicated to use than combining the product and chain rules.
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Old 2018-12-09, 10:00   #103
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Quote:
Originally Posted by jvang View Post
I’m not sure how someone came up with the quotient rule, which less complicated to use than combining the product and chain rules.
We can imitate what you just did, but with unspecified functions in order to show that it works in general.
Take any differentiable functions f and g, and let \(h(x)=\frac{1}{g(x)}\) and \(F(x)=\frac{f(x)}{g(x)}\).
Then \(h(x)=g(x)^{-1}\) so, by the chain rule,
\[ h'(x)=(-1)g(x)^{-2}g'(x)=-\frac{g'(x)}{g(x)^2}\]
and \(F(x)=f(x)g(x)^{-1}=f(x)h(x)\) so, by the product rule (and using the above expression for \(h'(x)\)),
\[ F'(x)=f'(x)h(x)+f(x)h'(x)=\frac{f'(x)}{g(x)}-\frac{f(x)g'(x)}{g(x)^2}
=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}. \]
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Old 2018-12-09, 13:17   #104
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Quote:
Originally Posted by Nick View Post
One way to explain it is that sets A and B have the same number of elements if you can pair off
the elements of A with distinct elements of B, with none left over on either side.
In this case, the function f(x)=100x associates each number between 0 and 1 with a
distinct number between 0 and 100 and none are left over.
The following, from Two New Sciences by Galileo Galilei may be of interest. At the time, the concept of "orders of infinity" was unknown. Thus, no distinction is made between the argument for the positive integers, and that for line segments. Also, negative square roots were not under consideration. The complete text here includes explanatory notes which I have omitted.

As a historical note, Galileo wrote Two New Sciences after being put under house arrest by the Inquisition. He had, many years previously, done a number of experiments, not yet published. He got out his old notes, and wrote them up.
Quote:
Salv. These are some of those difficulties that derive from reasoning about infinites with our finite understanding, giving to them those attributes that we give to finite and bounded things. This, I think, is inconsistent, for I consider that the attributes of greater, lesser, and equal do not suit infinities, of which it cannot be said that one is greater, or less than, or equal to, another. In proof of this a certain argument once occurred to me, which for clearer explanation I shall propound by interrogating Simplicio, who raised the difficulty. I assume that you know quite well which are square numbers, and which are not squares.

Simp. I know well enough that a square number is that which comes from the multiplication of a number into itself; thus four and nine and so on are square numbers, the first arising from two, and the second from three, each multiplied by itself.

Salv. Very good. And you must also know that just as these products are called squares, those which thus produce them (that is, those which are multiplied) are called sides, or roots. And other [numbers] that do not arise from numbers multiplied by themselves are not squares at all. Whence if I say that all numbers, including squares and non-squares, are more [numerous] than the squares alone, I shall be saying a perfectly true proposition; is that not so?

Simp. One cannot say otherwise.

Salv. Next, I ask how many are the square numbers; and it may be truly answered that they are just as many as are their own roots, since every square has its root, and every root its square; nor is there any square that has more than just one root, or any root that has more than just one square.

Simp. Precisely so.

Salv. But if I were to ask how many roots there are, it could not be denied that those are as numerous as all the numbers, because there is no number that is not the root of some square. That being the case, it must be said that square numbers are as numerous as all numbers, because they are as many as their roots, and all numbers are roots. Yet at the outset we said that all the numbers were many more than all the squares, the majority being non-squares. Indeed, the multitude of squares diminishes in ever-greater ratio as one moves on to greater numbers, for up to one hundred there are ten squares, which is to say that one-tenth are squares; in ten thousand, only one one-hundredth part are squares; in one million, only one one-thousandth. Yet in the infinite number, if one can conceive that, it must be said that there are as many squares as all numbers together.

Sagr. Well then, what must be decided about this matter?

Salv. | don’t see how any other decision can be reached than to say that all the numbers are infinitely many; all squares infinitely many; all their roots infinitely many; that the multitude of squares is not less than that of all numbers, nor is the latter greater than the former. And in final con- clusion, the attributes of equal, greater, and less have no place in infinite, but only in bounded quantities, So when Simplicio proposes to me several unequal lines, and asks me how it can be that there are not more points in the greater than in the lesser, 1 reply to him that there are neither more, nor less, nor the same number [altrettanti, just as many], but in each there are infinitely many. Or truly, might I notreply to him that the points in one are as many as the square numbers; in another and greater line, as many as all numbers; and in some tiny little [line], only as many as the cube numbers — in that way giving him satisfaction by putting more of them in one than in another, and yet infinitely many in each? So much for the first difficulty.

