Piper (my second grader) is learning fractions while we cook in the kitchen, and it is fascinating to see the ways in which fractions are not intuitive. For example, if you remove one egg from a four-egg recipe, it seems intuitive to her to remove one whole cup from the one-and-a-half cups of flour to match.
On the other hand, if you reduce a four-egg recipe to one egg, she gets the intuition right.
That matches the ancient history of fraction notation where Egyptians had a way to write "1/n" without generalizing to "m/n". They formed other fractions using sums (using only distinct n). That system is actually fully general but very complicated.
The ancient Babylonians had a more practical, but less general fractional decimal notation using base 60. (That is still used in minutes and seconds today.) And the Romans had a similarly practical but imprecise notation for "m/12".
The idea of allowing both numerator and denominator to be written was a Hindu innovation. Thanks to the Hindus, it is now possible to successfully pull off a cookie recipe no matter how many eggs are in the refrigerator.
Elementary school teaches you that the friendly way to write a big improper fraction like 3/2 is as a mixed fraction, that is, "an integer plus change" like 1½. Usually discussion of mixed fractions ends there. But mathematicians know there is a lot more fun you can have with mixed fractions if you restrict all numerators to "1".
A continued fraction is a mixed fraction where the numerators of the fraction parts are always "1". The trick that makes this interesting is that you are allowed to let the denominator be another continued fraction. For example, 2/3 can be written as a continued fraction by writing the reciprocal of 1½ like this:
The general procedure to turn any rational number into a continued fraction is this: First, separate and write any integer part bigger than 1. Then in the denominator write the continued fraction of the reciprocal of the fractional part.
That last one is a decimal approximation for the square root of 2. What do you think the exact continued fraction for sqrt(2) would look like?
Back in 2003 Warren studied consumer bankruptcy in The Two Income Trap. She discovered that overextended borrowers weren't being driven into bankruptcy due to unwise spending on luxuries, but in an effort to buy homes in better school districts. She says that the strongest predictor of bankruptcy is having a child and trying to educate that child.
The solution she suggested back in 2003? Improve public school districts to reduce the stark disparity between the schooling you can receive in different towns.
(It is pretty nutty that people spend all sorts of money on a house as a proxy for spending money on education. It has also occurred to me that what really ended the Great Depression was not the New Deal, but the GI Bill. Maybe we should pouring our stimulus into schools and students.)
This morning whitehouse.gov is still chronicling the waning days of the G.W. Bush administration.
As masses congregate in D.C. to watch Obama's inauguration, I await the changeover on the net.
My second-grade daughter is a very clever game-player.
She knows enough about Chess that she can play two chess games simultaneously. She is quite good at it - she likes to play one board as White and the other board as Black.
I'm not saying that she can win every game. But if you pit her against two grandmasters simultaneously, I'd bet you that she can beat one of them, or at least play to a draw, every time. Puzzling.
How does she do it?
Obama proclaimed yesterday, "To those who cling to power through corruption and deceit and the silencing of dissent, know that you are on the wrong side of history, but that we will extend a hand if you are willing to unclench your fist."
Who is he talking about? Obama never said, and that left us wondering aloud.
Today Chinese officials helped to elucidate things by censoring that particular quote from the state media translation of Obama's inaugural speech. The irony of their deceit, corruption, and silencing of dissent - it deserves a grim wince.
The true majesty of the incoming administration was on display the day after the inauguration, with Obama's first two actions. One, a revocation of Bush's executive order on the secrecy of old White House records, and second with Obama's issuance of a remarkable new executive order on White House ethics.
Both actions elevate the the legitimacy and authority of the office by proclaiming: the office of the Presidency belongs to the people, not to the person who happens to occupy it.
WSJ: Geitner accuses China on Thursday of "manipulating" its currency, a sharp rhetorical break from the Bush administration's approach to Beijing's controversial exchange-rate policy...
Bloomberg: China's propaganda machine responds: "China has never tried to gain advantage in international trade by manipulating its currency".
Ah, OK, now I get it. Maybe China's state banks are just run by a bunch of coin collectors who have built the world's largest $2T US currency collection because of its own intrinsic art value. It has got nothing to do with manipulating exchange rates. Maybe I should start my own RMB collection here in the States. Oh, wait....