[QUOTE=Numbers]Mally,

Para 1: Kindly explain what you mean by the run 0.0’123’4 1234… etc. and give us a larger ‘block’ as an example?

Basically, all I am trying to do is describe exactly the same notation you described. In the decimal 0.0123456712345671234567… we see that there is a pattern in that a block of 7 digits is repeated (or as you correctly pointed out, the period of the repeat is 7). As we discussed earlier, the ellipsis indicates that the expansion has not terminated. To write this expansion in a way that indicates both of these features, the repeating pattern and the continuing expansion, we put a dot over the first and last digit in the repeated block. I cannot write a dot over a digit on my computer so I am substituting the symbol ‘ for a dot, and placing the symbol immediately before the digit that has the dot over it. So that 12’34 means there is a dot over the 3. Thus, 0.0123456712345671234567… becomes 0.0’123456’7, where the first ‘ indicates that the 1 has a dot over it, and the second ‘ indicates that the 7 has a dot over it. All digits between and including those with the dots over them, are the repeated block.
If the expansion has a single digit that is repeated ad nauseam, 0.333333333… for example, the usual practice is to write two or three of the repeated digit and put a dot over the last. Using my symbology, this would be written 0.33’3 indicating that the final 3 has a dot over it.

Para 2) My phone number is 28360828. Would you say that 28360828 *2 =65671656 is not an operation on my tele.no.?

Yes, I would. Your phone number is not a number in the mathematical sense. It is merely a code written using numerical characters. If I were to dial 65671656 would I get two phone calls to you for the price of one? No, I would just end up talking to someone who is not you.
I was thinking primarily of programming, where variables are created to hold data of a specific type, and operations are only possible between variables of the same type. I might perform maths operations on a number variable that holds salary information (to sum the salary of all employees in a department) but there would be no point in summing their phone numbers because the result would be completely meaningless. So in this sense your phone number is simply a string of text characters that just happen to look like numbers. For this reason, the variable holding your phone number would be a string variable, and the value assigned to that variable would be considered a string rather than a number.

I take your point about quizzes taking longer to compile than to solve, and shall try to be patient but…

Patience is a virtue, possess it if you can,
It’s found seldom in a woman but never in a man.[/QUOTE]
:whistle:
I like the above verse.
Numbers:
Thank you for your clarification in both your posts. Now I understand you better than your previous posts.
1) I would prefer the symbol ‘ to be put before the 1st and after the last digits to mean ‘inclusive’ as in English. In your method viz. ‘123’4 it gives the impression that the 4 is not inclusive and is to be left out. I know what you mean but in writing it, it is confusing It would convey more meaning if written ‘1234’
2) This is a very moot point on what operation actually means
in maths
The RD Dict. in English gives the definition
‘Operation’:[ (5; math) Subjection of number or quantity to process affecting its value or form e.g. multiplication.]
:rolleyes:
Out of my 5 math dictionaries I pick an entry from one of them, best suited to our discussion.
Oxford Concise Dict. Of Maths: [An operation on a set S is a rule that associates with some number of S a resulting element. If this resulting element is always also in S, the S is said to be ‘closed under’ the operation. An operation that associates with one element of S a resulting element is called a ‘unary operation’; one that associates with two elements of S a resulting element is a binary operation.]
The above is self explanatory and I leave it up to you to derive a universal explanation from it related to maths. In other scientific disciplines such as QM etc the meaning is different in addition to the rules already established in Maths.
:unsure:
If I am not terribly mistaken I gather from your other posts that you are about to enter university soon. I presume you are ‘still wet behind the ears’
Even if you are not, you are, compared to my age!
‘Study the masters’ is a dictum worth following, and then compile your own work
Keep it up Numbers! You are definitely a precocious student and not a run of the mill type. You will go far if you ‘Go On’ :banana:
Mally :coffee: .

Wet behind the ears!
 

Mally,
It has nothing to do with the definition of “operation”, it is to do with the definition of the word “number”. Numbers are for two things, they are for counting things (5 apples, 12 eggs) and they are for measuring things (half a lollipop, 3 miles). What does your phone number either count or measure? It is not a number in the mathematical sense of the word.

