Pythagoras 700 BC

Almost every person with a high school education has heard of Pythagoras and his famous theorem a^2 + b^2 = c^2.

Or "a square plus b square equals c square"

and the most famous triangle is the 3-4-5 triangle

3^2 + 4^2 = 5^2

9 + 16 = 25

Of course in Pythagoras days they did not have algebra (this did not come along until the 1,000 AD with the Arabs and Hindu at least in Western Civilization history).

In the movie the Wizard of Oz, the scarecrow wants a brain (if you watch the movie closely, it is the scarecrow that comes up with most of the plans, so he does have brain). The wizard tells him the one thing he does not have is a Diploma.

So the wizard gives the scarecrow a diploma and he states:

"The sum of the square roots of any two sides
of an isosceles triangle
is equal to the square root of the remaining side."

OOPS !! what was that ? "square roots" ? I don't think so. Isosceles triangle ? Everyone knows you need a RIGHT triangle.

So the very first thing out of scarecrows mouth is WRONG !! (do you see the irony)

It is not the piece of paper that gives us brains, it is the 12 or 16 or 20 years of schooling.

Of course we all know the author did it this way because square root sounds more interesting then square. And Isosceles is a nice Greek word.

The other story about the 3-4-5 triangle has to do with the Governor of Florida, Jeb Bush (younger brother of President Bush and son of the President Papa Bush)

There was a news conference in Orlando about the FCAT's (these are statewide testing giving to high school kids that they must pass (whatever that means) in order to get a high school diploma). A college student asked Gov. Bush if HE had taken the FCAT's and he said YES !! (by the way he did very well on them...would love to see some newspaper guys take the test...lol)

She said that one of the questions on HER FCAT's was what were the angles of a 3-4-5 triangle.

Gov. Bush said well i know it is a right triangle and that there are 180 degrees in a triangle, but i can not figure out the angles without a calculator.

The liberal papers had a field day with that one. One paper asked the college student what the correct answer was, and she said 30-60-90 triangle. WRONG !!

So i wrote a little article for the Tampa Tribune.

1. you need trigonometry to find the angles of a 3-4-5 triangle. Trig is NOT on the FACT's.

2. the college student's answer of 30-60-90 triangle is wrong, because anyone who studied geometry and has a high school diploma should know that in a 30-60-90 triangle the side opposite the 30 degree angle is one half the length of the hypotenuse (this fact is on the SAT's.... taken before going to college).

Using this fact and Pythagoras, most high school students know that a 30-60-90 triangle would have to be something in the ratio 1 - square root 3 - 2. (you will probably need pencil and paper to verify this little fact)

Of course my article did go on to say that this problem could have been on the FCAT's because all it requires is a little elementary geometry. Since the FCAT's are multiple choice, if the choice were:
A) 30-60-90
B) 60-60-60
C) 45-45-90
D) 30-40-50 (most adults would pick this one)
E) NOTA (none of the above)

The answer would be E !! and I know Gov. Bush would have gotten this correct answer.

A few historical facts about Pythagoras. He lived in Italy. He had many followers and they tried to help the sick and the poor. It was said that Pythagoras had healing powers, probably knew something about plants and herbs. He also knew how to predict the weather and was said he could calm the storms. Lets just say he was well ahead of his time.

Geometry and the Greeks

What did the Greeks in BC think infinity was ? The smallest thing they could see was grain of sand, therefore there must be infinite many grains of sand on the earth.

But we now have a number that is a one followed by 100 zeroes called a "google".
(yep, that is where the famous internet search engine Google got its name.)

So 10^100 = a google. That is a big number. The USA national debt in 2010 is at 14 Trillion...and that is only 14 X 10^12 or $14,000,000,000,000.

There are only 6 Billion people on this earth...or 6,000,000,000.

And only 300 Million people in the USA...or 300,000,000.

Now if 10^100 is a google, what is 10 to the google power ?

That is called a "googleplex" !! and that is the largest name number we have.

How big is it ? It is so big, we can not even image how big it is.

The smallest thing we can see (with the aid of powerful microscopes) are molecules.

We know that our Sun is a just a star. And that there are Billions of stars in our galaxy. And that there are Billions of galaxies in our universe.

A googleplex is larger than ALL the molecules in our universe !!! (that just went over my head...lol)

So how big is the number of grains of sand on the earth ? A large number or small number ? Actually it is a very small number...we estimate it to be less then 10^22...a one followed by 22 zeroes. If we were God, we could hold it in the palm or our hand (actually someone mentioned under one of God's finger nails).

So in the movie Contact with Jodi Foster, Jodi travels to another world and is met by some being...the being represents himself or herself as Jodi's dead father, and they talk while walking down the beach. The being picks up a hand full of sand, the sand falls from his palm with little sparkling stars....and i stand up and yell, YES !! (this was making a reference to the Greeks and what they thought was infinity). I love those kind of moments in my life.

What does Geometry mean ? "Geo" means earth. "Metry" means measure. Therefore Geometry means measure of the Earth.

One of the original Greek mathematicians was Thales in about 800 BC. He gave us the first five theorems or properties. One of those we have already mentioned as the Height to Shadow concept which today we say "Rise over Run" which is of course Slope (I already blogged about that in "Extra Help" and discussion about equations of lines)

But lets revisit the idea of finding the height of a tree in the woods. We have our trusty yard stick that we carry with us (3 feet long).

If we place the yardstick in the ground near the tree we can measure a 1 foot shadow. Meanwhile we can also measure the shadow of the tree, it is 10 feet.

So if we use the ratio of Height/shadow = Height/shadow, we get

3/1 = height of tree/10

Therefore the height of the tress must be 30 feet tall. (Yes, yes get your paper and pencil). Now you can measure the height of the huge electric pole by the park and the that huge building down the street.

Thales in 800 BC measure the height of the great Pyramids in Egypt. And they call him a "god" because he did NOT have to climb the pyramids to do it.

Sherlock Holmes in one of his books used this concept to find a clue that was supposedly buried at the end of a shadow, but the tree had fallen down. Watson was impressed and once again Sherlock said "elementary my boy, quite elementary".

COMMENTS:

(Am forwarding this email to all 3 sons, Manny. You taught Richard in a GRE prep class. Of course you taught Doug at Berkeley. Mike and Julee's boys go to a math and science magnet school in Ocala. I've been really impressed by what they do in math--and maybe their parents will investigate. Thanks!!)