AAGT 1
 

:rolleyes:
As a sequel to’ Elementary Number quiz’ which was well received I am spurred on to send off some more from my formidable personal collection of puzzles and math recreational problems.
I have selected these puzzles from Series two ‘Can you solve these’ by David Wells. Out of the 112 problems given in this series I have chosen just 6 Arithmetical ones which are suitable for us. These are based on sound math ideas and thinking. I have also prepared from the series, problems in Algebra ,Geometry and Trig. for a later date. Hence I call it AAGT Quiz. So lets get started with AAGT 1 which deals with Arithmetic. If any one needs hints they are yours for the asking. Complete solutions will be given later as I have worked out most of them and have the correct answers.
AAGT 1.
A 1: What fractions when you turn it into decimal starts like this,
(a) 0.0204081632…….
(b) 0.0103092781……

A 2: EXACTLY 63.41463414 % of the people asked if they used ‘scrubbo’ soap powder replied yes.
What is the SMALLEST number of people who could have been asked the question?

A 3: Find a fraction which is greater than 7/17 but less than 5/12

A 4:What is the difference between ‘999999 thousand’ and 999899001 ?

A 5: (suitable for programming)
How many terms of this series,
1/2 +1/3 +1/4 + 1/5 +1/6 +…….. are needed to make the sum of the series greater than 2 and ½

A 6 ; What are the prime factors of the number , 22,438,769 ?
Have a good day, :smile:
Mally. :coffee:

2 - 3,170,731,707
3 - 83/200
4 - 998,999,109,999
5 - n = 19
6 - 22438769 = 53 × 67 × 71 × 89

BTW, may you post author, title and editor of some of your books??

Luigi

[QUOTE=ET_]2 - 3,170,731,707[/QUOTE]

Luigi,
I don't think so.

"EXACTLY 63.41463414 %" is "EXACTLY 0.6341463414 of the participants"
and that is "6341463414/10000000000" which reduces to
"3170731707/5000000000 of the participants"

Your number is the number that replied affirmatively. But the question asks for the minimum number polled.

Since there is no way for a fractional participant to reply, the number of participants must be a multiple of 5000000000.

That number is less than the current world population, but if you exclude the infants who are too young to reply, I doubt that there are enough left.

Wouldn't you like to have that kind of market penetration.

The continued fraction expansion of .6341463414 is 0, 1, 1, 1, 2, 1, 2, 1, 9380862 so I strongly suspect the desired fraction is 26/41 = 0.6341463414634146341463414634...
Hence a positive multiple of 41 people must have been polled.

Maybe Mally wanted to imply that 63414 is periodic.

Mally: I overlooked that you did not include the 1/1 term in the harmonic numbers, so that answer is wrong.

Alex

[QUOTE=ET_]4 - 998,999,109,999[/QUOTE]

How do you interpret "999999 thousand" as almost 10^12?
I interpret it to be (10^6-1) * 1000. (<10^9)

999,999,000 - 999,899,001 = 999,000 - 899,001 = 100,000 -1 = 99,999

[QUOTE=akruppa]The continued fraction expansion of .6341463414 is 0, 1, 1, 1, 2, 1, 2, 1, 9380862 so I strongly suspect the desired fraction is 26/41 = 0.6341463414634146341463414634...
Hence a positive multiple of 41 people must have been polled.
[/QUOTE]
But that is NOT what he ask. I agree that it is a more reasonable answer, but it is not "EXACTLY 63.41463414 %"

Particularly in light of the question immediately above it, I don't think that we can consider it a mistake that he phrased it as he did.

[QUOTE=akruppa]
Mally: I overlooked that you did not include the 1/1 term in the harmonic numbers, so that answer is wrong.

Alex[/QUOTE]

My answer is written as: Sum (1/n) (for n=2 to 19) > 2,5

Luigi

[QUOTE=Wacky]How do you interpret "999999 thousand" as almost 10^12?
I interpret it to be (10^6-1) * 1000. (<10^9)

999,999,000 - 999,899,001 = 999,000 - 899,001 = 100,000 -1 = 99,999[/QUOTE]

I'd like to see if you were responsive! :razz: :whistle:

My fault :blush: I'd take my time reading my own answers BEFORE publishing them.

