[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.



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