Home from Ann Arbor. Three daughters (plus two granddaughters and a great grandson) visited me for the purpose of helping me downsize: choosing things they wanted now or later. They made a big dent in stuff, but not as much as I had hoped. It leaves me free to get rid of the rest.
The Tea Leaf is marked and ready to pack for the auction of the Tea Leaf Club International in Kalamazoo, MI, on September, 29th. My girl friend is helping me and after the meeting we will continue to Niagra Falls (throuch Canada) and back through Ohio.
In early October I will join two nephews and their wives at a bed and breakfast (The Rise and Shine) in Maine. Friends of mine are the Innkeepers.
I hope to return to four colors soon.
Sunday, August 30, 2009
Sunday, August 16, 2009
My Life in Ann Arbor
I love this place. It has all of the amenities of a university town. My apartment is one block off of the University of Michigan campus. I'm within walking distance from almost everything I could possibly want: classes, computer lab, cafes, cleaners, book stores, coffee shops, movies, restaurants (including one Chinese), pizza, drug store, grocery store, ATM machines, clothing store, post office, libraries, shaded walks, barber shop, liquor store,, Art Museum, and Ben and Jerry's, The apartment has a broadband internet connection. There are several places where I can get a wireless connection for my netbook. I run into many people that I know (more than at home). Everyone is friendly. The weather has been to my liking. Unfortunately, the Ritz camera store where I got prints made closed. I can live with that. What else could I wish for? Well, for my girl friend to be here with me. (Gosh, I love that woman.)
UPDATE: I'm back home again.
Sunday, August 2, 2009
How much can you teach a computer?
How much math can you teach a computer? Consider finding an expression for the sum of the numbers from 1 to n.
I. Proof by induction: The computer should be able to do what students are asked to do, but this is not finding an expression, only verifying it.
2. Assume a quadratic form f(n) = an^2 + bn +c
A "little algebra" gives
f(n+1) - f(n) = 2an + (a+b)
the left-hand side of this equation, being the sum of the numbers from 1 to n+1 less the sum of the numbers from 1 to n is n+1. This gives us 2a =1 and a+b = 1 so a = 1/2 and b = 1/2. Our equation is now
f(n) = (n^2)/2 + n/2 + c
Setting n to 1 (or any number including 0) sets c to 0 giving our final equation
f(n) = n(n+1)/2
3. Manipulate the numbers using the distributive law. There are two cases depending on whether n is even (the simpler case) or odd. For the even case:
The sum of the numbers from 1 to n is equal to the sum of the numbers from 1 to n/2 plus the sum of the numbers from (n/2 plus 1) to n. The sum of the numbers from (n/2 plus 1) to n is the sum of the numbers from n to (n/2 plus 1). i.e. reverse order. Using summation pseudo-notation:
Sum(i=n/2 + 1 to n)(i) = Sum(i = 1 to i)(n+1-i)
Then
Sum(i=1 to n)(i) = Sum(i=1 to n/2)(i) + Sum(i=1 to n/2)(n+1-i)
=Sum(i=1 to n/2)(i+n+1-i) = Sum(i=1 to n/2)(n+1) = n/2 times n+1 . . . QED
The odd case is slightly more complicated or one could apply induction to the even case.
4. Graphical solution:
Take n vertical bars numbered 1 to n and set them side by side in ascending order. Each bar should be one unit wide and as many units high as its number. Draw a diagonal from the lower left corner to the upper right corner. The area under the diagonal is n squared over 2. Each of the n bars has half of a square unit extending above the diagonal making an area of n over 2. So n squared over 2 plus n over 2 is n(n+1)/2.
__________________________________________________
How much of this could the computer come up with. I'd like to see an AI expert teach it to learn math.
I. Proof by induction: The computer should be able to do what students are asked to do, but this is not finding an expression, only verifying it.
2. Assume a quadratic form f(n) = an^2 + bn +c
A "little algebra" gives
f(n+1) - f(n) = 2an + (a+b)
the left-hand side of this equation, being the sum of the numbers from 1 to n+1 less the sum of the numbers from 1 to n is n+1. This gives us 2a =1 and a+b = 1 so a = 1/2 and b = 1/2. Our equation is now
f(n) = (n^2)/2 + n/2 + c
Setting n to 1 (or any number including 0) sets c to 0 giving our final equation
f(n) = n(n+1)/2
3. Manipulate the numbers using the distributive law. There are two cases depending on whether n is even (the simpler case) or odd. For the even case:
The sum of the numbers from 1 to n is equal to the sum of the numbers from 1 to n/2 plus the sum of the numbers from (n/2 plus 1) to n. The sum of the numbers from (n/2 plus 1) to n is the sum of the numbers from n to (n/2 plus 1). i.e. reverse order. Using summation pseudo-notation:
Sum(i=n/2 + 1 to n)(i) = Sum(i = 1 to i)(n+1-i)
Then
Sum(i=1 to n)(i) = Sum(i=1 to n/2)(i) + Sum(i=1 to n/2)(n+1-i)
=Sum(i=1 to n/2)(i+n+1-i) = Sum(i=1 to n/2)(n+1) = n/2 times n+1 . . . QED
The odd case is slightly more complicated or one could apply induction to the even case.
4. Graphical solution:
Take n vertical bars numbered 1 to n and set them side by side in ascending order. Each bar should be one unit wide and as many units high as its number. Draw a diagonal from the lower left corner to the upper right corner. The area under the diagonal is n squared over 2. Each of the n bars has half of a square unit extending above the diagonal making an area of n over 2. So n squared over 2 plus n over 2 is n(n+1)/2.
__________________________________________________
How much of this could the computer come up with. I'd like to see an AI expert teach it to learn math.
Saturday, August 1, 2009
Luncheon Encounter
I often meet Mike for lunch at Panera Bread, normally arriving before he does. I was standing in line when the man with a white mustache behind me asked how fast the line moved. I told him fairly fast as there were three cashiers. Just then three in front of us left to go elsewhere so he said it that made it move faster. Then he jokingly asked the others in front of us if they wouldn't like to go elsewhere too. (None did.) When I got my food I selected a table for four. When he got his he asked to join me. (It was rather crowded.) I agreed, of course. He was attending a conference for teachers of physics. He was a retired theoretical physicist from the University of Arizona. I had been wondering how they knew the speed of light and how they knew it was constant and took the opportunity to ask him. The speed can be measured and the original way used mirrors on mountain tops. Maxwell's equations told them it was constant. I didn't get an answer for the reason why it is the same for all observers. Mike joined us then and on hearing the man's occupation, asked me if I had asked the man my questions, which I just had. Mike introduced himself and the man said they called him J D. Mike told them about his work in statistics for the social sciences in which J D showed considerable interest.
Later I looked for him on google. After a couple of false starts I used "physics faculty" with "University of Arizona" which gave a list from which I picked out J D Garcia. From this I located several citations including two which included a picture verifying it was the correct identification.
Later I looked for him on google. After a couple of false starts I used "physics faculty" with "University of Arizona" which gave a list from which I picked out J D Garcia. From this I located several citations including two which included a picture verifying it was the correct identification.
Subscribe to:
Posts (Atom)
