News for number lovers

January 25, 2011

When I read Norman Juster’s The Phantom Tollbooth, I always find myself curiously annoyed that the princesses Rhyme and Reason (who need to be rescued from the Castle in the Air to bring sanity back to a land in chaos) determine that words and numbers are equally important. (I am relieved to learn from an interview with Juster about his famous book that he wasn’t trying to make any grand point regarding that conflict – like most of the book, it was just about having fun with language and ideas.)

I’ve always loved words (and obviously Juster has also). Numbers can occasionally be interesting, but mostly they’re just useful tools. Without numbers, I couldn’t be writing this blog post because there wouldn’t be any computers. Without the technology that numbers enable, we’d probably all be cave dwellers. But I could at least share stories around the fire. Without words, there would be no human society (at least not as I think of society), and no use for technology even if it existed.

Recently I took a quiz that is supposed to be able to give some indication whether a person has an autism spectrum disorder. While I don’t have autism, my score is closer to that of many autistic people than to the average non-autistic person. I’m not good at chitchat or social situations in general, I prefer a library to a party, and I notice patterns in things a lot (I’m not sure if I can quite say “all the time” but it seemed close enough).

But I cannot say I am exactly fascinated by numbers, as people with autism spectrum disorders often are. (My younger son, who is considered autistic, isn’t much of a numbers person either. I help him almost every week with his homework for his Extended Learning Program math class, and while the math is easy enough for me, I have trouble figuring out how to help him arrive at the answers himself.)

All that said, after reading an article about an important mathematical discovery, I decided I needed to post this if only because I figured it would interest my sister Margaret (in case she hasn’t already read this). For myself, I find myself reacting about as I would to the discovery of some new species of beetle – well, that’s nice, and I’m sure it’s important to some people, but I just can’t get too excited about it.

Frankly, I’d never even heard the term “partition numbers” before – or if I had, it had completely escaped my notice. Even after reading the article, I am uncertain what makes them so important. And I can’t muster the interest to figure out for myself why they are, though I’m more than willing to believe that this discovery about them is as breathtaking as the article says it is.

I did enjoy the article though – after all, it’s full of words.

Whether and how to change the flag

July 2, 2010

Even since I first heard of Puerto Rico when I was a child, I’ve heard arguments about whether or not it should become our 51st state. As with other complex and controversial topics, the arguments I hear or read often sound convincing – until someone else presents a contrary view.

On the whole, I tend to lean toward agreeing with the proponents of statehood, both for reasons of principle and pragmatism. As to principle, why should citizens of this country not have the same kind of voting rights and elected representatives at the federal level? The practical reasons have to do with the economic boost that statehood proponents believe would occur, as it has with other states that entered the Union.

Trying to predict economic outcomes, of course, is difficult at best. But I do think that statehood proponents have a point when they point out the flaws in the economic arguments of opponents to statehood. The latter group claim that since rates of poverty are so high in Puerto Rico, having Puerto Ricans pay federal income tax would generate little revenue, while more tax dollars from the existing fifty states would flow into Puerto Rico.

The question is whether the current state of the economy in Puerto Rico would persist. The opponents of statehood seem to assume that it would. The proponent of statehood point to studies that purport to show that the island’s economy would experience a significant boost. People who know far more about economics than I do can’t agree on the matter, so I’m not going to try to render an opinion. But I do know that the economy is so complex, influenced by so many interdependent factors, that you can’t change a few factors and expect the others not to change also.

The purpose of this post isn’t to argue for or against statehood, however. If the subject interests you, there are a variety of website that discuss the matter. The U.S. Council for Puerto Rico Statehood is – as the name says – for statehood. No Statehood for Puerto Rico and ProEnglish oppose it. This one gives a fairly balanced view, I think, of the issues from both perspectives.

