I loved this little problem from math class.

0.999… (with the ellipses representing ‘repeats forever’) equals one.

The first time most people see this it doesn’t quite compute, since conceptually people think of 0.999… as ‘always getting closer and closer to 1 – but not quite reaching it – for infinity’. Wrapping your head around the idea that ‘infinity’ can equal a finite number can make your stomach feel a bit queasy.

The first proof I learned was straight algebra:

1) Let x = 0.999….
Multiple both sides by 10 (if you do the same thing to both sides of the equal side, the equation stays equal). So:

2) 10x = 9.999…

Now, let’s subtract equation 1) from equation 2). The left side is easy: 10x minus x is 9x. For the right side, we could look at it digit-by-digit. For the ones digit (the number just to the left of the decimal point), 9 minus 0 is clearly 9. For the tenth digit, .9 minus .9 is zero. Continuing, .99 minus .99 is 0, .999 minus .999 is zero, and we can continue into infinity – in other words, the entire right side of the decimal point in equation 1) is exactly the same in equation 2), which means that subtracting one from the other is 0. So we’re left with the left side of the decimal point, which we said above was 9.

So: 
   10x = 9.999…
-      x = 0.999…
————————–
     9x = 9

Divide both sides by 9, and we get x = 1.

And x = 0.999… = 1.

Cool, huh?

But my favorite proof was actually one I didn’t learn about until later, even though it’s much easier.

First - remember that 0.5 is the same thing as 1 divided by 2, or 1/2.  1 divided by 4 (1/4) is 0.25, etc. It turns out that one divided by nine (1/9) is equal to 0.111…

It’s clear that 2 times 1/2 is equal to 1. It’s obvious that 4 times 1/4 is equal to 1. So clearly, 9 times 1/9 must equal to 1.

Likewise, 9 times 0.111… must equal 0.999….

But 9 times 0.111… is the same thing as 9 times 1/9, and we saw above that 9 times 1/9 was equal to one. Ergo, 9 times 1/9 = 9 times 0.111… = 0.999… = 1.

Note that you could to this using thirds: 1/3 is equal to 0.333…. Three times 0.333… is equal to 0.999…, and we know that 3 times 1/3 is equal to 1, so 3 times 0.333… = 0.999… = 1.

Don’t forget to bring this up at your next cocktail party! You’ll be a hit!

London School of Economics

Some fascinating podcasts on the London School of Economics Events page. Recently I’ve been listening to Nobel Prize winner Paul Krugman during his three-day event in June 2009, where he dissects the global economic crisis:

The Return of Depression Economics Part 1: The sum of all fears

The Return of Depression Economics Part 2: The eschatology of lost decades

The Return of Depression Economics Part 3: The night they reread Minsky

These are great lectures – Krugman is a good speaker, offering both interesting insights dashed with equal parts history and humor. One drawback to listening to these podcasts, however, was that on occasion it was clear that Krugman was referring to some slides which he was obviously using with his lecture. While the podcasts were great, it would be nice if we had a videocast of the event.

Well – seek and ye shall find: Scroll down to June 2009 for all three of Krugman’s lectures in full-color video.

Dan Carlin: Common Sense and Hardcore History

Dan’s an interesting guy. His Common Sense focuses mostly on current events. I don’t necessarily agree with everything he says, but he is entertaining.

But I find myself listening to his Hardcore History series more. While he would be considered an ‘amateur’ historian, I’ve been impressed at his topic subjects. He has touched on some relatively easy-to-overlook topics. Plus, since he’s such a good presenter, you can really get caught up in his breathless descriptions of events. I love how they include faint-yet-distinctly-there sound effects in the background – for example, you could faintly here horses and elephants in the background as he was discussing Hannibal crossing the Alps.

MIT Open University

I admit, with no sense of shame, to being a bit of a math nerd. I still find myself gravitating to the mathematics section whenever I’m in a book store of any significant size. So imagine my joy when I discovered that MIT has entire courses online – with video! Brilliant! Now – lots of these lectures are way over my head – Electcromagnetics and Applications, anyone? – but I found the Single Variable Calculus course a lot of fun. Like most math teachers the instructor has a geeky sense of humor that only other geeks could appreciate. And the audio sometimes does not catch the questions being asked by the students, which can make it hard to keep up sometimes. But the lecture notes are pretty good, and the video work is excellent. Be warned – the related rates topic kicked my butt back in high school and it will do so again now….

….and already ruling the roost <g>

Making a mixtape for someone you like – how quaint.

I’ve only done it once. It must have turned out pretty good – I’m married to her.

I’m sure heads of state have spent less time and energy on S.A.L.T. negotiations or GATT trade agreements compared to the time and effort we spent on deciding on a name for our baby! Part of the problem was that we had a couple of names that we all kinda liked, but that were relatively common, while one of the names we all liked in particular was a bit unusual. Ultimately we went with the unusual name. So without further ado, our baby’s name is (drum roll):

ADAIR ANTHONY (First name, middle name)

Adair (a-DARE) is a somewhat common last name but very rare as a first name. The name is Scottish/Celtic in orgin, and means ‘from the ford by the oak trees’, which is kinda neat since there is a small tree-lined creek right behind our house. The story behind the name for us, however, is that the name pronounced in Japanese sounds like ‘thank you’ in Turkish…and we did spend Easter holiday last year in Istanbul. Hmm.  -

Anthony is my middle name, and I’ve always liked it.

