Saturday, 31 March 2012
Friday, 30 March 2012
Thursday, 29 March 2012
Almost Fibonacci Collection
Fibonacci Statue in Pisa, *plus.maths.org |
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181... Ok, You get the idea...
I just came across a couple of odd references to things that produce Fibonacci like sequences, at least for a short while...
It started when I posted a note on the 89th day of the year (You do the math) that 89 is the fifth Fibonacci prime and the reciprocal of 89 starts out 0.011235... (generating the first five Fibonacci numbers) (which I found at the Prime Curios page. (Ok footnote here, I didn't realize until just recently that if the nth Fibonacci number is prime, then n is also prime, with one exception. but not the converse)
Don S McDonald (@McDONewt) then sent me another... 69 Choose 5 = 1 1 2 3 8 5 13 which is so nearly perfect...
Now I'm thinking there must be more of these... so fire away... Send them in and I'll make a collection we can all share..
In the comments, Joshua Zucker pointed out that the decimal expansion of 1/89 goes well beyond 0,1,1,2,3,5.. The next digit is nine, instead of 8, because it includes the tens digit fromthe 13 that would follow, and the following digit is the sum of the 3 from 13, and the 2 from 21, and you could continue this way indexing the next Fibonacci term one to get more. When Joshua said all the digits, he didn't mean ought to thousands of digits, since the period of the fraction is only 44 digits.
Wednesday, 28 March 2012
Tuesday, 27 March 2012
Monday, 26 March 2012
Sunday, 25 March 2012
Saturday, 24 March 2012
Friday, 23 March 2012
Thursday, 22 March 2012
Wednesday, 21 March 2012
Tuesday, 20 March 2012
Monday, 19 March 2012
Sunday, 18 March 2012
Saturday, 17 March 2012
Friday, 16 March 2012
Thursday, 15 March 2012
Pi Tag
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862
8034825342117067982148086513282306647093844609550582231725359408128481117450284102701
9385211055596446229489549303819644288109756659334461284756482337867831652712019091456
4856692346034861045432664821339360726024914127372458700660631558817488152092096282925
4091715364367892590360011330530548820466521384146951941511609...
If you pick some number, any number, and look for it somewhere in the sequence of digits of pi, it will show up... I can't prove that, but I believe it.... For example, I was born in October, the tenth month. The digits 10 show up starting at the forty-ninth decimal place of pi (I ignore the 3 before the decimal point just to avoid counting complexities).
Now if you take the location as a new number, and look for the place where that occurs in pi, you create a sequence of numbers that must do one of three things. Either they keep going, jumping up and down maybe and go off to infinity (or at least to a really big number), or they cycle through the same set of numbers, or they simply go to a single number which happens to be in the position of its numeral value.... like 1. The first number after the decimal point is one, so we just keep going back to one...
But if you start with four, which is in position two, and two is in position six, the string looks like this
4 , 2, 6, 7, 13, 110, 174, 155, 314, 2120 , 5360... eventually I got to 119,546 and I give up on that one... maybe it actually gets smaller... One of those big numbers might turn up in place 141 or 1415 or 14159 and each of those would take us back to position one, but I'm willing to say, for the moment, if they go past 1000, they've gone out of bounds... Folks with big computers and the patience to program them may find a way to take this out to a plethora of digits, but I'm working with one crunch at a time...
As you go through the simple numbers it seems that all of them run off to big numbers, then 14 shows up and it goes to position one.
But if you try 19, you find it is in position 37. 37 turns up in position 46, but 46 occurs in position number 19, and we have a cycle... trivia alert, 19 is the smallest number in Pi Tag that has a non-trivial (goes to one and repeats ones forever) cycle.
I'm really hoping I can find a two -cycle... a number that gives a number that brings us back where we started... a leads to b, leads back to a.... Sort of a buddy system... there could be lots of them...
If you want to play around with this, there is a pretty nice web site that lets you put in any number and see what position it shows up in... and it handles really big stuff, I put in my social security number just to see if it's location was bigger than the number or smaller...turns out it only goes up to 200 x 106 and my number didn't show up by then...
The page is here if you want to give it a go.
It's pretty easy to check because as soon as a number goes to a number lower than itself, you know what it will do (you've already checked that number) for example, 23 is in position 16, but I know 16 goes off to above 1000, so stop and go to the next number...
And if you find a two-cycle before I do, or any other cycles, drop me a note..... and now back to work...
Wednesday, 14 March 2012
Tuesday, 13 March 2012
Simon Stevin's Non-fraction method of Decimals
------------------------------------------------------------------------------------------------
A while back, John Cook at the Endeavour Blog posted about a comment by Keith Kendig in his book, Conics.