Think of it like this: All numbers are imaginary. To prove this just ask yourself the question, when was the last time I stubbed my toe on a seven?
What makes a number a valid subject for mathematical operations is not what it looks like on the page, but the idea that it conveys; how many things it has counted or how far it has measured. By these criteria your phone number is not a number.
There is a number 28360827 + 1 that can be subjected to mathematical operations, of course there is. But as long as it remains 28360827 + 1 it is not your phone number. The two have separate and distinct identities. One of them is a number and the other is just a recognition code for your telephone that for the sake of convenience we write down using numerical characters.

p.s. Can I borrow your wife to help me with my homework, she is obviously much better at counting than me? :redface:

[QUOTE=Numbers]What does your phone number either count or measure? It is not a number in the mathematical sense of the word.[/QUOTE] Careful there, young whippersnapper, phone numbers may no longer "count", but they used to. Now, admittedly, you could not "add" two phone numbers, but you could add a pulse to most phone numbers and get a new phone number.

Basically I agree with your concept. But I tried to look up "number" in my algebra texts and could never find a definition for a number. I have "Cardinal numbers", "Rational numbers", "Algerbaic numbers", "Transcendental numbers", etc. But I cannot find the definition of a "number". It appears that, rather than "numbers", you are talking about the properties of a Group formed by a set of numbers and certain operators.

Where does it say that in order to be "a number", that it MUST form such a Group?

Whippersnapper replies...
 

Wacky,
It was never my intention to become embroiled in a deeply philosophical discussion about the precise definition of “number”, it just sort of led almost by default out of an inept attempt to define a string (see previous posts in this thread). However, since we are here …
Before we get too deeply into this let’s just remind ourselves that in his monumental Principia Mathematica, Bertrand Russell took 345 pages to define the number 1, so if we are to come to any kind of consensus in my lifetime we need to agree some ground rules. I am quite happy to assume the role of the student who is (metaphorically) trying to punch his way out of the corner, but you must promise not to punch too low (low is good, but not too low).

Number is not defined in my mathematics dictionary either, which I found quite odd until I tried to define it myself. So, since you mentioned them, let’s start by looking at Groups.

"Where does it say that in order to be "a number", that it MUST form such a Group?"

I didn't say that, you did.

A group is a set G closed under operation o such that:
1) for all a, b and c in G, a o (b o c) = (a o b) o c,
2) there is an identity element e in G such that a o e = e o a for all a in G,
3) for each a in G, there is an inverse element a1 in G such that a o a1 = a1 o a = e

Since operations o on a telephone number are not valid (because the result is meaningless) I would contend that none of the above apply to telephone numbers, and therefore, if (big if) it is a requirement that a number be a part of such a group that a telephone number is not a number in the mathematical meaning of the word. And anyway, what exactly is the inverse of a telephone number?

Groups are only one type of set; rings and fields are others, but since elements in these are also validated by operations they are also invalidated by the arguments applicable to groups. On thinking about it further I see that rings would form a special case because there is no telephone number 0, and if I’m not mistaken a ring must have a zero element. I’m also sure that there must be a joke in there somewhere but I can’t see it.

A set in general can quite legitimately contain non-mathematical objects (the set of objects on your desk, for example). But it seems to me that unless we can specify one or more of the following, a. Operators to act upon the elements of the set; b. Quantity or magnitude of the elements in the set; then those elements remain non-mathematical objects. And since I have already argued that neither of those apply then telephone numbers are not numbers in any mathematical sense.

Your turn. :banana:

and another thing...
 

Before the bell sounds, I would just like to add that in the same way that I can refer to teaspoon * 3 to mean three teaspoons without inferring that a teaspoon is a number, I can also say telephone number * 3 without inferring that a telephone number is a number. So there is some sense in which quantity can apply to a telephone number (my phone book has 250 telephone numbers per page) without implying that it is anything more than a non-mathematical object.

I have also realised that I essentially side-stepped your question. "Where does it say that in order to be "a number", that it MUST form such a Group?" My original response to that sounds like the kind of thing that Cheesehead would say (no offence intended) and adds nothing to the discussion.

Let T = the set of Mally’s telephone number.

An operation on a set is worth considering only if it has properties that lead to useful or interesting results. I cannot conceive of any useful or interesting results that could possibly emanate from a set with only one element, or even of any legitimate mathematical statement that could be made about such a set. From this we can conclude that one of the basic minimum requirements for a set to have properties that lead to useful or interesting results is that it should contain more than one element. Which I think answers your question. For an element or object to be a number it would have to be a member of something, and whether you call that something a set, a group or a banana is irrelevant.

So, what other elements can go in the set T?

Let T = the set of all domestic telephone numbers in Mumbai, India

or, let T = the set of all telephone numbers in India

or, let T = the set of all telephone numbers in the world

All of which would be valid sets. But, save for examples of the type quoted above (let n be the number of telephone numbers on a page) there are no useful or interesting results coming out of such a set.

