Proof without words

Today I came across William Dunham's awesome book titled The Mathematicial Universe with the following fascinating picture. It is a wordless proof.


This is silently proclaiming that 1+2+3+...n = (n)(n+1)/2! (Ok, that's not a factorial, but excitement!)

So the story is about the young Gauss who had a teacher that made him add 1+2+3+...100 so to keep the students busy all day. Gauss figured out that you can pair up the first and last number, 1 and 100, add them up so you have 101. The next pair, 2 and 99 also add up to 101. There are 50 such pairs so the answer is instantly 5050.

Let's decipher the picture.

Each white column is indicating 1,2,3... When you add them all up, it covers exactly half the area in the n by n+1 grid!

The author calls it immensely elegant. And I agree! Oh my I want to frame this!

Math is not always dry and boring.

Here is another one. It is saying adding odd numbers 1+3+5+7+9... you can always get a perfect square.



I noted that when I was a kid! One day I got so bored that I start counting tiles on the floor and noted the pattern.

So how do you prove it, in words?

Odd numbers can be represented by 2k+1, where k is an integer.
So 3 = 2(1)+1, 5=2(2)+1, 7 = 2(3)+1, and so on.

Let's add 1+3+5, we have 1+2(1)+1+2(2)+1, this is 2(1+2)+2+1
Let's add 1+3+5+7, we have 1+2(1)+1+2(2)+1+2(3)+1 = 2(1+2+3)+3+1.

See the pattern? Add n odd numbers, we get 2(1+2+3...n)+n+1.
Now apply the formula: 2(n)(n+1)/2 + n + 1, we have n(n+1)+n+1 = n²+2n+1. That is a perfect square.

Truck crashes into Chinatown Train Station

A truck slammed into the Chinatown Train Station during evening rush hour last Friday, killing 2 and injured a dozen.
See this Tribune article for details.

The weather was nice during the day before the accident. A heavy thunderstrom striked on that evening, as if heaven is weeping for the poor victims.

I got off at the same station just 1 day before. It can happen to ME.

The station is located at the T-intersection from the 90/94 exit and Cermak. Vehicles must turn left or right after exiting, or slam into the train station. In the ancient study of Feng Shui, that is a terrible spot, as it constantly seems like someone is coming toward you. Feng Shui specialists may advice you to put a mirror or something to reflect it somehow. While there is not really scientific basis of Feng Shui, the designer of the station should have used some precaution. A barricade like those surrounding government buildings should have been there to minimize potential damage.

I don't really blame the CTA for this. There IS plenty of space to turn left or right. Who would have thought that someone slam right into it?

After this incident, please, put barricades on any station that is in potentially dangerous T-intersection.

If the truck has smashed into the beam supporting the bridge to the other side of Cermak, it may kill just the reckless driver, or collapse the bridge, none of which are nice scenarios.

This is a terrible accident.

The Last Lecture

You may have heard about the life touching Last Lecture by the dying computer science professor Randy Pausch over the internet. Search for it in youtube if you like. Ok, here is the link. Today I saw the book form. This last lecture is not a computer science lecture, rather, it is a lecture of life. Over 6 million people has watched it! (That's about the population of the very crowded Hong Kong!)

The video lecture is full of witty charm. No wonder 6 million people viewed it.

In the last days of life, the professor's main concern are his kids. He need to jam pack what he would teach his kids for the next 20 years into a small book or lectures. In the lecture (book), he shared a lot about his life: childhood dreams and all that... he wanted to be in zero gravity and play for NFL, etc.

One page caught my attention: He "celebrated" the quadratic formula by painting it on a wall when he was young. I like the formula too, but didn't write it on my wall. However, I wrote about it in this article.

The lecture (and book) is jammed packed with life lessons: following childhood dreams and seizing and enjoying every moment of life. The positive attitude he had is such great impact on his viewers, especially his kids when they are old enough.

Here is his quote: "We cannot change the cards we are dealt, just how we play the hand."

* * *
If the professor lives to be 100+ years old, even if his researches rock the academic world, he probably would not have captured as much attention. Old saying: it is not how long you live, it is how you live that matters. Unfortunate folks with health issues are often comforted with that old saying.

How come we are more interested to hearing from people who are dying and not as interested to those who are not? Why did the Last Lecture capture so much attention?

People like and want inspiring stories, and don't mind spending about $15 to buy it as an gift item. Professor Pausch's ability to communicate his life lessons truly flourished. I am sure his impact would last, especially within the minds of his young children.

What do YOU have to say in your last lecture? We all must die some day.

Although it is very unfortunate that the very talented professor has terminal disease, he is also in some ways fortunate.

The professor at least knows his life will end in matter of months and had a chance to write the lectures. (I sure hope some breakthrough in medicine or a miracle somewhat would save him!) For far too many unfortunate people, death comes too soon too suddenly.

The professor has an audience (his students, family, etc). Unfortunately, not everyone has loyal audience. I hope you do.

You can also use your blog as your lecture hall. The entire web is your audience.

Rounding

"I won that double-digit victory that everybody on TV said I had to win, and the voters of Pennsylvania clearly made their views known -- that they think I would be the best president and the better candidate to go against Senator McCain," Clinton said Wednesday morning in a CNN interview.

So the numbers for Pennsylvania are as follows:
Obama: 1,042,297 45%
Clinton: 1,258,245 55%

But if you look (divide) closely, Clinton has 0.546934157 and Obama has 0.453065843.
Hmm, seems to be less than the 10% double digit victory.

Let's subtract: the answer is 0.093868315.

Let's round it to the nearest percent... ask an elementary school student. Will the student say 9% or 10%?

Another trick to deceive poorly math educated Americans?

About the "clearly made their views known" comment, even with a solid 55%, is it big enough a majority to be a really clear winner? 10% is THE smallest double digit victory and the winner is almost there.

Wanted: Political Wisdom

BBC article: http://news.bbc.co.uk/2/hi/americas/7359957.stm

As the candidates appeared on the US talk show circuit on Tuesday morning, Mrs Clinton was asked how she would respond if Iran launched a nuclear attack on Israel, and replied with a stark warning.

"If I'm the president, we will attack Iran... we would be able to totally obliterate them," she said.

"That's a terrible thing to say, but those people who run Iran need to understand that, because that perhaps will deter them from doing something that would be reckless, foolish and tragic," she told TV channel ABC.

In response, Mr Obama said: "Using words like 'obliterate' - it doesn't actually produce good results, and so I'm not interested in sabre-rattling."

He said only that Iran should know he would respond "forcefully" to an attack on any US ally.

That is a tough question to answer. I admit I have to look up obliterate to figure out the exact meaning of the word, but I can tell from context: total destruction.
Cold sweat come down as I imagine total annihilation taking place as nuclear bombs explode all over.

Isn't it horrible to wipe out the ENTIRE country to retailiate?

Political wisdom and patience are the requirements for a world leader. It is important to choose the right words. I like Obama's answer. He didn't give special status to Israel, and forcefully say he would respond if there is attack on any ally.

Let's wait a few more hours to see who wins in the Pennsylvania contest.