It happened when I started learning about decimals in school. I knew then that ten has one zero, a hundred has two, a thousand three, and so on. And then this teacher starts saying that tenth doesn’t have any zero, a hundredth has only one, a thousandth has only two, and so on. … Only much later did I have enough perspective to put my finger on the problem: The decimal point is always misplaced!
John demonstrates the proposed solution as well.
The proposed solution is to put the decimal point above the units position rather than after it. Then the notation would be symmetric. For example, 1000 and 1/1000 would look like this:
Of course decimal notation isn’t likely to change, but the author makes an interesting point.
I then commented on John's blog that, in fact, Simon Stevin, who probably is more responsible than anyone else for introducing decimal numbers into the west had used a very similar approach as the one suggested by Keith Kendig. The image below is from De Thiende, which translated into English appeared as Decimal arithmetic. In fact, Stevin didn't think of his method as using fractions at all. In fact the English Translation in the full, self-advertising manner of books of the period, was Decimal arithmetic: Teaching how to perform all computations whatsoever by whole numbers without fractions, by the four principles of common arithmetic: namely, addition, subtraction, multiplication, and division. (My emphasis)
So, as I mentioned at John's blog, "He seems to have viewed the values as integers, much as we now think of minutes and seconds as integers. Few people consciously think of 3 minutes as 3/60 of an hour in regular computations. This was the view that Stevin took. He did not even use fraction names for the place values, but referred to them as prime, second, third, etc. (It has been often suggested that the use of ', ", etc for the minute and second in time, 12 23' 13", date back to the Greeks measure for angles of arc, but Cajori finds no evidence of their use prior to the 16th Century. The names minute and second came from the Latin for "minor part" which gave us minute, and the "second minor part" which gives us seconds.)
![]() |
Decimal fractional numeration and theGlen Van Brummelen |
The notation, in spite of the objections of folks like Mr. Kendig, didn't seem very useful, and so in 1612 Bartholomaeus Pitiscus opted for the decimal point we use today, and when it was used by John Napier, well here we are.
His decimal points first appear in his 1608 edition of Trigonometria in the added trigonometric tables and can also be found in the 1612 edition. (The word 'trigonometry' is due to Pitiscus and first occurs in the title of his work Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus first published in Heidelberg in 1595 as the final section of A Scultetus's Sphaericorum libri tres methodicé conscripti et utilibus scholiis expositi. In 1600 a revised version of Pitiscus's work was published in Augsburg as Trigonometriae sive de dimensione triangulorum libri quinque. )
Now if all Simon Stevin had done was introduce a really nice book on decimal fractions, you could give him a pat on the back and a big
"ataboy" and send him on his way, but:
He was big on hydrostatics, handy if you are Dutch, and in fact he made many improvements in the Dutch windmill pumping system. He also figured out that water pressure depended on height, and not on the shape of the container, and he was the first to explain the tides as the effect of the moon.
And for a walk on the wild side, he invented those land yachts you see racing up and down beaches. The one he made for the Prince of Orange was said to outrun the horses (with 26 passengers!).
he was one of the first to write about the equal temperament musical scale related to the twelfth root of two, which he seems to have gotten from Galileo's father.
And in math, he was the first in the west to write about a general solution to the quadratic equation; which alone should make him a name known to high school students and teachers.
In the Stevin plaza on the statue of Stevin is an image that many find curious. A nice explanation below comes from the Futility Closet of Greg Ross.
"Drape a chain of evenly spaced weights over a pair of (frictionless) inclined planes like this. What will happen? There’s more mass on the left side, but the slope on the right side is steeper. Simon Stevin (1548-1620) realized that in fact the chain won’t move at all — if it did, we could link the ends as shown and produce a perpetual motion machine."
This is remembered as the “Epitaph of Stevinus.” Richard Feynman wrote, “If you get an epitaph like that on your gravestone, you are doing fine.”
the statue in the middle of Simon Stevinplein in Bruges.
It is important to distinguish between the (re)invention of the decimal point, and the creation of decimal fractions.
In a mercantile context, Leonardo of Pisa had come close to a pure representation of decimal
fractions in his 1202 Liber abaci by extending from metrological considerations, where (for instance) he represented the number
71.7579463519 as 9 1 5 3 6 4 9 7 5 7 71.
10 10 10 10 10 10 10 10 10 10
Note the fraction is read from right to left and if you read the value reading only one upper digit at a time the first approximation is 7/10. When you add in the five you multiply all the tens below and right of it to get 5/100 more or 75/100.
Monday, 12 March 2012
Prime Curios, Plum Wonderful
One of the things I post each day on my "On This Day in Math" blog is something about the number that day of the year is. For instance yesterday was the 71st day of the year. I checked the Prime Curios page and it told me (among a bunch of other stuff that I will mention later) that 712=5041 = 7! +1! ...
