Multiplication with lines

Another topic my wife showed me. Multiplication by drawing lines. As she pointed out, this would get cumbersome (or worse) with numbers of several digits, but it’s interesting anyway.

Here’s a video.

Here’s a written example.

Why does this work? Nobody knows! OK, somebody does. This site also suggests how to do this multiplication when some of the digits are zero. How do you draw a “missing” line? Dots easy!

Earliest (almost) Easter

The following came from my wife via a friend via who-knows. I’ve added some material and changed an incorrect date.

Easter this year is on Sunday March 23, 2008.

* As you may know, Easter is always the 1st Sunday after the 1st full moon after the Spring Equinox.
* (The date of the Easter full moon is determined by the motions of a mathematical “moon”; these motions approximate the movements of the real moon, but discrepancies occur for the sake of simplicity in the rules. Any such discrepancies are viewed as unimportant. For clarity, the calculated date is called the Paschal full moon.)
* This dating of Easter is based on the lunar calendar that Hebrew people used to identify Passover, which is why it moves around on our Roman calendar.
* Based on the above, Easter can actually be one day earlier (March 22) but that is pretty rare.

This year is the earliest Easter any of us will ever see the rest of our lives! And only the most elderly of our population have ever seen it this early (95 years old or above!). And none of us have ever, or will ever, see it a day earlier!

Here are the facts:
* The next time Easter will be this early (March 23) will be the year 2160 (152 years from now). The last time it was this early was 1913 (so if you’re 95 or older, you are the only ones that were around for that!).
* The next time it will be a day earlier, March 22, will be in the year 2285 (277 years from now). The last time it was on March 22 was 1818.
* So, no one alive today has or will ever see it any earlier than this year!

The odds are considerably better for witnessing a late Easter. Many people are still around from the last time Easter fell on April 25, an event which took place in 1943, and a good many people here today will likely still be around when Easter next falls on April 25, which will occur in 2038.

This ignores the Eastern Orthodox calendar, which is still based on the Julian calendar.

See also http://www.snopes.com/holidays/easter/earlyeaster.asp for more info.

Celebrate “Pi Day”

Should be a national holiday!

Enjoy Pi Day.

Elections and Mathematics

If you’re interested in elections and mathematics, you need to read this book. If you wonder about the practical applications of math to “real life”, you need to read this book. If you’re a politician who ignores math in solving social problems, you need to read this book. If you’re just curious about some things, you should read this book.

“What is the book?”, you ask. The book is For all practical purposes : mathematical literacy in today’s world found in my local library.

There are several chapters related to voting and apportionment. (There are many chapters unrelated to voting, but all relating to mathematics and the “real world”, such as management, statistics, computers, size and shape, etc. The subtitle for the book is “Mathematical Literacy in Today’s World.)

This particular edition is an update of the 1987 edition. A new chapter on “Electing the President” has been added. Other chapters have been revised. There is a lot of online material available through the publisher, consisting of exercises, answers, and various Java applets, video clips, and probably more. There was a video series based on the original book which was shown on TV. My library does not have the videos, but clips are supposed to be available online. The ones about voting and apportionment are easy to follow, and show mathematical difficulties with various systems.

If you were going to develop a system of voting, what would you require? You might want fair and equitable. You might want one that isn’t easily manipulable mathematically. What properties would you want in a voting system? And what systems satisfy those properties?

All voting systems have difficulties, especially those in which there are more than two candidates, or in which more than one person/idea will be chosen in a given category.

Read about the Condorcet Winner Criterion, Plurality Voting, manipulability, Borda Count, the Hare System and Monotonicity, Approval Voting (voting for several in order to choose one), Arrow’s impossibility theorem (any voting system can give undesirable outcomes, even if you’re not Al Gore), May’s theorem, Pareto conditions, Sincere voting, Weighted Voting systems, Banzhaf Power Index, other indexes of a voter’s power, and so on.

Voting is more complicated than an “X” in a box or a purple finger.

What about choosing the number of representatives that a group (say a state) gets? Not so easy there, either. Fair division (say cutting a cake) and apportionment are difficult areas. Various apportionments systems have pros and cons. Some elections have been swayed because of this. Is it fair how many representatives a state gets? What do you do with a fractional representative? What does “fair” mean? Read how Hamilton, Jefferson, Webster, and others have approached apportionment. During the primary season, is it “fair” that the winner of a state gets all the delegates? Should they be divided according to how well a candidate does?

Caution: I was disappointed with the chapter on “Electing the President”. Not the math part, but the bias. If this weren’t a new chapter, I’d recommend the previous edition of the book. The chapter “Social Choice: The Impossible Dream” starts with a picture of Al Gore, not George W. Bush. Gore’s picture on the next page is bigger than Bush’s. Not a big deal, but a bias, I think. The 2000 election did have difficulties, but mostly not mathematical. The section “Is There a Better Way to Elect a President?” asks a legitimate question. No voting system is perfect, but maybe there is a method that would satisfy more people (but would still have to be “easy” to implement in a nation of 300 million people.) A comment in that section says that mathematics may show possible reforms “that may ameliorate some of the problems that plague our current system.” I thought “plague” is too strong. There are lots of things that could be done before changing the current system of voting.

In short, this is an informative book with sections on voting and apportionment. Something like this should be required reading in Civics classes, and should be known in a general way to more voters. Being an informed voter may mean more than knowing the candidates’ positions.

Secret Codes – Updated

Earlier there was Dad’s Secret Code. Then there was Mom’s Secret Code.

But before all that, there was Fletcher Pratt’s Secret and Urgent: The story of codes and ciphers. This was one of the book highlights of my boyhood days. This is an interesting, and to some of us, exciting book to read. It gives some history of codes and ciphers, and also gives some examples. Since it was originally written during World War II, it doesn’t have much on WWII code efforts. That would have to come later in other books. I wasn’t the only one impressed by this book.

I was going to suggest some related books on codes and ciphers, but at the moment my memory won’t cough them up.

Anyway, start with Pratt’s book and work toward the present. There are some interesting books that do give insight into WWII and more recent code work. It is an amazing area of mental work. Some have suffered mental breakdowns because of the stress involved. But it is also a fertile area in recent mathematical work. Modern computer communication and data storage would not happen without information encodings of various kinds.

It’s an interesting area of study, with problems for all levels of skill and knowledge. Now, if I could only break Dad’s Secret Code.

[Update: Two books that I couldn't recall earlier are: The Code Breakers by David Kahn, and The Code Book by Simon Singh. A search on Amazon.com for "Code Breaker" as a title yields several books. A search by subject should give even more. Unir sha ernqvat nobhg pbqrf naq pvcuref!]