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Continued Fractions

Elementary school teaches you that the friendly way to write a big fraction like 3/2 is as an improper fraction, that is, "an integer plus change" like 1½. Usually discussion of improper fractions ends there. But mathematicians know there is a lot more fun you can have with improper fractions if you restrict all numerators to "1".

A continued fraction is an improper 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:

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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 factional part.

You can use the form above to try any rational number as a continued fraction. For example, try 3/5, 355/113, or 1.414.

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?

Rationals are Nonobvious

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.

Yellowstone Swarm

Is the world's largest supervolcano about to erupt? An unprecedented cluster of several hundred small earthquakes has shaken the Yellowstone caldera in the last few days.

Three times in the last 2.1 million years, the Yellowstone volcano has an eruption about 1000 times larger than Mount St Helens, covering much of North America with a thick layer of ash.

As Yellowstone geologists explain,

"Precursors to volcanic eruptions include strong earthquake swarms and rapid ground deformation and typically take place days to weeks before an actual eruption."

The quakes continue and you can follow them in real time. A GPS graph measuring uplift is here.

Since 1975, when geologists discovered that Yellowstone caldera had risen 28 inches in 50 years, the caldera has been closely monitored. Since 2004 the uplift accelerated to 3 inches per year.

It has been about 640,000 years since the last major eruption. Time for another burp?

(Update: my father-in law who is a geologist says, probably not. He says the earthquakes in the current swarm are small potatoes.)

RMB in the NYT

The Chinese exchange rate is in the spotlight today in the New York Times.

In Spending We Trust

Obama seems to underestimate local government's ability to be clever in finding ways to spend money with strings attached. The incoming administration's promise of "shovel-ready" investments is bad news for America: it means that hundreds of billions of dollars of transportation investments are about to be wasted on a patchwork of short-term repair and repaving projects.

Foreign Affairs on China

In the current issue of Foreign Affairs, Roger Altman pens an analysis of the Great Crash of 2008 and explains why China is both a causer and a beneficiary of the crisis.

Junichiro Obama

The country faces an historic economic collapse caused by an asset-bubble crash and years of uninspired leadership. Now a charismatic national leader campaigning on a platform of change wins a surprise victory in his party. His overwhelming popularity sweeps a new generation of liberal democratic leaders into power. What happens next?

Of course, I am describing Japan in 2001, and we know what happens next: over the next few years, the country emerges from an historic deflationary slump and reforms a number of ossified institutions. Junichiro Koizumi was Japan's Obama. For a possible preview of the beginning of the Obama administration, look a couple years ahead in this 2003 Asia Times analysis of Koizumi's reelection.

Helicopters Take Off

Here is a nice Forbes article that anticipated Bernanke's zero-interest rate announcement. It notes that the Fed meeting was just a way for to signal that his monetary helicopters are flying.

Once the Fed decides to mint money, it doesn't actually have to do anything exotic to push rates negative if it wants. The Fed could start buying back treasuries for higher than face value instead of issuing new debt, as happened organically in the short-term treasury markets recently.

But there is probably a better way to drop cash into the American economy. The trade deficit tells us that there is no fundamental shortage of consumer spending in the U.S., even with the current crazy contraction. The American shortage is in production, manufacturing, infrastructure and business investment.

Here's to hoping that Ben doesn't give $5,000 away to consumers as Forbes anticipates. We are better off sending minted money to industry and infrastructure - it is the difference between giving a man a fish and teaching him to fish.

Amazing Transparency

Have you been following Obama's change.gov website? It is an incredible vision of what 21st century government should look like. The incoming administration has been posting every recommendation submitted by interest groups, providing a public discussion forum for each one. The comment threads seem noisy and uninformed sometimes, but it is clear that the website is also attracting knowledgeable people who are scrutinizing and challenging the proposals.

I have posted my two cents on afterschool education, transportation policy, and historic preservation.

I've also tried to raise my concern that China's trade policy is the largest obstacle standing in the way of U.S. economic recovery plans.

Change.gov is turning out to be a remarkable experiment. If you have ever thought that the world should be run differently, you should pitch in.

I hope the Obama team finds a way to actually improve its governance by tapping into the community that it has incubated.

A Reflective Phobia

If you have hippopotomonstrosesquippedaliophobia, you have a fear of long words.

To name your fear is to conquer it.

(Anthony brought this home from fifth grade.)

Trade Deficit Arithmetic

My kids will tell you that 40 > 14. The Bush administration doesn't seem to care. The $14b auto bailout football should not be dominating the Treasury agenda today, because the real problem is in China's record $40b subsidy of the low Chinese yuan this month. That represents a rate of 480 billion dollars of disincentives for non-Chinese manufacturers every year.

Perfect Numbers

When playing taxman, there are numbers like 6 where the taxman will tie your first move. Another number where this happens is 28 (try it out). When a number equals the sum of its proper factors, it is called a "perfect number".

Perfect numbers are surprisingly few and far between, and mathematical knowledge about perfect numbers is also surprisingly incomplete.

The first few perfect numbers are: 6, 28, 496, 8128, and 33550336; and they quickly get big enough to stretch the limits of computers. Only a few dozen perfect numbers are known. And while mathematicians since the Greeks have suspected that there are an infinite number of perfect numbers, that conjecture has never been proven. Futhermore, all known perfect numbers are even, but it has never been established that an odd perfect number is impossible.

Despite all the mathematical intrigue, any middle schooler can do the computations to understand the pattern underlying the known perfect numbers. All you have to do is try writing the first few perfect numbers out in binary.

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