As a teacher, I loved that kind of stuff because it was so easy to generalize problems from it. Kids do a lot of simple math exploration looking for patterns like this, and I think a lot more happens when they are doing math that they want to do.
So why mention that now. Well I was just reminded that Chris Caldwell from Tennessee Martin, (just down the road from my home here near Possum Trot, Ky) and his coauthor, G. L. Honaker, Jr. who run the Prime Curios page, also put out a book a couple of years ago (2009) called (wait for it) Prime Curios! The Dictionary of Prime Number Trivia.
So I'm plugging the book because I hack off his page regularly and think every math teacher in the country ought to have a copy on his bookshelf for kids to finger through, "Happy birthday kid, you were born on the XYZ day of the year...did you know...."
Anyway, just to give you a feel for a day, here are some things from the page on the prime number 31, because it has a note about a most unusual speed sign.
"The speed limit in downtown Trenton, a small city in northwestern Tennessee, is 31 miles per hour." They should know about this because Trenton is only a little down the road from Martin, where Chris teaches.
And the little teapot on the sign? Well, Trenton also bills itself as the teapot capital of the nation. The 31 mph road sign seems to come from a conflict between Trenton and a neighboring town which I will not name ,...but I will tell you they think of themselves as the white squirrel capital. Yeah, I know you city folks don't get off the freeway, and think I'm kidding, so here's their "Welcome sign".
Some other good notes from the book on 31:
The sum of the first
31 odd primes is a prime square.
(ask the students to find out that sum and that prime and watch how they attack it to learn a lot about how they attack problems)
The sum of digits of the 31st Fibonacci number is 31.
The big "31" sign made its debut at all Baskin-Robbins
stores in 1953, offering customers a different ice cream for every
day of the month. Note that 31 is the largest prime factor of 1953."
Ok, where else can you find neat details like that... or like these... and remember, these are all on the SAME page:
"There are 31 milligrams of cholesterol in a tablespoon of butter."
"31 and the 31st prime are both Mersenne primes."
"The smallest prime that can be represented as the sum of two
triangular numbers in two different ways (21 + 10 and 28 + 3)"
and that's part of the stuff that's on ONE of the pages... you want this book... it's just a fun book.
As a teacher, I loved that kind of stuff because it was so easy to generalize problems from it. Kids do a lot of simple math exploration looking for patterns like this, and I think a lot more happens when they are doing math that they want to do.
So why mention that now. Well I was just reminded that Chris Caldwell from Tennessee Martin, (just down the road from my home here near Possum Trot, Ky) and his coauthor, G. L. Honaker, Jr. who run the Prime Curios page, also put out a book a couple of years ago (2009) called (wait for it) Prime Curios! The Dictionary of Prime Number Trivia.
So I'm plugging the book because I hack off his page regularly and think every math teacher in the country ought to have a copy on his bookshelf for kids to finger through, "Happy birthday kid, you were born on the XYZ day of the year...did you know...."
Anyway, just to give you a feel for a day, here are some things from the page on the prime number 31, because it has a note about a most unusual speed sign.
"The speed limit in downtown Trenton, a small city in northwestern Tennessee, is 31 miles per hour." They should know about this because Trenton is only a little down the road from Martin, where Chris teaches.
And the little teapot on the sign? Well, Trenton also bills itself as the teapot capital of the nation. The 31 mph road sign seems to come from a conflict between Trenton and a neighboring town which I will not name ,...but I will tell you they think of themselves as the white squirrel capital. Yeah, I know you city folks don't get off the freeway, and think I'm kidding, so here's their "Welcome sign".
Some other good notes from the book on 31:
The sum of the first
31 odd primes is a prime square.
(ask the students to find out that sum and that prime and watch how they attack it to learn a lot about how they attack problems)
The sum of digits of the 31st Fibonacci number is 31.
The big "31" sign made its debut at all Baskin-Robbins
stores in 1953, offering customers a different ice cream for every
day of the month. Note that 31 is the largest prime factor of 1953."
Ok, where else can you find neat details like that... or like these... and remember, these are all on the SAME page:
"There are 31 milligrams of cholesterol in a tablespoon of butter."
"31 and the 31st prime are both Mersenne primes."
"The smallest prime that can be represented as the sum of two
triangular numbers in two different ways (21 + 10 and 28 + 3)"
and that's part of the stuff that's on ONE of the pages... you want this book... it's just a fun book.
Sunday, 11 March 2012
Saturday, 10 March 2012
Friday, 9 March 2012
Thursday, 8 March 2012
Wednesday, 7 March 2012
Tuesday, 6 March 2012
Sunday, 4 March 2012
Saturday, 3 March 2012
Friday, 2 March 2012
Thursday, 1 March 2012
Subscribe to:
Posts (Atom)