Luigi

[QUOTE=Wacky]But that is NOT what he ask. I agree that it is a more reasonable answer, but it is not "EXACTLY 63.41463414 %"

Particularly in light of the question immediately above it, I don't think that we can consider it a mistake that he phrased it as he did.[/QUOTE]

Yes, the answer to the question strictly as stated (with fully capitalised EXACTLY and all) isn't 26/41. But, as you noted, this answer fails a simple plausibility check. In such cases I think it is not unreasonable to assume that the author of the puzzle goofed.

Alex

Plausibility Check
 

Question 2.

The answer is 41, of whom 26 said yes.

How much more plausible can it get? :showoff:

[QUOTE=Numbers]Question 2.

The answer is 41, of whom 26 said yes.

How much more plausible can it get?[/QUOTE]

You have missed the point. 26 of 41 is = 63.41463414634146341463414634... %. That is a bit more than "EXACTLY 63.41463414 %" which was requested.
To get the EXACT answer, you would have to have sampled almost EVERY person on Earth. Since there are a number of individuals who would be unable to answer, it is quite likely that you could not have obtained the required number of responses.

Go back and read the dialog between Alex and myself.

AAGT1
 

:smile:
Thank you Luigi, Richard, Alex, and Numbers (welcome to M/Forum)

Luigi:Pl. refer to my AAGT1. These are from 'Can you solve these' Series 2 by Davis Wells. In all there are 3 series. Editor Jean Slack. Publishers Tarquin Pub. I got them from FOYLES Lon.

Background of David Wells:
Born 1940, and has the rare distinction of being a Cambridge scholar in maths and failing his degree.
While at University he was British under 21 Chess champion.
He is the inventor of games “Guerilla” and ‘Checkpoint Danger’
He is puzzle editor of ‘Games and Puzzles magazine’ and ‘Problem Solver’ for secondary pupils. He has published several books most of which I have.

Regards the Quiz I find there has been no response to Ques.A1
These are the correct answers leaving out Ques A1.
2) 26/41
3) one of the many fractions 169/408 or 84.5/204
4)99,999
5)18
6)53,67,71,89

My Comments:
I enjoyed the dialog this quiz generated.
Yes Alex the idea conveyed is that decimal 63414 is periodic in 26/41

I agree with you Richard that the actual percentage is greater than the one given.
Perhaps if it was written 63.41463414* ( dot symbol of recurring decimal) it would be more correct. This is an excellent point!
Consider if one more person was asked, the fraction would then be 27/42 which is 0.642857142 and this is greater than 0.63414* (dot)
We cant consider a fraction of a human being!

Numbers: you found no difficulty in accepting the fraction and I’m glad.

Since this problem generated a controversy I put down the Hint given by David Wells

Hint:‘The decimal 0.xyz xyz is equal to the fraction xyz/999
Similarly the decimal wxyz wxyz =wxyz/9999
Ans:41. As a fraction the percentage given is as 7046/11111 which cancels down to 26/41. So the smallest number of people who could be asked the question is 41 of whom 26 replied Yes”

Pl. Note: A1 has not been answered.
Hint: Restrict your search to fractions where the numerator is 1. Because the first decimal starts 0.02 the first fraction must be between 1/50 and 1/33
Now could you crack it out?
Mally :coffee:
P.S, Yes Alex you have it

[QUOTE=mfgoode]Regards the Quiz I find there has been no response to Ques.A1[/QUOTE]
Personally, I considered them trivial and a "warm-up" for the remaining questions.
[QUOTE]
I agree with you Richard that the actual percentage is greater than the one given.
Perhaps if it was written 63.41463414* ( dot symbol of recurring decimal) it would be more correct. This is an excellent point![/QUOTE]
The typographical character ellipsis ('…') is the proper method to indicate that something continues. Although I don't think that it is customary to use multiple ones, you did use it in problem A1. In the above quote, I see an asterisk.

I tried to read something into your use of the term EXACTLY. It looked like a trick to me. Had you simply said "63.41463414…% of the people", or even "63.41463414% of the people", I would interpret it to mean that you were looking for a fraction with the smallest denominator whose decimal expansion matches the digits given.