What I found interesting this evening was a far easier question: How could we rearrange the stars on our flag to add in one more? Fifty-one is three times seventeen, but it would hardly work to have three rows of seventeen stars. You could split seventeen into eight and nine, and have six rows of eight and six row of nine, but then you wouldn’t have the nice symmetry of today’s flag, with longer rows of stars at both top and bottom.

Of course, if Puerto Rico became a state, might there be other territories desiring the same status? How would you make a flag with fifty-two stars, or fifty-three? Fortunately, a mathematician and a computer can offer practical solutions to these questions far more easily than economists and politicians can answer the thornier questions regarding statehood.

Check here for an interactive flag calculator that lets you see possible configurations for anywhere from one to one hundred stars – with three exceptions for which there are no valid patterns (at least not using the six most common star configurations). Many numbers offer two or more possible patterns (try clicking on the long, short, alternate, equal, wyoming, and oregon buttons when they are not grayed out).

Not a day over 30

January 14, 2010

[Published 1/E/7DA]

I am 30 years old today. And as my sister is reading this and will no doubt protest at this statement, let me point out that I did not say what numbering system I am using.

I first learned about alternatatives to the decimal system when I was in fourth grade. The class was divided into two groups for math, the “Houghton-Miflin” group, which used a blue textbook published by Houghton-Miflin, and the “Addison-Wesley” group, which used a green textbook published by Addison-Wesley. I was in the Houghton-Miflin group, which was for the more advanced students. While the teacher (whom I can’t remember at all, as math was the one subject where we left our regular classrooms and went to another teacher’s room) worked with the Addison-Wesley group, those of us in the Houghton-Miflin group worked largely on our own.

That was fine until we got to a chapter on base 8. I was good at math and had never been stuck on any concept before, but I simply could not make sense of this strange system. I don’t know if Houghton-Miflin didn’t do a good job of explaining it, or if I just had some kind of mental block. I simply could not make any of the problems come out with the answers given in the back of the book.

Fortunately a classmate (he had dark curly hair, but I have no idea what his name was, though I think it started with an “A”) came to my rescue. He showed me a way to do the problems that enabled me to get the right answer. It had something to do with subtracting or adding two at certain points, and it worked, as I now was able to get the answers in the back of the book. But I had absolutely no idea why it worked that way.

Some fifteen years later, I began studying computer science, because I wanted to become a computer programmer. One of the skills we had to master was converting numbers between our familiar decimal system, and the binary system used by computers. Because binary numbers are awkward for humans to read and write (the year 2010 in binary is 11111011010), programmers use octal (base 8) or hexadecimal (base 16) representations of these numbers.

If you’re not familiar with different numbering systems, I’m not sure whether I can explain it better than my Houghton-Miflin textbook did. Here’s a site which explains them, though as I already understand it I can’t say whether this is a good introductory explanation. In a nutshell, just as the decimal system has the “ones” place in the first position on the right (assuming only whole numbers with no decimal point), then the “tens” to the left of that, then the “hundreds” then “thousands” and so on, each “place” being ten times the one to its right because it is base ten, any other number system works the same way but multiplying by whatever base you are using.

So binary has the ones place, then the twos place, then the fours then the eights and so on. The number we call 14 (I like that number, as I tend to get presents on that day of the month at the beginning of each year) would be represented as 1110, because it is made up of one “eight” (2 x 2 x 2) plus one “four” (2 x 2) plus one two (and no ones). In octal, that same number would be 16, because there is one eight (one in the eights place) and six ones.

Of course it gets confusing trying to discuss these numbers using digits from the decimal system. The number that we call fourteen in decimal couldn’t be called fourteen in octal, but it wouldn’t make sense to call it sixteen either, even though we write it as 16. Because the “teen” comes from the word ten. So if we actually used octal in everyday life, perhaps 16 would be called “six-ayet” or something like that. And the number 60 (six times eight plus zero times one, or 48 in decimal notation), might be called “six-ets.”