At the beginning I wished we had thought of something other than telephone numbers, which is becoming quite boring and repetitive to type, but by now I am just glad we didn’t come across this topic via Massachusetts Institute of Technology Student Enrolment Numbers.

NB: This was composed before I saw your reply #38

I made the claim that you were requiring "numbers" to be capable of forming a group because you seemed to be requiring that they have those properties such as closure under addition, etc. If I am not mistaken, all, or at least most, of those things that we commonly call "numbers" (real, Cardinal, etc.) do actually form groups.

You also imply by your language that they must have the traditional "addition" operator. However, if I am not mistaken, a group is defined on a set (of any kind of things) and an operator (not necessarily "summation").
So having PARTICULAR operators is not requisite to "mathematical sense"

I'm a bit rusty on my "set theory", but I think that "vector field" is more likely to apply to phone numbers. In the old days of rotary dials and electro-mechanical "step" offices, "dialing" a number actually involved turning a dial which upon release caused a string of pulses as it momentarily broke the electrical circuit on the phone line pair. The equipment in the central office would, quite literally, count the pulses and position a commutator (or later the relay bank equivalent) to the corresponding count. A pause between digits signified that you were continuing to the next dimension in the vector space. In any of those dimensions, you could add a pulse in a meaningful way.

If you want to stretch the terms, there has always been an "operator" which can perform many transformations on phone numbers. And, if you know the codes, there is an identity element (the identity element doesn't have to be "one") in the call-back relay.

But my philosophical point was that your claim that phone numbers were not "numbers, in a mathmatical sense" seemed to be based on the absence of certain properties that I cannot support as being, by definition, necessary.

It is analogous to saying that a watermelon isn't a watermelon if it doesn't have seeds. (A plug for the Luling "Thump Fest" this weekend) A seedless melon may not be of much use in the "Spitting Contest", but, ice-cold and accompanied by a Shiner beer (from the nearby Spoetzl Brewery), it still makes for some mighty fine eating on a hot Summer afternoon.

So come on down. There's a bit of shade under the big Oak tree. We can pitch a few horseshoes and contemplate just how many of those beers we need to sample for "quality assurance". :)

AAGT; Philosophy and Numbers
 

:whistle:
Wacky I think your are right on telephone numbers, they were pulsed and still are. I enjoyed the repartee though but could not take part as my monitor packed up (monsoons you know!) :sad:

Numbers: "All is Number" Pythagoras :rolleyes:
Educational reading
[url]www.math.tamu.edu/~don.allen/history/pythag/pythag.html[/url] - 33k
Mally :coffee:
P.S. Numbers I have a better idea. Lets swap: my wife for your girlfriend!! :innocent:
Mally :coffee:

AAGT 1 Ugly method!
 

tom11784:A6- factors of 22438769?
'again by an ugly method' unquote
:rolleyes:
By paper and pencil and perhaps a pocket calcuator there are two methods known as 1) Fermat's method for two factors when the number can be expressed as the sum of the difference of two squares
2)Euler's method when the odd numbers can be put as the sum of two squares. However these are at times tedious but they do work out.
A ready method when theory is not required just click on this site.

[url]http://wims.unice.fr/wims/wims.cgi?session=S4615FD9BD.1&lang=en&cmd=reply&module=tool%2Falgebra%2Ffactor.en&calc=factor&formula=22438769[/url]
or click on Google - Factoris :surprised
Mally :coffee:

[QUOTE=mfgoode]
A ready method when theory is not required just click on this site.

[url]http://wims.unice.fr/wims/wims.cgi?session=S4615FD9BD.1&lang=en&cmd=reply&module=tool%2Falgebra%2Ffactor.en&calc=factor&formula=22438769[/url]
or click on Google - Factoris :surprised
Mally :coffee:[/QUOTE]

WIMS error
There is an error in your request to this WIMS site.

Your request contains a user identification error. Are you trying to connect to somebody else's session?

:whistle:

Luigi

[QUOTE=ET_]WIMS error
There is an error in your request to this WIMS site.

Your request contains a user identification error. Are you trying to connect to somebody else's session?

:whistle:

Luigi[/QUOTE]

:surprised I have checked the site and its okay with me. As a matter of fact both sites respond beautifully.
Maybe its the other way around. Someone is trying to connect with me .
I have suspected this for a long time but I have no fear if anyone does attempt to do so as my mail is above board. Maybe Ill change my password!
Thank you Luigi. I would try Factoris any time for primes and I am the first if Im not mistaken to introduce this web site to Mersenne forum :smile:
Mally :coffee:

Now it works again (from my office).

Factoris is a beautiful piece of math I found online, I often enjoy it. :razz:

Luigi