EXACTLY why was the particular wording chosen?

:)

[QUOTE=Wacky]Personally, I considered them trivial and a "warm-up" for the remaining questions.

The typographical character ellipsis ('…') is the proper method to indicate that something continues. Although I don't think that it is customary to use multiple ones, you did use it in problem A1. In the above quote, I see an asterisk.

I tried to read something into your use of the term EXACTLY. It looked like a trick to me. Had you simply said "63.41463414…% of the people", or even "63.41463414% of the people", I would interpret it to mean that you were looking for a fraction with the smallest denominator whose decimal expansion matches the digits given.

EXACTLY why was the particular wording chosen?

:)[/QUOTE]

:smile: : Thank you wacky for your observations and comments.

[QUOTE=Wacky]Personally, I considered them trivial and a "warm-up" for the remaining questions.]

Yes they were meant to be. The general rule, but not necessarily universal, is the practice to, as you say, 'warm up'. All text books and exam papers start with the simple problems first and then move onto 'stiffer' ones towards the end.
This rule was not followed in the quiz I compiled in my selection. The word 'trivial' is a relative term. I suppose thats why it was left out in the first
response? Still it is not answered, except by Alex, which has not been posted.

Quote:[The typographical character ellipsis ('…') is the proper method to indicate that something continues. Although I don't think that it is customary to use multiple ones, you did use it in problem A1. In the above quote, I see an asterisk.]Unquote.

:sad: Yes. You are correct in ('...'). IMHO multiple ones are used and I have used it in A1. Thats exactly how David Wells printed it and its not my make up.
I used the non conventional * for recurring but qualified it with explaining that it was a dot symbol for recurring decimals.
If you know how to write the dot on the rt. hand top of a number which is popularly used for recurring decimal please let me know as I am not aware of how to print it from My MS office keyboard or if there is another recognised symbol that is used instead?

Quote[I tried to read something into your use of the term EXACTLY. It looked like a trick to me. Had you simply said "63.41463414…% of the people", or even "63.41463414% of the people", I would interpret it to mean that you were looking for a fraction with the smallest denominator whose decimal expansion matches the digits given.]unquote.

No wacky I am not an oriental yogi doing the Indian rope trick!
Please go back to my original Thread and you will find it is exactly as you say- 'Exactly 63.41463414%' This is how David Wells wrote it and thats what I copied. May I add that ('...') is not the same as the symbol written for the recurring decimal. The first term you use denotes a continuation of numbers one by one in a series like 12345..... Whereas the dot that is used is for a recurring block of digits like .63414.
Still wacky I might be wrong but by the preceding discussion I have learnt some finer points of math symbolism.
I would like to reiterate that the wording of the entire set of 6 questions is not by me but by David Wells. He does not normally answer questions on correction of his puzzles. I know because tho' he gives his address he does not acknowledge them
Thank you once again.
Mally :coffee:

[QUOTE=mfgoode]I suppose thats why it was left out in the first
response? Still it is not answered, except by Alex, which has not been posted.[/QUOTE]
If you wish it posted:
A1:[SPOILER] 1/49 [/SPOILER]
A2:[SPOILER] 1/97 [/SPOILER]

[QUOTE]If you know how to write the dot on the rt. hand top of a number which is popularly used for recurring decimal please let me know[/QUOTE]
Short of having a font which includes the particular glyph, I don't know how to reproduce it in any standard manner. Some programs are able to do sub- and super- scripts, but that is not standard and usually, at best, only approximates the desired effect.

As for the notation itself, I am not familiar with it. My initial reaction is that it is ambiguous. Does 0.010203* mean 0.01020303… or 0.0102033333… or something else?

[QUOTE] Quote[I tried to read something into your use of the term EXACTLY…

I would like to reiterate that the wording of the entire set of 6 questions is not by me but by David Wells.
[/QUOTE]

Sorry, please substitute "the author's" for "your" above. In any case, I do not feel that the problems are stated in a sufficiently precise manner.

But I do think that they are good problems. And I thank you for bringing them to us.

AAGT 1
 

[QUOTE=Wacky]If you wish it posted:
A1:[SPOILER] 1/49 [/SPOILER]
A2:[SPOILER] 1/97 [/SPOILER]

As for the notation itself, I am not familiar with it. My initial reaction is that it is ambiguous. Does 0.010203* mean 0.01020303… or 0.0102033333… or something else?