Last fiddled with by Dr Sardonicus on 2018-12-09 at 13:21
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Old 2018-12-09, 15:24   #105
jvang
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Quote:
Originally Posted by Nick View Post
We can imitate what you just did, but with unspecified functions in order to show that it works in general.
I wasn’t sure of how to set up the general case, but I see that using h(x) to represent g(x)^-1 helps a lot. Makes sense now!

Quote:
Originally Posted by Dr Sardonicus View Post
The following, from Two New Sciences by Galileo Galilei may be of interest. At the time, the concept of "orders of infinity" was unknown. Thus, no distinction is made between the argument for the positive integers, and that for line segments. Also, negative square roots were not under consideration.
I don’t know what you mean by “orders of infinity,” either. Does that means that some infinities are more infinite than others?

A side note: on this day in 1968 (thanks Wikipedia!), Douglas Engelbart provided a 90 minute public demonstration of computer components that are essential to modern personal computing:
Quote:
Originally Posted by Wikipedia
windows, hypertext, graphics, efficient navigation and command input, video conferencing, the computer mouse, word processing, dynamic file linking, revision control, and a collaborative real-time editor.
The presentation is now commonly referred to as “the mother of all demos,” as a variation on “the mother of all battles.” It was coined during the 1990s and the Gulf War, where the latter phrase was translated from a warning in Arabic.

It seems that many of the presented technologies had already been developed several years before, but Engelbart was the first to put them all together in a public demonstration. At the time, computers were only used for calculations and other number crunching tasks, but Engelbart had an idea that computers could also be used to augment people's minds. He wanted to create a machine that could be used interactively to “augment their intelligence,” by which I assume he means be more productive or efficient.

The article mentions, among other things, that Engelbart's team custom made two modems running at 1200 baud (a unit of data transfer speed? Something to do with dial up?) and that they had a live video feed of Engelbart working at Menlo Park, 30 miles away. How did they do live video back then? I’m somewhat sure that it couldn’t have been over a telephone line...
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Old 2018-12-10, 09:31   #106
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Quote:
Originally Posted by jvang View Post
The article mentions, among other things, that Engelbart's team custom made two modems running at 1200 baud (a unit of data transfer speed? Something to do with dial up?)
The baud rate of a communication line gives the number of line signal transitions per second.
If the line has only 2 possible states then this is the same as the number of bits per second,
but some communication channels have more than 2 states, and then the number of bits per second is higher than the baud rate.
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Old 2018-12-10, 14:43   #107
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Quote:
Originally Posted by jvang View Post
Does that means that some infinities are more infinite than others?
The usual method to show this is to assume there exists a one-to-one relationship between the positive integers and the real numbers between 0 and 1 - so there is a listing of the the first, second, third, etc. Now construct the number whose decimal expansion at all positions is either 2 or 3 (other choices work). Decimal position "n" of this constructed number is determined by looking at the nth decimal position of the nth number on the list, and making the constructed number different. This is, the nth position of the constructed number is 2 unless that matches, in which case the constructed number is 3.

Given the list, this constructed number is a well defined number between 0 and 1 that is not on the list - a contradiction. You could make a new list with this number, but then you could construct a new number not on the new list. The list cannot exist - so there are more real numbers between 0 and 1 than there are integers. Some infinities are larger than others, and not just in the "trivial" sense that they are super-sets (like the integers contain all the even integers, but the two sets are the same size).
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Old 2018-12-10, 14:46   #108
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Originally Posted by jvang View Post
<snip> I don’t know what you mean by “orders of infinity,” either. Does that means that some infinities are more infinite than others?
<snip>
Well... yes. As indicated above, we say that two sets are "the same size" ("same cardinality") if there is a one-to-one correspondence between them.

A general result about sets shows that there is always a "bigger" set than a given set S.

Let S be a set, and 2S (sometimes, Pow(S)) the set of all subsets of S, or "power set" of S. (This set is postulated to exist for any "given" set S.) The power set is sometimes formulated as the set of all functions

 f:\;S\;\rightarrow\;\lbrace 0,1\rbrace

Each function in this set corresponds to a unique subset of S, namely

\lbrace s \; \in \; S \; \mid f(s) \; = \; 1\rbrace

A standard, easy-to-prove result is

Theorem: Let S be a set. For each function from S to 2S, there is at least one element of 2S which is not in the range of f.

Proof: Let

 f:\;S\;\rightarrow\; 2^{S}

be a function with domain in S, and range in 2S. (The function

f: \; s \; \rightarrow \; \lbrace s\rbrace

is such a function, which is even one-to-one with domain all of S, showing that 2S is "at least as big" as S itself.) Now let

A(f) \; = \; \lbrace s \; \in \; S \; \mid \; s\; \notin \; f(s)\rbrace

If A(f) = f(a) for some element a in S, then, is the element a in A(f) = f(a)? Well, if it is, is isn't, and if it isn't, it is. Uh-oh, trouble! Conclusion: A(f) is not in the range of f, and the proof is complete.