After deciding on the name, I did a quick Google search to see if there was anyone out there with that name – found lots of people with Adair as a last name, but only turned up one first name…but it was pretty impressive – Lord Adair Turner is the chairman of the FSA, the UK equivalent of the SEC in the US. So a big name in UK finance, with a title of nobility to boot! Hopefully that’s a good omen!

…another Urista to unleash upon the world! Meet ‘Baby’, born 12:10am on Monday, January 4, the Year of the Tiger. He was 3,070 grams (6.7lbs) and healthy and perfect in all respects:

It was a long, 24-hour labor, but Mom & baby are doing fantastic! We don’t have a name yet, so he’ll be ‘Baby’ for a while yet.

Two interesting articles in this week’s Nikkei Veritas. (http://veritas.nikkei.co.jp/index.aspx – all in Japanese, and you’ll need to buy a subscription to read the whole edition).

First – the feature article spends about four pages talking what Japan might look like for the typical household in 2020. It doesn’t paint a pretty picture – average annual GDP growth of only around 1%, average number of people paying into national pension/health insurance per senior citizin falling from 2.82 in 2009 to 2.05, a 4 million person decline in the total workforce, number of nursing care providers needed to take care of an aging population increasing from 1.2 million in 2008 to 2.14 million…

The article suggests a number of possible proposals to address these problems: Pushing back the retirement age to 67 or even 70 (one assumes that this would be in tandem with a hike in the minimum age at which one can receive benefits), or raising the consumption tax to at least 10%. Another big initiative is to get senior citizens and women into the workforce. This, of course, has two drawbacks – first, having more women in the workforce probably won’t do Japan’s already-low birth rate any favors. Second, Japan already faces a shortage of nursing/health care providers – having more senior citizens and women leave the home and move into the workforce isn’t going to help matters. We saw above that Japan will need another 1 million care providers by 2020, and that’s without having any increase in women/senior citizens participating in the labor force.

Finally -irony of ironies – one major topic was expanding the tourism trade to attract 20 milloin people to visit Japan by 2020, more than double the current level.

I say that is ironic because NO WHERE does the article mention the best and most obvious choice for a country with a dwindling workforce and rapidly aging population: Immigration. Tourists, yes! Come spend your money here. Just don’t stay too long. That immigration wasn’t even mentioned suggests Japan would rather fade into an irreversible decay of sub-standard living standards and global irrelevance before it let more foreigners in. The biggest danger I see is that the everyday man on the street seems to think that ‘no economic growth’ will mean things just stay as they are – which they might IF the country wasn’t already facing a declining workforce and aging population.

Even without going into the deep economics of it, immigration makes sense conceptually. For example: For the first 20 years or so of their lives, children are a massive fiscal drain for a country – children need health care and education, but selfishly don’t work to offset the costs. In return, however, they work for 40 years or so after getting out of school, and pay in to the pension/healthcare system. Once they retire, however, they have to depend on the country again for 20 years or so.

Immigrants, on the other hand, tend to be young adults – just at the starting points of their careers. And some (many?) end up going back to their native lands at some point. In other words – immigrants often make a big fiscal contribution to the country by being here in there working years, but are often not a drain on the country because they aren’t here as children and/or retirees. If Japan is serious about tackling is problems of future growth, immigration is going to have to be on the ticket.

The second article is by Takeshi Nakano, a To-dai grad and former MITI (MIET) wonk currently working as a professor at the University of Edinburgh. He argues that free trade is wrong, and that protectionism is good – that protecting local industries results in thriving local eocnomy, and that thriving economy is what boosts trade. In other word, free trade is a result of – not a cause of – economic growth.

Somewhere, Adam Smith and David Ricardo are spinning around in their graves as 250 years of economic thought are flushed down the drain.

Nakano’s first evidence is the 19th century: an era of global free trade from 1860 through 1892 or so, coincided with the long global depression (dates vary, but generally run from 1870 through the early part of the 1890s for Europe, and much shorter for the US, 1872-1879), and that countries only pulled out of the (series of) recessions from 1892-94 as European governments started implementing protectionist policies. He also says that overall trade flourished over the latter half of the 19th century – and that trade increased fastest in countries with the most protectionist policies, but that England – which maintained the most open trade policy – suffered the most severe downturn. Finally, he says that Paul Krugman agrees with this view, in a Feb 1 2009 NY Times piece (full piece here: http://krugman.blogs.nytimes.com/2009/02/01/protectionism-and-stimulus-wonkish/)

His thinking is shaky at best. The late 19th century depression is mostly considered to be a fall in prices, not a depression-level fall in actual output. Secondly, he is so far wrong on his dates that one can only conclude he is being deceptive on purpose: many nations did indeed turn to protectionism as the recession worsened – but far, far earlier than Nakano says. Germany and France dropped their free trade stance in the late 1870s, for instance. One could suggest that these protectionist movements were a reason the recession lasted so long in the first place. Nakano also fails to note that this was long before Keynsian economics came along to change in thinking regarding government involvement in national economic strategy.

