Sunday, 31 August 2008

The Question is, why didn't China win ALL the Gold medals

If the olympics were won by talent, then China would probably win all the gold medals. They have over 1.3 Billion people. Just assume a normal distribution and there should be LOTS of great runners, swimmers, weight lifters and yes... ping-pong players. The fact that they don't win them all tells us something (I'm not sure what) about the impact of money and training facilities. I was looking for some statistics on population and gold medal performance (I don't think there is any honest statistics on the MONEY spent per athelete), and I came across a page from Punch, Nigeria's most widely read newspaper (It says so on the banner) that listed some interesting facts from the China Olympics which I have copied below. They had more, in fact, one for every American Gold medal, (36 if you weren't paying attention), but I found some more interesting than others.

It’s the first time since 1936 that a country other than the United States or the Soviet Union has led the medal count. Hmmm, does this mean the cold war IS officially over???

Per capita, China won one gold medal for every 25 million people in the country. The United States’ per capita rate was one gold medal for every 8.5 million. The tiny island nation of Jamaica, which won a staggering six gold medals in Beijing, had a per capita rate of one gold for every 450,000 residents. Had China won at that rate, the country would have earned 2,889 gold medals.

Six countries won their first ever Olympic medals: Afghanistan, Bahrain, Mauritius, Sudan, Tajikistan and Togo. Panama and Mongolia won the first gold medals in their respective histories (they had won other medals before, but never gold)

India has 17% of the world’s population. They won 0.31% of Olympic medals.
China: 19.8% of population, 10.4% of medals.
United States: 4.6% of population, 11.5% medals.

Jamaica: 0.041% of population, 1.15% medals.

Iceland was the least populous country to win an Olympic medal.

Pakistan was the most populous country not to win an Olympic medal (164 million residents, sixth-largest nation in the world).

More proof that boxing is dead in the United States: the country earned just one medal (a bronze) in the 12 boxing events.

From 1980 to 2008, Jamaica won three Olympic gold medals. In a span of six days in Beijing, Usain Bolt won three.

And finally, if Michael Phelps were a country of his own, he would have finished tied for 9th in the gold medal count, ahead of countries including France, Netherlands, Spain, Canada, Argentina, Switzerland, Brazil and Mexico.

A question of my own... Softball was eliminated from the summer Olympics because the US won it so steadily. Can we assume that ping-pong and diving may soon be on their way out? The sad truth is that the political legacy we leave around the world will impact our atheletes on the Olympic fields.

Tuesday, 26 August 2008

two times two is five??? by Lewis Carroll

I recently came across an example of Carroll's cross between wit and logic in a 2008 Gresham lecture by Robin Wilson:

Honoured Sir,

Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.

If x and y are each equal to 1, it is plain that

2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0.

Hence 2 × (x2 - y2) = 5 × (x - y).

Now divide each side of this equation by (x - y).

Then 2 × (x + y) = 5.

But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5.

Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.

I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll.