Hexadecimal gets more confusing, because the ones place can have number greater than nine. After all, the next place over is the sixteens place, and what we call sixteen would be written as 10. So how do you write the numbers we know as ten through fifteen? I don’t know who came up with the solution, but it is to use the letters A through F. So twenty-six in decimal becomes 1A in hexadecimal: one sixteen and ten ones. Forty-seven becomes 2F: two sixteens and fifteen ones.

That (2F) was my age yesterday. But today, adding one more year, the F becomes a 0, carry one (just like adding 1 to 9 in the decimal system) to the sixteens place, and my age is 30. Of course, it doesn’t look as good when I write my age in octal: I am now 60 (six eights and zero ones). And forget about binary – I don’t want my age to be 110000!

Games: Cranium Family Fun Game and Rack-O

September 6, 2009

Al asked for a Family Night, and the holiday tomorrow allowed for staying up late this evening playing games. So we gathered around the game table downstairs, and we picked out Cranium Family Fun Game as one mostly likely to work well for all ages and provide lots of fun and laughs.

Like most of Cranium’s games, this one has a number of different kinds of activities. Depending what color you land on, you pick out a card from one of four decks: Creative Cat, Word Worm, Data Head, or Star Performer. We quickly agreed that Data Head was the easiest category, generally depending more on knowledge than ability. Recognizing common objects from photos showing just a small detail is probably the hardest in that category, while the true/false questions and multiple choice were usually easy for all of us.

Word Worm is my favorite category, as I really like words. Spelling words backwards is not very challenging for any of us, but finding six words starting with six different letters (roll the letter dice to get your letters) in six specified categories can be quite a challenge. So much so, in fact, that we never managed before time was up.

Creative Cat and Star Performer require more ability and creativity, and generally are where the laughs come in. How do you pantomime playing musical chairs, or doing instant messaging? My husband had somewhat more luck acting out being a race car driver, and later being a waitress (the latter was quite memorable and will probably continue to generate laughter whenever we remember it).

I had to crab walk around the room with a plastic frog on my belly. There was some question as to whether the frog was still on my belly, as it slid down near my hip, but as I made it around the room, panting with the effort, my husband decided I had accomplished it. On other rounds, we raced around the house collecting items, such as something made only of cotton (a T-shirt), or something with batteries (a remote control). Kyra helped with this category, providing both something alive, and something for a dog to fetch.

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An odd day

May 7, 2009

I feel kind of sorry for math teachers. I mean, what can you do to make math exciting to most kids? You have some students who hate math because they don’t understand it, and other students like me who think it’s boring because it’s easy. And then there are a few who really do think math is a lot of fun, but I think the number of number-lovers is pretty small.

Foreign language teachers get to serve new foods and teach new games. (The high school German club did a German night at my son’s elementary school, and he has now decided he wants to learn German instead of French.) English teachers can teach books that are exciting to read, or have the class put on a play. Social studies classes can reenact historical events or have debates over issues. Science teachers can demonstrate amazing experiments, or set up a science fair.

But what can math teachers do that raises the profile of their subject? On March 14, my older son told me how ridiculous he thought it was that they had a school assembly to recognize Pi Day. In math class (and maybe some others), there were “pie” activities – including pie to eat, which of course he did enjoy. I pointed out that math teachers don’t get a lot of chances to put their subject on center stage, and take advantage of the opportunity when it comes. He did not change his opinion (and he’s a straight A+ math student).

A math teacher in California is doing his best to promote math with Odd Day. Today’s date, 05/07/09, is one of six dates each century that is made up of three consecutive odd numbers. He suggests some ways to celebrate Odd Day:

It’s a great day to do your odds ‘n ends, give a friend a high-five, root for the odds-on-favorite, read the Wizard of Odds, watch the Odd Couple, say aaaahd in the doctor’s office, look for sea odders, find that missing odd sock, and beat the odds.

