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 | |
| a2 | = | ab | Multiply both sides by a |
| a2 − b2 | = | ab − b2 | Subtract b2 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.
9 Dec 2009 at 9:05 am [Comment permalink]
If a2 = ab
Then if we subtract b2 from both sides, it will be
a2 - b2 = ab - b2
not
a2 - b2 = a2 - b2 as you have written.
Of course, this is not the fatal error.
gail
9 Dec 2009 at 9:06 am [Comment permalink]
You made some mistakes in your "proof" that could throw off the point of it all not to divide by zero. Check your work.
9 Dec 2009 at 9:07 am [Comment permalink]
Please note, line 3 on the right side is typed incorrectly-it should have been ab - b2
Shirley
9 Dec 2009 at 9:11 am [Comment permalink]
Gail, Joani and Shirley: Thank you for pointing out the error. (Doh!!).
I have corrected it in the post above.
9 Dec 2009 at 9:15 am [Comment permalink]
I read the two proof which verify 2=1 there is a number of algebraic mistakes for example your supposition said
a=b,but you divide both side by (a-b) dividing a number by zero is impossible so your conclusion is wrong and in the 2nd numerical examples also a number of error this is math a full of rule,do not cheat as but teach as.
9 Dec 2009 at 9:27 am [Comment permalink]
Hello Wesen. Thank you for your feedback - I have achieved my objective => you thought about it!! 🙂
BTW - there are hints that something is wrong in each "proof". For starters, I used inverted commas around the word "proof", which suggests it is not really a proof.
Next, I put the following, which also suggests something is going on:
"We start with a = b and end up with a remarkable proof – or is it?"
Finally, I put "Can you see what’s wrong in each proof?".
The intention was certainly to teach you, not cheat you.
9 Dec 2009 at 11:03 am [Comment permalink]
thanx for the news article. it helped me get to know about other cultures.
and also in math tip (a), if a=b then a-b=0. therefore you cannot divide by zero. so the result is wrong.
9 Dec 2009 at 11:31 am [Comment permalink]
@Joani - your comment is rather harsh. Do I gather you never make mistakes?
9 Dec 2009 at 2:18 pm [Comment permalink]
Dear all Greetings!
This news letter gave me a lot of useful information on calender dates etc. Thanks.
Correection
In Math proof of 1=2, the third step right side it should be
ab-b2
Thanks
9 Dec 2009 at 3:56 pm [Comment permalink]
Thanks for all the articles it helped me in some ways but i am just too busy that I cant handle all those but i read them and i even applied some in dealing with my students especially 2 =1. Please don't get tired of sending your articles for me to know your culture. I really appreciate those very much.
I am holding an office . I do teach and at the same time manage my people. I am the head of our department. So my time for teaching and opening my mail is divided. I hope you understand and thank you very much.
10 Dec 2009 at 8:10 pm [Comment permalink]
SquareCircleZ
Oh, I'm sorry. Maybe I was too terse. I was in a hurry, and had to rush out the door. Please don't take how I said it in a bad way. I sat down with my kids and went through the proof and caught the mistake and thought someone might need to correct it on line for others, but I realized I had run out of time myself to figure out how to write out an equation with the squared terms, etc.
Please don't take an offense. It was meant to be helpful.
Joani
10 Dec 2009 at 8:13 pm [Comment permalink]
@Joani: No problems! You are right - I should have checked it more closely. Anyway, Graeme was more upset than me!
@Chandra: Yes, thanks - already noted and fixed (please see above comments)
10 Dec 2009 at 8:15 pm [Comment permalink]
Mr.murray,
Greetings to you, once again am very grateful for this edition of interractive maths, particularly iam astounded about the proof of 2=1, my question is, how can it be explained or applied in practical terms i.e our day to day lives? also help explain in simple language about the fact that negative x negative=positive, i shall be grateful if my concerns are addressed. Iam looking forward to another edition of inteerative maths.
10 Dec 2009 at 8:19 pm [Comment permalink]
@Richard: The proof of 2=1 is not a proof at all! (Please see the above comments which explain the error in the first one.)
Please don't apply it in your life because it is not true!
On your last point, here is an explanation of product of negatives numbers giving a positive number:
Integers (see lower down the page)
10 Dec 2009 at 8:27 pm [Comment permalink]
Richard, it surely DOES apply to life! The lesson is, think about what you see. It is obvious that 2 does not =1, but often, things are not so clearly understood and we do not question what appears before us. The classic "if it's too good to be true, then it's probably not" is a slight twist. Think about things before acting, speaking, taking stands, criticizing, forming opinions, etc. Analyze the "facts" that brought forth the apparent conclusion. We do WAY too little of that these days.
11 Dec 2009 at 9:57 pm [Comment permalink]
Thank you sir....
I couldn't believe the proof of 2=1..............
Maths is really amazing subjects......!!!!!!!!!!!!!
11 Dec 2009 at 9:58 pm [Comment permalink]
The Topic For The Month Is So Cool. Thank You For Sending Me. It Will Really Help Me!!!
13 Dec 2009 at 4:02 am [Comment permalink]
Hello every one. The news letter was amazing and interesting. I read the proofs you did. I did notic somthing in proof 2 (which is not a proof!!!) I noticed that you shoould put all the possibilties of the signe when you took the square roots of both sides((we know that the square of +5 and -5 is always 25)).In line #4 the possiibilites which give the true statement are when you take the left side positive and the right side negative or the other way around. I hope am right...
13 Dec 2009 at 9:23 am [Comment permalink]
You are right, Musab! I was getting concerned since no-one had pointed out what was wrong with the second "proof".
@Gail: Thanks for your input. I was too hasty when I said don't use it in daily life! Yes, many examination systems encourage rote learning and little thinking, and I agree that is a shame.
14 Dec 2009 at 11:38 am [Comment permalink]
Hello sir
I have a doubt in the second not a proof "proof".
Is it the implication of third step from second step which is wrong or is it taking the roots which misleads us.I got this doubt because we have both real roots in this case which means both can be considered as a solution.
I hope I have put myself in proper words.
15 Dec 2009 at 8:39 am [Comment permalink]
hi murray thank you so very much for the years supply of info.i do hope to make a more positive contribution in the followingyear.god bless and seasons greetings
27 Dec 2009 at 9:54 pm [Comment permalink]
Sir,
Your statistics about How Many People Like HISTORY , is mighty interseting.
Could one trouble you to have a survey about What People Think About The History of MATHEMATICS.
I think,a knowledge of mathematics is incomplete without a background idea of the history of mathematics and the persons who created mathematics.
Anyone can pick good math with a bit of effort and some patience, BUT to create MATHEmatics is not everybody's gift.
More about this, later.
Compliments of the season,from the Indian Himalayas
ahmed
28 Dec 2009 at 3:27 pm [Comment permalink]
You're welcome, Ahmed.
On another matter, is it true that the glaciers are retreating where you are?
29 Dec 2009 at 9:50 pm [Comment permalink]
Regarding the 1st proof, we've considered a=b which means a-b=0. now in the 5th step cancelling a-b on both sides will mean 0/0 which is mathematically an indeterminant form. Hence the proof is absurd.
30 Dec 2009 at 3:34 am [Comment permalink]
HI sir .I am a student from 2nd year engineering dept .
As you know it is very necessary for me to understand well .So i would to say thank you very much .
HAVE A NICE TIME !!