Thursday, 14 August 2008

Etymological Musings

Sometimes in the summer (oh, sweet summertime, where have you flown.... school bells tinkling in my near future...and I am SOOOOO NOT ready) I find time to wander through the blogs of folks who understand and explain word origins. Recently I came across a couple of interesting words that I thought I would share. I meant to do this a couple of days ago, but got busy and put it off... and one of the words was about that very thing.
Everyone knows the word procrastinate, but not everyone know that the word literally means to move forward to tomorrow. The Latin root, I learned, is procrastinare, and the pro prefix means forward. Crastinus means tomorrow, and together they represent that evil practice of putting off today's job until tomorrow. Cicero, in an attack on Mark Antony said, "... slowness and procrastination are hateful". But I came across an even more appropriate term for my summer behavior,perendinate
which means to put something off until the day after tomorrow, which is the literal interpretation of perendie. There is a lot more about procrastination and its interpretation in many languages and cultures at Ben Zimmer's SLATE.
If you watched the National Spelling Bee, you might have heard another interesting word, hyphaeresis, which is an example of what it means. It is the name for the act of leaving out a letter or syllable in a word; in hyphaeres the omitted letter is the "o" from hypo, which is Greek for below, as in hypodermic (below the skin) and hypotenuse, the longest side of a right triangle which literally means "to stretch below".
As long as I am amusing you with interesting words (I AM amusing you...right???) I will add one more.."stamina". OK, you know what stamina means, or at least how it is commonly used now...but do you know how it is related to stamin, the sexual part of a flower (Yeah, I said the magic word and NOW you are interested..)? Here, said better than I could ever do, are the words of the original blogger
"Before stamen meant "The male or fertilizing organ of a flowering plant," it meant 'the warp in an upright loom' (the Latin word stāmen is from the Proto-Indo-European root *stā- 'stand'), and from there it came to mean (in the OED's words) "The thread spun by the Fates at a person's birth, on the length of which the duration of his life was suppose[d] to depend. Hence, in popular physiology, the measure of vital impulse or capacity which it was supposed that each person possessed at birth, and on which the length of his life, unless cut short by violence or disease, was supposed to depend.' (1709 Tatler No. 15.1 'All, who enter into human life, have a certain date or Stamen given to their being, which they only who die of age may be said to have arrived at'; 1753 L. M. Accompl. Woman I. 246 "Bad example hath not less influence upon education than a bad stamen upon the constitution.") Hence the plural stamina meant "The congenital vital capacities of a person or animal, on which (other things being equal) the duration of life was supposed to depend; natural constitution as affecting the duration of life or the power of resisting debilitating influences" (1701 C. WOLLEY Jrnl. New York 60 'Such as have the natural Stamina of a consumptive propagation in them'; 1823 GILLIES Aristotle's Rhet. I. v. 180 "If the stamina are not sound, disease will soon ensue"), and finally the modern sense "Vigour of bodily constitution; power of sustaining fatigue or privation, of recovery from illness, and of resistance to debilitating influences; staying power" (1726 SWIFT Let. Sheridan 27 July Wks. 1841 II. 588/1, "I indeed think her stamina could not last much longer when I saw she could take no nourishment"). This was originally construed as a plural, but by the nineteenth century careless writers were using it as a singular (1834 M. SCOTT Cruise Midge viii, 'Why, Sir Oliver, the man is exceedingly willing,.. but his stamina is gone entirely'), and this rapidly became standard."

2015:  As a footnote, my brother-in-law worked for several years in the Pondicherry area of India and he says it was common there for folks to use the word "prepone" as a variant of postpone, to move a meeting forward.  Makes perfect sense if you realize that the "pone" is from the Latin "ponere", to put or place, so prepone, just means to put it forward.  A check of the OED suggests that this is only used in India, but it seems like a perfectly good word.  Tell  your students you want to prepone their test two days.  That should get their interest in vocabulary pumping. 

Tuesday, 12 August 2008

NY Times Editors Need Math Help

The article was on August ninth in the Times, with a title of Chinese Basketball Builds Toward Podium... It was a nice article, polite, balanced, and informative...and then it had a paragraph that said
"With 300,000 million people playing basketball across the country — roughly the same number as the population of the United States — and with money being poured into the sport at the youth level, China is aiming to make the medal round at these Olympics."
Ok, now look up the population of China... and the USA.... Keep this in mind the next time the TIMES decides to publish an article on the inability of America's youth to do mathematics.

Monday, 11 August 2008

Things I learned Late in Life about Repeating Decimals and Prime Divisors

In my last blog I mentioned that if you write out any three digit number and repeat it two times it will be divisible by 7, and 13. I didn't mention it, but it will also be divisible by 11. The question of course is a) why, and b) how does that help us understand divisibility and primes. The easy reason is that any such number, xyzxyz, is going to have a factor of 1001 (xyz) and since 7, 11, and 13 all are factors of 1001, they will divide into it evently. Now if you repeat a two digit number 3 times (xyxyxy) then you get a number that is divisible by 7 and 13, and 37, but not 11. I'm not sure it works for all numbers, but I have noticed some patterns in what I am going to call the "modular sequence" of prime numbers. Becuase it is short and easy to check on your calculator, I will use 7 and 13 for examples, and then show some extensions for others.... so here goes.