Sorry, please substitute "the author's" for "your" above. In any case, I do not feel that the problems are stated in a sufficiently precise manner.

But I do think that they are good problems. And I thank you for bringing them to us.[/QUOTE]
:smile:
Thank you for the interest you take in such elementary problems compared to all the advanced ones our members are doing in finding primes.
No Im not annoyed and dont need an apology.

The meaning is not ambiguous. if you get a 'run' like wxyz wxyz where wxyz repeats itself then the dot on the rt. hand corner of the number indicates that wxyz repeats and need only be written as wxyz' where Im putting the comma for the actual symbol which is a dot actually
Eg: 1/7 =142857 14287 142857 ....(to use your symbol . )
Then it need only be written 142857' using ' for the symbol.
Wacky I would like to request you to please post a thread on the common math symbols and their substitites which can be adapted to the standard keyboard. I see the symbols used by akkruppa and others which I cannot understand or interpret in the std. math notation.
Thank you
Mally :coffee:

I understand that you are using a symbol indicates that the preceeding group repeats (indefinitely). What I don't understand is where the group starts.

As for commonly used substitutions, I have no list. And I fear that any attempt to compile one would be rather incomplete because I tend to not even realize as unusual things that are personally common. So if you are confused about some terminology, please inquire. I'm sure that someone will be happy to give you an explanation.

Blocks of recurring digits
 

My understanding is that if what Mally refers to as a "run" but which I prefer to think of as a block of digits in the decimal expansion of a fraction repeats, there is in standard notation a dot over the top of the first and last digit in the block. Therefore, (using the symbol ' to indicate that the following digit has a dot over it) 0.0123412341234 could be written as 0.0'123'4.

However, as you have both found, it is not easy to represent a digit with a dot over it and use of the ellipsis ... (which is "usually" only three dots) has become more common. It is already used extensively in the notation for a series where, for example (2 x 1) + (2 x 2)... ...(2 x(y -1) + y means that the series continues indefinitely in the same manner until the term or terms found after the second ellipsis.

In the representation of the decimal expansion of a fraction, however, the elipsis would merely indicate that the expansion has not terminated and would not be interpreted by anyone I know of as meaning that any part of the preceding expansion should be repeated.

Moving on to what I think is a slightly more important point, I think the poster of a question (and that includes me, as I recently found to my cost) has a responsibility to ensure that their question is posed in such a way that it's meaning is unambiguous and clear. Falling back on the defence that "this is how it appears in my book", or as I did "this is how it was told to me" is disingenuous at best. Asking good questions is as much an art form as answering them, which is why quizzes are so popular and so much appreciated. Thank you, Mally, may we have the second round now please.

A 1: What fractions when you turn it into decimal starts like this,
(a) 0.0204081632…….
(b) 0.0103092781……
[spoiler] a) 1/49 (=2/98), b) 1/97[/spoiler]

A 2: [b]EXACTLY[/b] 63.41463414 % of the people asked if they used ‘scrubbo’ soap powder replied yes.
What is the SMALLEST number of people who could have been asked the question?
[spoiler] 5,000,000,000 - this is not the same as saying your calculator shows...[/spoiler]

A 3: Find a fraction which is greater than 7/17 but less than 5/12
[spoiler] so many to choose from - 169/408 is the midpoint[/spoiler]

A 4:What is the difference between ‘999999 thousand’ and 999899001 ?
[spoiler] 999999000-999899001 = 99999 [/spoiler]

A 5: How many terms of this series,
1/2 +1/3 +1/4 + 1/5 +1/6 +…….. are needed to make the sum of the series greater than 2 and ½
[spoiler] well I agree with the previously posted value of 19 but am not pleased with my method of arriving at that answer[/spoiler]
[spoiler]anyone get a nice solution not involving programs or lots of artihmatic?[/spoiler]

A 6: What are the prime factors of the number , 22,438,769?
[spoiler] 53x67x71x89 - again by an ugly method[/spoiler]

A 3: Find a fraction which is greater than 7/17 but less than 5/12

I am surprised nobody took the easy way out. While 169/408 certainly works, (5+7)/(17+12)=12/29 is much easier to find. Of course, proving this works generally probably is not easier than finding 169/408...