Worse, he completely (deliberately?) mis-reads Krugman’s article and cherry-picks his quotes. Krugman is clearly noting that the extreme circumstances of the time and the short-term/long-term trade-offs and risks. Protectionism in the form of subsidies and tax incentives is NOT the same thing as restricting imports and increasing tariffs.

Finally – Nakano simply fails to adhere to one the most basic principals of economic thought: What is good for one might not be good for the whole. If I stand up at a football match, maybe I have a better view (to the detriment to the people behind me). If everyone in the stadium stands up, I may be no better off – or indeed, I maybe worse off. Germany and the rest of Europe may have benefitted in the late 19th century specifically because the UK and (for the most part) the US continued to buy their goods, even though they did little to return the favor.

Nakano’s thinking is clearly that protectionism is better for me, as long as you keep buying my stuff.

A classic from the Straight Dope:

Dear Cecil:

Over one hundred years ago, on September 21, 1897, a little girl with great doubts asked the editor of the New York Sun for the answer to a question that had been bothering her. There was no Straight Dope then, so she had to settle. The Sun came up with an answer, a good answer, the correct answer. But folks have forgotten it, or no longer believe it. The man who answered her question was just a staff writer who got the assignment from his boss. He wasn’t the World’s Smartest Human, like you are. He didn’t command the respect that you do. So, I hope you won’t mind settling this question once and for all, for all the little Annies, Ryans, Joshes, Megans, and Tammys in the world. If I may paraphrase:

Dear Cecil: I am 47 years old. Some of my friends on the Straight Dope Message Board say there is no Santa Claus. JKFabian says, “If you see it in the Straight Dope it’s so.” Please tell me the truth. Is there a Santa Claus?

— Ranger Jeff, The Idol of American Youth

Dear Jeff:

Let’s just say his existence can’t be definitely ruled out.

I’m not saying there aren’t improbable aspects to the story. You have x number of kids (even leaving out the Muslims, Shintoists, Hindus, animists, etc., who one presumes get shafted, giftwise), you have y time per visit, you have z average distance between domiciles, you have an earth of known diameter, and you have 24 hours in the day. It doesn’t add up. You have the problem of access to the gift-giving venue in the absence of chimneys with fireplaces, unless we’re assuming that Santa Claus oozes through the keyholes in the manner of the critter in The Abyss, which is not a pretty picture. You have the problem of what in all likelihood is the earth’s single largest concentration of toy manufacturing facilities in a polar region remote from resources of every type (cold excepted), that’s so carefully camouflaged as to be invisible to satellite surveillance, and that produces no detectable emissions. Although now that one thinks about it, there’s that ozone hole over the south pole. Hmm.

On the other hand, consider the following:

• A great many seemingly improbable events do in fact occur. Florida winning the World Series. Cleveland winning the World Series. Compared to this, what is the accurate delivery of zillions of packages in the course of a single night?
• Besides, Fed Ex does it. So what if we’re talking Memphis and drivers in baseball caps rather than the north pole and elves? It’s the principle of the thing.
• OK, so there’s a certain amount of mortal participation involved. Perhaps, as a parent, you’ve personally done your bit to help Santa and thought you did so of your own accord. The ants in the anthill probably think they’re doing it on a whim, too. But looking at the matter objectively, we can’t deny that a larger purpose is at work and that we are in the service of an agency greater than ourselves.
• You mean the IRS.
• I mean the impulse to be generous. Three hundred sixty-four days out of the year humankind commits all manner of heinous acts. On the 365th day we give toys to the kids. I’m not saying that the latter compensates for the former. I’m not saying Adolph Hitler wouldn’t have given presents to his children, if he’d had children. But come on, it’s got to count for something. The giving of gifts in such a way that no credit will devolve upon ourselves is sufficiently at odds with our routine behavior as to be accounted a mystery, and we may as well give that mystery a name. Santa Claus it is.
• Besides, to believe in Santa Claus is to believe in magic. The belief in magic in many respects is a pernicious thing. Because of it you’ve got countless multitudes thinking that aliens abduct people, that Elvis is alive, that you can earn big money stuffing envelopes in your home, and that the TV preacher can cure you if you send him 50 bucks. A certain class of persons, of whom your columnist is one, will go through their lives attempting to extinguish these foolish hopes. No doubt in the main it is good that we do so. But even the sternest among us remembers the wonder we felt as children to think there was a force having a kindly interest in us that wasn’t bound by the rules of this drab world. Wherefore if there’s someone who’s going to say flat out that Santa Claus doesn’t exist, it’s not going to be me.