IntMath Newsletter - Proofs, holidays, looking back
By Murray Bourne, 08 Dec 2009
In this Newsletter:
1. Math tip (a) - Proofs that 2 = 1
2. Math tip (b) -- Holidays, calendars and math
3. IntMath Poll --- History
4. Most popular articles during 2009
5. Latest from the Math Blog
6. Final thought – Perfection
1. Math tip (a) – Proof that 2 = 1
Let's perform some mathematical magic. We start with a = b and end up with a remarkable proof - or is it?
a | = | b | |
a^{2} | = | ab | Multiply both sides by a |
a^{2} − b^{2} | = | ab − b^{2} | Subtract b^{2} from both sides |
(a − b)(a + b) | = | b(a − b) | Factor both sides |
a + b | = | b | Divide both sides by (a − b) |
b + b | = | b | Since we started with a = b |
2b | = | b | |
2 | = | 1 | Divide both sides by b |
Here's another "proof" that 2 = 1. This time, it only involves numbers.
4 − 6 | = | 1 − 3 | True statement |
4 − 6 + 9/4 | = | 1 − 3 + 9/4 | Add 9/4 to both sides |
(2 − 3/2)^{2} | = | (1 − 3/2)^{2} | Factor both sides |
2 − 3/2 | = | 1 − 3/2 | Find square root of both sides |
2 | = | 1 | Add 3/2 to both sides |
Can you see what's wrong in each proof? These are great examples to use with students who have been using algebra for a while. They certainly challenge thinking!
2. Math tip (b) – holidays, calendars and math
December is full of significant religious holidays. Did you know that a lot of mathematics was developed to improve the prediction of these important cultural, religious and commercial events? You could make a lot of money (or perhaps avoid getting your head removed) if you correctly predicted a significant moon rise or alignment of planets.
Bodhi Day - Buddha's Enlightenment - 8th day of the 12th lunar month (fixed as 8th Dec in Japan).
Hanukkah (Festival of Lights - occurs on the 25th day of Kislev, which is the 3rd month of the "civil" year. In the Western calendar, this falls on 11 Dec 2009. The Muslim astronomer and mathematician al-Khwarizmi (they guy who gave us algebra) studied the Jewish calendar in the 9th century. By then, it was mathematically derived (not based on observation only).
Al Hijra - Muslim New Year, is the first day of Muharram, the first month in the Islamic calendar (18 Dec 2009). This month moves throughout the year, since the Islamic calendar is lunar (12 moons per year) and therefore 11 or 12 days shorter than the solar year (based on 1 revolution around the sun per year).
Christmas (birth of Christ, 25th December). Christmas occurs on a fixed date each year, not like Easter (and the Jewish Passover), which are based on lunar calendars and so change each year.
Kwanzaa (for African Americans, celebrating year-end harvest festivals, fixed on 26th December).
Here's the article I wrote last year about the famous Christmas song: The Twelve Days of Christmas – How Many Presents?
3. Latest IntMath Poll – History
The last IntMath Poll asked readers about their feelings regarding history.
My feelings about learning history are:
46% | It’s OK |
26% | It’s my favorite subject |
23% | I hate it |
5% | I’ve never studied it |
Total votes: 2300
Poll date: Nov/Dec 2009
It's good the majority of readers like history (or find it “OK”). I used to hate history when I was a student - how stupid was that? We need to know our history to understand why people do what they do.
Math histroy is important, too. If you have an understanding of the story behind the math you are studying, it helps you to understand it (and apply it).
The latest IntMath Poll asks about the technology you use to draw graphs. You can respond on any page in Interactive Mathematics.
4. Most popular in 2009
To round up the year, I thought I'd share with you the articles that attracted the most readers and comments during 2009.
Many readers find these valuable:
This tutorial helped a lot of people:
Here's some interesting math history:
Here's some topical real-life math about the swine flu and global warming (as I write, world leaders are deciding our future in Copenhagen. Let's hope sanity prevails.):
Here's some very useful free math software:
This was the most popular Newsletter this year:
This one continues to attract a lot of attention, especially on Thursdays and Fridays.
And finally, this question continues to stump people:
5. Recently in the Math Blog
Here are the latest articles from squareCircleZ:
a) Friday Math Movie - Toys that make worlds
The creator of SimCity and Spore gives an entertaining overview of the games he's created.
b) Friday math movie - Science of Watchmen
This video explains some of the math and science behind the movie Watchmen.
c) $4 billion for math and science
Obama pledges big money for science and math education.
d) Friday Math Movie - Imagine Leadership
Leadership changes our nations. Check out this movie for some real life stats examining how we can improve our world.
e) Calculating takeoff speeds
Here's an application of polynomials to a real-life situation - what speed can an aircraft safely take off?
6. Final thought – Perfection
I'm surprised how many people do a task until it is just "good enough". I believe it's important to do things as well as we possibly can — and that includes doing math, of course!
The following quote comes from Vince Lombardi, a respected American football coach.
Perfection is not attainable, but if we chase perfection we can catch excellence.
Keep your feedback coming - I love to hear from you.
Here's wishing everyone a peaceful and enjoyable end of year holiday season and may 2010 be a great year for you all.
See the 25 Comments below.