The Easy Way Out
 

Any two fractions can be thought of as being a/x, and b/y. What you did was convert them to fractions with the denominator x+y. This gave you (if you will forgive the rather ugly notation)

a((x+y)/x) /x+y and b((x+y)/y) / x+y

So any whole number between these two numerators will give a fraction that is > a/x, but < b/y.

Converting that to your case we get:

a((x+y)/x) = 11.9411… lets call this k

b((x+y)/y) = 12.0833… let’s call this t

And since t > a+b, a + b is an integer < t.

Of course, this is not a proof, this only helps us understand what happens. But it is now obvious that if y is a factor of x+y then t will be an integer so that t will not be < a+ b (it might = a+b, but it won't be < a+b).

So, your method works as long as y is not a factor of x+y, and as you point out, it is rather easy to do.

Oops
 

The second last paragraph of my previous post should obviously say:

Of course, this is not a proof, this only helps us understand what happens. But it is now obvious that if y is a factor of x+y then t will be an integer so that t will not be > a + b (it might = a+b, but it won't be > a+b).

Apologies for any confusion.

[QUOTE=Wacky]I understand that you are using a symbol indicates that the preceeding group repeats (indefinitely). What I don't understand is where the group starts.

As for commonly used substitutions, I have no list. And I fear that any attempt to compile one would be rather incomplete because I tend to not even realize as unusual things that are personally common. So if you are confused about some terminology, please inquire. I'm sure that someone will be happy to give you an explanation.[/QUOTE]
:smile:
Wacky: I have referred to David Wells books and came to the following conclusions.

Numbers’ notation of 2 commas (for want of a better ‘glyph’) at the beginning and end of the string is more correct than one comma though the example he has given is erroneously written and misleading.

The word ‘string’ or ‘run’ denotes order and is preferable to ‘group’ or ‘block,’ in this context, as we are dealing with a particular order of integers which repeats itself indefinitely.
A single comma (or dot) above an integer is a single repeated digit eg: 7.9999’ is 7.9’

The number of digits recurring in a string is called the period.
Thus 1/7 =0.142857 ‘142857’ etc. 0.1/7 =0.0142857 is a period of 6 not 7 as zero is place value and not repeated.
100/7 =14.2857. In this case ignore the decimal but see the order is maintained i.e. the period starts from 142857 and not after the decimal 0.2857.
Since 142857 is a string recurring indefinitely it can be written as ‘142857’

Take 1/13 =.076923 076923. Here the zero is also recurring and is not only a place value but also a digit and so it has to be written ‘076923’ such as 1/13 = 0.’076923’.This means that the string repeats itself indefinitely.
One must make sure the string repeats indefinitely

I revert to AATG1 and take fractions given there which are not fully worked out
Thus 1/49 = 0.020408163226 …till last digit. This is an example of a trimorphic number in which the powers of 2 appear in sequence but eventually over lapping so that the pattern although there still cannot be seen
1/97: The period is a max length of 96 starting with the powers of 3.
All this was tedious but hope it helps out.

Numbers:
1) 0.0’123’4 for 0.1234 1234 1234 is written wrongly as 4 also repeats itself too so the string is ‘1234’

2)I have mentioned my authority and background. That should be sufficient. Pl. Note Akruppa (“The author has goofed”). David Wells is a math’cian to be reckoned with!
No offence meant.


3) Its better to use the * for the ‘x’ for multiplication. Your . example is most confusing

4) please avoid circumlocution. This is maths, where the max. amount can be explained in the minimum terms unlike English.
Mally :coffee:

[QUOTE=mfgoode]:
though the example he has given is erroneously written and misleading.
[/QUOTE]

I believe that if you read my post again you will see that I said "using the symbol ' to indicate that the following digit is repeated". I am therefore using the symbol in a consistent manner before the digit to be repeated. Whether you find this misleading or not is a personal matter, but it is certainly not erroneous.

I deliberately avoided the use of the word "string" because this usually denotes a character or variable that is not a number, a phone number for example, on which no numerical or arithmetic operations will or can be performed.