The modular sequence of 7 is 5,4,6,2,3,1... Ok.. what does that mean... look at the 2, it is in the hundreds place... Suppose I take a number, 243 and find the remainder on division by 7. (in this case 243 mod 7 == 5). If you increase the number by 100, the remainder will increase by 2.. (343 mod 7 == 5+2 == 0; and 443 mod 7 == 0+2==2); and similarly an increase in the value of the number by 1000 would increase the remainder by 6 (or reduce it by 1, which is the same thing.... imagine that 13 divided by 7 is one with a remainder of six, but you could think of it as two with a remainder of negative one).

Now how does that help us explain the fact that 7 divides xyxyxy and xyzxyz. One way to find the remainder that I mentioned earlier is to use the dot product of the vector x,y,x,y,x,y times the modular sequence, 5,4,6,2,3,1. You end up with 14x + 7y which is of course divisible by 7 because both the 14 in the first term and the 7 in the second are divisible by 7. In the same way, the dot product of xyzxyz and 546231 will give us 7x+7y+7z, which is also divisible by 7. And just as an added treat, xxxxxx will also be divisible by 7 because the totatl of all the numbers in the modular sequence is 21, so the dot product would be 21 x.

Ok, a little more about the modular sequence to help you see the patterns that I think are true. THe modular sequence is the same length as the repeat cycle of the unit fraction 1/p. For example, 1/7 and 1/13 both are repeating decimals of length 6. 1/7 = .142857.... and 1/13 = .076923...The modular sequence for 13 is 4, 3, 12, 9, 10, 1. If we add up all the numbers we see that the sum is 3 (13) = 39 [remember that the sequnce for 7 summed to 3(7)] So let's form a rule.. it works, you can check it later. THe sum of the terms in the modular sequence will be p(n)/2 where p is the prime number in question, and n is the length of the sequence. [A similar formula holds for the sum of the digits in the repeating cycle of 1/p, the sum of the digits will be 9d/2... every repeating cycle has a digit sum which is a multiple of nine].

If the length n is divisible by some number x, then the sum of every xth digit in the modular sequence will add to a multiple of p. This is what we saw with every second digit of the sequence for seven (5+6+3=14 and 4+2+1=7) . Since the length of 1/13 is also six digits, we can do the same with the alternate digits of the modular sequence for 13 ( 4, 3, 12, 9, 10, 1) 4+12+10 = 26 and 3+9+1=13... (not able yet to explain the 2:1 ratio, or if it persists in all even length sequences). If we add every third digit for seven we get 5+2=4+3=6+1. For 13 we get 4+9=3+10=12+1. A similar idea works in the repeat cycle of any prime for which the length of the cycle is divisible by two; all the numbers in the last half are the nines complement of the numbers in the first half... 1/7 = 142857... look at 142 and then 857; 1+8=9 and the same for each digit. That was a trick I showed earlier to complete long strings on your calculator.

Ok, how about a harder one.. 1/17 has a repeating decimal 16 digits long, .0588235294117647 (for decimals 1/p if you want to see the back half of the sequence, do (p-1)/p on your calculator... look at 16/17 on your calculatror for instance) First, confirm that the corresponding numbers in each half add up to nine. 0+9, 5+4, etc.. Now in the modular sequence for 17, they will add up to 17, so when we have the first half, we can just use compliments to get the second half. FOr 17 the sequence is 12, 8, 11, 13, 3, 2, 7, 16, 5, 9, 6, 4, 14, 15, 10, 1. Take a moment to see that the corresponding parts in each half add up to 17....

Now my conjecture is that if the length n is divisible by d, then any number of d digits repeated enough times to make d digits will be divisible by p, in this case, 17. If this is true then in the modulus the numbers located d units apart must add up to a multiple of p. OK, so 2 divides 16, so every other number should add up to a multple of 17; 12+ 11+3+7+5+6+14+10=68 which is 4x17, and 8+13+2+16+9+4+15+1=68 also. Ok, it works for two, but four also divides 16, so every fourth number should add up to a multple of 17 as well. The first set would be 12+ 3+ 5+14=34 which is 2x17 (are YOU seeing a pattern?)... the second is 8+2+ 9+ 15= 34.. you try the other two.