It follows from this result that 2S is "strictly bigger" than S itself.

Now, let S be the set of positive integers. Define a function from 2S as follows. If A is a subset of S, let

f(A) \; = \; \sum_{a\in A}2^{-a}

Thus, f(A) is a real number between 0 and 1. Unfortunately, f isn't quite a one-to-one correspondence. The finite, nonempty subsets and the complements of finite, nonempty subsets represent the same numbers, namely the set of rational fractions f, 0 < f < 1, with denominator a power of 2. One representation ends in all 0's and the other in all 1's. Luckily, this set is "only as big" as the set of positive integers, so the conclusion still follows, that the set of real numbers in [0,1] is "strictly bigger" than the set of positive integers.

Aside:

The cardinality of the set of positive integers is often indicated by \aleph_{0}, "Aleph-null." This may defined as the cardinality of the set of all finite ordinal numbers(*), and as such is the first infinite cardinal number. The next cardinal number, \aleph_{1} is the cardinality of the set of ordinal numbers of sets which are either finite or of cardinality \aleph_{0}. The hypothesis that

\aleph_{1} \; = \; 2^{\aleph_{0}}

is called the continuum hypothesis, and (if memory serves) has been shown to be logically independent of the usual axioms governing these things.

(*) An ordinal type is set S with a linear ordering (like the usual ordering of the integers, the rational numbers, or the real numbers). An ordinal number is an ordinal type in which the linear ordering is also a well-ordering; that is, in which each nonempty subset has a least element under the ordering. The usual ordering of the positive integers is a well-ordering, but the usual linear orderings of the set of all integers, the rational numbers, and the real numbers, (or of the positive rational or positive real numbers) are not well-orderings.

Last fiddled with by Dr Sardonicus on 2018-12-10 at 14:51
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Old 2018-12-11, 00:51   #109
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I’m kinda confused now, with this whole “orders of infinity” thing. wblipp's response was slightly understandable, but I’m completely lost on subsets/power sets/rational fractions

The main hindrance to my understanding of wblipp's post is the constructed number. Can you elaborate on what you mean by all of the decimal positions, and what would be on the list in the first place?

I don’t even know what’s confusing about Dr. Sardonicus's post, it’s just way too complex...

I tried using Nick's infinite coin analogy as an explanation to my friend, but we got hung up on the first case, where the coins aren’t numbered. I didn’t have much time to present it, so there may have been a misunderstanding or something

Often I find that some things just make sense; Nick's coins immediately clocked for me. When I try to explain things like that, and it doesn’t immediately make sense for others, I have no clue how to explain it. For a really basic example, how would you explain negative numbers to someone who has never thought of them? Someone from several hundred years ago? I don’t remember when negative numbers became widely accepted, although I do remember that Euler pioneered imaginary numbers (or did he invent the notation?), maybe that’s a better example
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Old 2018-12-11, 05:56   #110
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He means, assume you can "count" all real numbers in (0,1) interval (i.e. put them in a correspondence with the natural numbers).
Then you have them in some arbitrary order:

1 : 0.104752003005.....
2 : 0.379805307156.....
3 : 0.952057123459.....
4 : 0.195305712345.....
5 : 0.619527542131.....

etc

Then you construct the following number (see the underlined positions): if the nth decimal in nth number in the series is not 2, your number has a 2 there, but if it is 2, then your number has a 3 there:

0.22323.....

Now, you can prove easy that this number is not in your list, because it is different of any number in the list (they differ at position n, at least - it is not equal to the first, because the first position in the first number is not 2, but in your number it is 2, it is not equal to the third because the third has a 2 on the third position, but your has a 3, etc).

This idea is kind of "the same" as when you prove there are infinite prime numbers, just assume you put all in a set and construct one which is not is the set. The method is called "diagonalization" and it should worth mentioning that it is not accepted by all mathematicians (some say the method is not valid, because things get strange when you go to infinities! - fortunately for this particular task, there are other ways to prove it).

You can search the web for Cantor sets, alef-zero, etc.

By the way, the set of rational numbers (Q) has the same cardinality as the set of natural numbers (N), i.e. you can have a bijection from all naturals to all fractions, i.e. there are as many natural numbers as there are fractions with natural/integers numerators and denominators. When I was in grade 11, the teacher shown us the counting method, I had a hard time to grasp this - think about it: all pairs of integers are countable, then all triples are countable (just make a cube and start from a corner, instead a square) - all quadruples are countable... all n-uples are countable, but some guy comes and tells me that 2^n is not... Grrrr... .

Last fiddled with by LaurV on 2018-12-12 at 07:19 Reason: spaciation... fonts... grrr....
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