I made a brief attempt to write an Odd Ode, but it is more odd than ode:

This poem has five
Syllables in line one, then
Seven in the next, nine in the last.

As my younger son is always looking for games to play, I told him about Odd Day. With typical enthusiasm, he suggested we dance in the street, as that would be odd. I agreed it would – but we didn’t. We did give each other nine high fives, hopped seven times on one foot, and tried to spin around nine times (but stopped after five because we were both getting dizzy).

We counted to seven in three languages (English, Spanish, and German), and thought of odd foods to eat. We made up a story about visiting a planet with odd-looking creatures (three heads with five eyes on each head, five arms with three fingers on each hand, and three legs with one toe on each foot). Now we’re going to play Go Fish with only odd cards – but I’m not sure whether we can collect pairs or if we have to collect three of each number…

By the way, had you noticed that “perennial” and “student” each have an odd number of letters? So do these words…
Happy Odd Day!

Statistically speaking

March 10, 2009

One of the most interesting classes I took in grad school (for my MBA) was statistics, even if it was also one of the more difficult courses. I finally learned how to estimate probabilities rather than trying to tally up all the different possible outcomes. I learned what margin of error meant, and standard deviation, and other terms I had seen in reference to various statistics without knowing what they means.

Unfortunately I have trouble remembering much of it now. Just a few days ago I was arguing with my husband over the probability of getting the large straight in Yahtzee if you got 1, 2, 4, 5, and 6 on your first roll. If you re-roll the 1, on the next roll you clearly have a 1 in 6 chance of getting the 3 you need. He argued that since you get two chances, that would mean a 1 in 3 chance of getting the large straight. Since you only use the third roll if the second one failed to yield a 3, and the third roll also gives a 1 in 6 chance of a 3, I say the overall chance of getting the large straight is lower. But I can’t remember how to figure it out.

Of course, it’s not particularly important to be able to calculate the probability exactly. We both know that the probability of his getting the numbers he wants on my handheld Yahtzee is less than for me to get them, because the game likes me better. (Honestly – I’ve had up to four Yahtzees in one game.) But it is just a game, mostly a way to pass the time when there’s not a book handy to read.

There are other areas where understanding statistics is more important. Today’s Wall Street Journal has an article on misleading numbers in advertisements, “In Ads, 1 Out of 5 Stats is Bogus.” (Personally, I would have guessed that more like 4 out of 5 stats are bogus.) Just this afternoon, I brought in the mail and found a piece from an auto insurance company. Four out of five times, they claimed, their rates beat the competition. Before throwing it out, I puzzled briefly over the possibilities.

Perhaps they didn’t bother giving a quote when it was clear their rates would be higher, and they only calculated the percentages for the cases where they did give a quote. Perhaps they were only referring to a certain category of customers or type of insurance, where they do have lower rates than the competition. Whatever the case, I am fairly certain they manipulated the numbers somehow to give a misleading impression.

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Dated Scripture

October 17, 2008

Today is one of those dates that reminds me of a Bible verse. When I was a teenager I memorized a lot of Bible verses, first in the youth group at church, then in Bible school. And learning the reference was just as important as learning the verse word-perfect. I don’t remember them all now, but there are quite a number that I can still quote word for word (some of them in the King James Version I used at the time), together with the chapter and verse.

Today is 10/17 – which reminds me of this verse: “Faith comes by hearing, and hearing by the word of God.” Romans 10:17

If you never had to learn Bible verses – or if those chapter and verse numbers didn’t stick so well – my association of dates and verses probably seems very strange. Maybe it is. But that’s how my memory works. Refer to the 23rd day of March as 3/23, and it just pops into my head: “For all have sinned and come short of the glory of God.”

If you do know some of the commonly memorized verses, see if you can guess what verses I associate with these dates:

January 6
January 8
January 9
February 8
March 16
March 20
April 12
May 8
June 4
June 23
August 28 (also my husband’s birthday)
November 25

Keep reading to see my answers

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