If you want to play around with this some more, as I do, Here is some information of interest:

The length of the divisors, and some of the repeating digits and modular sequences that I have done so far out of the first few primes

n a(n) repeat seq mod seq

2 0

3 1 .3 1

5 0

6 1 6 4

7 6 142857 5,4,6,2,3,1

11 2 09 10,1

13 6 076923 4, 3,12,9,10,1

17 16 0588235294117647 2, 8, 11, 13, 3, 2, 7, 16, 5, 9, 6, 4, 14, 15, 10, 1

19 18 052631578947368421 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5,10,1

23 22 0434782608695652173913

29 28 0344827586206896551724137931

31 15 032258064516129

37 3 027 26,10,1

41 5 02439 37,16,18,10,1

43 21

47 46 0212765957446808510638297872340425531914893617

53 13

NOtice that the numbers whose sequence lengths are not divisible by two (such as 31 or 41) do not yield complimentary halfs, but they should still obey the division rules. (I hope that pqrstpqrstpqrst will be divisible by 31 since it repeats a sequence of five digits three times,) Have fun...

Thursday, 7 August 2008

Out of the Mouth of a Donkey?

Got an e-mail a couple of days ago from an Indian Mathematician named Sajeev Singh. Sanjeev asked me to let people know about his work at Total Gadha where he posts notes about mathematical ideas with some comments etc. I took a look and found some really well done posts on some interesting ideas about math that is accessible to high-school and college age folks. Gadha, by the way, if my memory serves me in old age, means Donkey, but I'm not sure how that reflects in the local culture. One more reason to visit India some day.

One of the notes he posted took me back, yet again, to the question about the 6! or 720 six-digit numbers that can be made using only the digits 1,2,3,4,5, and 6 once each. I realized that with a single brief mental calculation you could confirm that a) none of them were prime, and b) none of them were perfect squares. Why NOT?? (think, then read)

The reason has to do with the digit root; what used to be called casting-out-nines(and some of us still do call it that). If you add the digits, 1+2+3+4+5+6 the total is 21, and if you add 2+1 you get 3, so the remainder when any of the 720 numbers is divided by nine is going to be three, try it. 243561, for example, when divided by nine has a quoatient of 27062 with a remainder of three. Any number that has a digit root of three is divisible by three. All 720 of the six-digit arrangements are divisible by three. But how do I know none of them is a perfect square???? Same answer, the digit root is 3. THe digit root of a square number is the square of the digit root of the square root. Let's look at some numbers and their squares:

12 = has digit root of one

22=has digit root of four

32=digit root of nine

42=16, so digit root of seven

52=25 so digit root of seven

62=36 so digit root of nine

72=49 so digit root of four

82=64 so digit root of one

92=81 so digit root is nine

Keep goin.. these are the only four numbers you get for the digit root of a square,... ever...square 37?? digit root is one... square 155?... digit root is four.. you do these in your head.. To find the digit root of the square of 23 you add the 2 and 3 to get five, so the digit root of 23 is five. That means the digit root of 23 squared is the digit root of 5 squared.... seven. 23 2 = 529. Add the digits you get 16 and add those digits you get...yep, 7. What does NOT ever happen is a digit root of 3; so none of the 720 numbers are square numbers. Can you prove than none of them are perfect CUBES????

Two things better than being reminded of something you once knew are to learn something you never knew, and even better, to be stimulated by a question to discover something new on your own.... and both those things happened after going to TotalGahda. Rather than throw all that at you and once though, I will give you a chance to experiment with one.

Think of a two digit number, now write it down three times xyxyxy and divide by 7. What is the remainder? Wait, divide it by 13.... NOw what is the remainder? I just discovered this morning that this works.. and proved a general version of it which I will include next time with a conjecture about a couple of extensions to all primes... not sure if or who might have done it before... and I can add that if you write the two digit number out eight times, it will be divisible by 17. It will also work if you write down any four digit number four times, or any eight digit number twice (wow, what a clue)

If you have trouble dividing big long numbers and finding the remainder, try using the Google search engine. Just enter something like "4512451245124512 mod 17 =" in the search box and the Google calculator takes over and gives you the remainder (modulus)

Ok, IF you get stuck on that one, you can try this, write down any THREE digit number, and copy it again...xyzxyz to make a number... now divide by seven again, and by 13???? The reason this one works may be a little easier(I've know it has been around in calculator books before).... see if you can figure it out. I'll be back in a couple of days with the additional material, and if you KNOW this has been done already, especially if you have a citation, Please advise.