Your point about the use of * instead of x is well taken, and I will try to remember.

I'm sorry if you find my speech circumlocutious; I shall try, but I am afraid that this is how I talk. It is not easy to unlearn the lessons of a lifetime.

Now, amid much appreciation for the previous two quizzes which have both stimulated interesting and educational repartee, can we have the next round now, please. Or are you not just a fine quizmaster but also a torturer?

[QUOTE=Numbers]Any two fractions can be thought of as being a/x, and b/y. What you did was convert them to fractions with the denominator x+y. This gave you (if you will forgive the rather ugly notation)

a((x+y)/x) /x+y and b((x+y)/y) / x+y ...[/QUOTE]
even easier... assuming a,b,x,y all positive ... take numbers c=a/x and d=b/y with c<d so a=xc, b=dy

we then see that (a+b)=(xc+dy) from which we find the inequalities

(a+b)/(x+y) = (xc+dy)/(x+y) > (xc+cy)/(x+y) = c = a/x
(a+b)/(x+y) = (xc+dy)/(x+y) < (xd+dy)/(x+y) = d = b/y

Thus a/x < (a+b)/(x+y) < b/y as desired.

For completeness, we can allow a,b to be negative:
If a<0<b we have trivially since smaller numerator, larger denominator (in magnitude)
If both of a,b are negative multiply by -1 to get the first case.

and prettier, too
 

Congratulations tom11784, much prettier than mine (I did say mine was ugly) and the simple stipulation that c < d obviates the necessity of dealing with the obscure case where x+y = 0(mod y). Very elegant, I wish I had thought of it myself. :bow:

AAGT 1
 

[QUOTE=nfortino]A 3: Find a fraction which is greater than 7/17 but less than 5/12

I am surprised nobody took the easy way out. While 169/408 certainly works, (5+7)/(17+12)=12/29 is much easier to find. Of course, proving this works generally probably is not easier than finding 169/408...[/QUOTE]
:smile:
Very good nfortino! I'look into the proof.
Mally :coffee:
Quote
A 5: How many terms of this series,
1/2 +1/3 +1/4 + 1/5 +1/6 +…….. are needed to make the sum of the series greater than 2 and ½
well I agree with the previously posted value of 19 but am not pleased with my method of arriving at that answer
anyone get a nice solution not involving programs or lots of artihmatic?]unquote]

The min. of terms required is 18 :rolleyes:
Mally :coffee:

AAGT1
 

[QUOTE=Numbers]I believe that if you read my post again you will see that I said "using the symbol ' to indicate that the following digit is repeated". I am therefore using the symbol in a consistent manner before the digit to be repeated. Whether you find this misleading or not is a personal matter, but it is certainly not erroneous.

I deliberately avoided the use of the word "string" because this usually denotes a character or variable that is not a number, a phone number for example, on which no numerical or arithmetic operations will or can be performed.

Your point about the use of * instead of x is well taken, and I will try to remember.

I'm sorry if you find my speech circumlocutious; I shall try, but I am afraid that this is how I talk. It is not easy to unlearn the lessons of a lifetime.

Now, amid much appreciation for the previous two quizzes which have both stimulated interesting and educational repartee, can we have the next round now, please. Or are you not just a fine quizmaster but also a torturer?[/QUOTE]
:smile: Numbers:
Thank for your response and comments.
At times healthy criticism and discussion leads to enlightenment more than flattery.
I am here to pick up whatever math gems that can come my way so am open to debate and even to ridicule provided the purpose is served. Many of my previous posts should prove this fact.

For instance I overlooked your dialog with nfortino and tom 11784 and almost replied in the negative. I then realised I was completely in the wrong and learned something of value. Now where would I pick up this very astute observation in any text book? And so it goes.
Now to reply to your post
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?

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

Para 3) Its good and I’m glad to see your readiness for adaptation.

Para 4) Your use of the word ‘circumlocutious’ I found to my surprise is an archaic word that is not even mentioned in my RD Great Encyclopaedic Dictionary. All the same it’s in use! Great!