Monday, 4 August 2008

Repeating Decimal Periods and Patterns

In my last post I asked, "IF you try to find the decimal number for 1/17 on the TI-84 it says 1/17 = .0588235294... and that's all you get. Now the challenge.... can you write out the complete decimal expansion (one complete cycle of the repeating expansion) of 1/17 using only what the calculator gave you and simple mental arithmetic...?"

The repeating portion of a decimal fraction is called the repetend, and the number of digits in the repeating cycle is called the period or length of the repetend. For 2/7, as shown above, the repetend is 285714 and has a length of 6. To figure out the original question I asked, it helps to know a rule about the length of repetends; for any prime number p > 5 the length of the repetend of 1/p is a factor of p-1. This seems first to have been stated by Johann H. Lambert in 1769. I find it odd that we had discovered so much about Primes so early, and this little tidbit came so late. For seven above, the length of 1/7 (or n/7) is six digits long. In looking at 1/17, the rule says that the period of the repetend must be a factor of 16, so the length will be 2, 4, 8, or 16 digits. Looking at what is given, we realize that the period can not be any of the smaller numbers, so it must have a 16 digit cycle, but how do we find the other digits?

Look back at 2/7 and another interesting property of repeating decimals may pop out at you. This one was published by Henry Goodwyn in 1802. He noted that if you divide the period into a first half and a second half, the sum of the digits in corresponding positions add up to nine.. For 2/7 above the repeat cycle is 285714 so the halfs are 285 and 714. Note that 2+7 = 8+1= 5+4 = 9, and Goodwyn showed this is true for ALL repeating decimals..... so it must be true for 1/17. If we know the period is 16 digits, then the first eight are 05882352 and to find the next eight, we just use the nines compliment of each number in the first eight, so they are 94117647, and we see that 1/17 = .0588235294117647...Now suppose we suddenly needed 2/17, we could double the string above, or just double the first half, and use the rule to complete the cycle.

In the age of computing when we can get out to a couple of hundred digits with almost any software, it might not seem like much of a tool....but sitting around drinking beers at the lake, and someone asks, "by the way, what is the repetend of 1/23?", you can whip out your pencil and do the first 11 digits and then coast the rest of the way adding nines-compliments.

I feel like I should have picked that one up somewhere in the seventh grade, but stumbled across it a few days ago reading an old book on Primes by the late George Loweke, who taught at Wayne State Univ. before retiring a few miles up the coast of Lake Michigan from me. Thanks George, but you should have mentioned this one to me sooner.

Friday, 1 August 2008

It is sort of like casting out sevens

Sometimes I enter the math contest they host at the Wild About Math blog site, and usually am among the people who get a correct answer, but haven't been the lucky name pulled from the hat yet. A recent problem about divisibility got me thinking about testing by divisibility by sevens again.

The problem was actually about divisibility by eleven, and asked
Consider all of the 6-digit numbers that one can construct using each of the digits between 1 and 6 inclusively exactly one time each. 123456 is such a number as is 346125. 112345 is not such a number since 1 is repeated and 6 is not used.

How many of these 6-digit numbers are divisible by 11?

The answer, of course, is none. If you don't know the rule for testing eleven see my blog on casting out nines and elevens. My interest was piqued by a resonse by Jonathan, of the jd2718 blog made these observations about the 6! or 720 possible numbers formed with the six digits as described:

As a consolation, all 720 of them are multiples of 3.

Half of them (360) are even (multiples of 2)

One in 6 (120) are multiples of 5.

Eight in thirty (192) are multiples of 4.

None are multiples of 9.

Fourteen of 120 (84) are multiples of 8.

Multiples of 7? New puzzle, good place to stop.

It is Not surprising that Jonathon covered all the other one digit divisors but did NOT test seven. Seven is the one that books on mental math simply say "divide by seven"; but thinking about it, it shouldn’t be too hard

I wrote recently about a mental test for divisibility by seven but it did have one flaw. It worked fine to tell you a number was divisible by seven, but if the number was not divisible by seven it did not give the correct modulus. Casting out nines will always give you the correct modulus. The sum of the digits of 2134 is 10 which has a digit root of 1, so if you divide 2134 by 9 you get a remainer of one. My rule for seven didn't give the true remainder.

The remainder when you divide a number by seven (or any other number) is frequently called the modulus. I've mentioned this before, it is just a way to divide integers into sets. Odd numbers are all equivalent to 1 mod 2 (they all leave a remainder of one when you divide by 2) and even numbers are all equivalent to zero. If a number is equivalent to zero mod (something) that means it is divisible by that (something). One of the things that makes them effective is the simple rule that if a+b=c then for any modular index n(the number you are dividing by) c mod n = a mod n + b mod n. For example 25 mod 7 = 4 and 25 = 20 + 5 by 20 mod 7 = 6, and 5 mod 7 = 5... 6+5 = 11 which is equivalent to 4 mod 7. So now we have the basics.

Every increase of one in the units increases the modulus 1, and every increase in the tens increases the modulus three, (for example 21 mod 7 = 0; 31 mod 7 = 3,and 41 mod 7 = 6). I figured out that the rest would be an increase of two in the hundreds, six in the thousands (think minus one) four in the ten-thousands (minus three?) and five in the hundred-thousands (minus two….) so the sequence is 546231… and what a coincidence, if you multiply 5(-2)+4(-3)+6(-1)+2(2) +3(3) + 1(1) …YOU GET -14, which is zero mod 7, the sequence of moduli used in each place makes it a multiple of seven…(546231 = 0 [mod 7]… but wait, 231 by itself is a multiple of seven, since4 23 - 2(1) =21… and 546 is also. [54 - 2(6)=42]..

It is probably easier on retrospect for that particular number to just square each digit in the sequence and discard any sevens (kind of like casting-out sevens) 25+16+36+4 + 9 + 1 is simplified to 4+2+1+4+2+1 and it is obvious the mod is seven…

To check other numbers, you can just write them out multiplied times the correct modulus for that place value and probably check it in your head, but you would have to check each one… so making up a sequence, 234561, we would think 2×5=10 (the first digit times the first modulus) drops to 3, plus 3(second digit)x4(second modulus) is 15 which drops to modulus of one, now add 4(6) to get 25 and drop to mod 4, then add 2×5 to get 14, and we are at mod 0 [so 234500 is divisible by seven] and we know that 61 is NOT divisible by seven, so we can be assured that 234561 == 61 == 5 mod 7… ok, not EASY, but certainly could be done sans calculator…. [footnote, for those of you who have just gone through an pre-calc class and somewhere along the way they taught you about vectors, you probably imagined that you would never run across another dot product in your life, but you just did. IF you think of the sequence of modular values as one vector (5,4,6,2,3,1) and the six-digits of the number as a second vector, then the dot product is an integer that has the same modulus as the original six digit number....come on... that's pretty cool use of vectors!!!]

It is easy enough to write remember the modulus up to ten-thousands (6231) to make this pretty useful as a factoring tool if you really needed the correct modulus. If the number is 4723 we just think 4(6)=24--> 3 + 7x2---> still 3, + 2(3)=9 -->2 and then + 3(1)= 5 so 6231 divided by seven leaves a remainder of five.

Ok, so if that seems too hard, here is a different problem I want to explain next time... IF you try to find the decimal number for 1/17 on the TI-84 it says 1/17 = .0588235294... and that's all you get. Now the challenge.... can you write out the complete decimal expansion (one complete cycle of the repeating expansion) of 1/17 using only what the calculator gave you and simple arithmetic... don't do the long division (you don't have to if you know something neat that I will show.. Try looking at 1/7 and 1/13 etc and see if you can find a fact, 1/11 is an incredibly interesting one to look at.... LATER!!!