Para 5) No, I’m neither a good quiz master not a torturer. But you must realise, Numbers, that it takes a longer time to compile a quiz than to solve it. And then I should have the right answers and be able to work the questions out and be able to explain them. So please have patience. Even then I goof them up sometimes.
I follow Gauss’ motto “Pauca sed matura”. I carefully study a post before I reply. You must excuse me if I take long to reply as I refrain from ad lib answers extempore. :rolleyes:

BTW; I am still groping over your sequence and ladder problems/threads and tom11784's proof. :unsure:
Mally :coffee:
P.S. I have combined two of your previous posts in one. Hope its not confusing.

More questions from the quizmaster
 

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.

AAGT1 Question 5
 

This is not intended to be a proof, but it is the simplest way I could find of summing a series of fractions.

Let x = positive integer
Let each term in Mally’s series be 1/x(n)

Let m(n) be the Lowest Common Multiple of x for n terms of the series

So if you had only the three terms 1/2 1/3 1/4 then m(3) = 12

To sum this series we would calculate a(m/x), but since a always = 1 we simply sum the series m/x which gives us the numerators for a series of fractions (m/x)/m

Then you simply sum the numerators until your answer > 2.5m

Most answers in this thread are that 19 terms are required, but Mally seems to think it is 18. So, we set n = 19 which gives m = 232792560. Therefore 2.5m = 581981400. Summing the terms in the series we find that after 18 terms the answer = 580842597 and after 19 terms the answer = 593094837 so 19 terms are required to exceed 2.5m.

I’m with the 19’s

[QUOTE=Numbers]This is not intended to be a proof, but it is the simplest way I could find of summing a series of fractions.

Let x = positive integer
Let each term in Mally’s series be 1/x(n)

Let m(n) be the Lowest Common Multiple of x for n terms of the series

So if you had only the three terms 1/2 1/3 1/4 then m(3) = 12

To sum this series we would calculate a(m/x), but since a always = 1 we simply sum the series m/x which gives us the numerators for a series of fractions (m/x)/m

Then you simply sum the numerators until your answer > 2.5m

Most answers in this thread are that 19 terms are required, but Mally seems to think it is 18. So, we set n = 19 which gives m = 232792560. Therefore 2.5m = 581981400. Summing the terms in the series we find that after 18 terms the answer = 580842597 and after 19 terms the answer = 593094837 so 19 terms are required to exceed 2.5m.

I’m with the 19’s[/QUOTE]
1/2 + 1/3 + ... + 1/18 + 1/19 > 2.5, correct ?

How many *terms* are there in this series ? :whistle:

[QUOTE=axn1]1/2 + 1/3 + ... + 1/18 + 1/19 > 2.5, correct ?

How many *terms* are there in this series ? :whistle:[/QUOTE]

:whistle:

AAGT
 

[QUOTE=ET_]:whistle:[/QUOTE]
Thanks ET for coming to the rescue. :bow:

You know, Luigi, I asked my wife for 40 yrs how many terms are there between 2 and19. She counted on her fingers as she distrusts calculators and including both the 2 and 19 she said 18 and thats how I gave my answer! :smile:
You see I have realised she is an authority in more things than just counting!

But jokes aside both you and akruppa slightly cloaked your answers and so it became what we used to call Euclids 6th theorem (in my text book) and my geom master gloated at it and called it 'pons asinorum' He was Irish! :whistle:
Coming back to the present my grateful thanks to Akruppa for tipping me off and he was the first to answer all questions correctly within hours of my thread.
This is what he says on the problem.
Quote :5)[These are the harmonic numbers H_n which grow like log(n). H_7 ]= 2.59..] :surprised
Thanks to the wrong answers I woud have overlooked his point.
I took out my scientific calcuator Sharp El-506R and fed in 18 as In 18
and lo and behold got 2.890371758 > 2.5 :grin:
Thanks to Numbers and axdf1 ? to point the way.
Mally :coffee:
P.S. thats the answer in the book.



.
l

[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

AAGT 1
 

:smile: Attn;Numbers.
In Series 3 'Can you solve this' by David Wells, he represents repeating digits by a line on the top of the digits covering the entire string like 1/7 = 142857
142857 with a straight line on the top of the first set. I cant reproduce it here but I guess you will get what Im saying. This usage escaped our discussion.
Mally :coffee: