First off where did humans get the idea of 360 degrees in a circle ?
What else in our lives has approximately 360 things in it ? Give up ? Days in a year !!
Yep we got our 360 degrees concept from the 365 days in year. What did they do with those extra 5 days ? Party or fiesta, the same way we celebrate New Years and 4th of July. (by they way in the business world there are 30 days in EACH month and 360 days in a year)
So what is half of this 360 degrees circle ? Yep, a line and therefore 180 degree in a line. And what is a quarter of a circle ? (You know North-West or South-East angles). Yep, that is 90 degrees. (and we call those perpendicular lines)
So a straight line has 180 degrees. Makes sense right ?
Now lets say we have 2 parallel lines (see the post on Euclid and parallel lines)
And these two parallel lines are crossed by a third line. (check out picture on side bar)
If two lines are parallel, then the alternate interior angles are equal.
(if line one is parallel to line two, then angle d = angle e)
Ok now you will need your own paper and pencil. Draw a triangle. Then draw a line through the top point of your triangle parallel to the base of the triangle.
Label the angles of the line on top, angle 1, angle 2 (the top angle of the triangle) and angle 3.
Do you agree that angles 1 + 2 + 3 = 180 degrees a straight line ?
(you might need your child or grandchild to help you with this. They have done this several times in school, i hope.)
Now comes the tricky part. Label the two "base angles" inside the triangle on the bottom, angle 4 and 5.
Here comes the magic of Euclid and his proofs. (Don't be scare you already got a 100 for the effort to read all this and wanting to just learn. Remember when we stop learning, we stop growing and we collect moss like a rock.)
So since the lines are parallel, the angle 1 = 4 and the angle 3 = 5 (these are called alternate interior angles like the picture shows on the side bar d = e).
So now in the equation 1 + 2+ 3 = 180, we can replace the 1 with the 4 and the 3 with the 5.
And WHALA, we get angles 4 + 2 + 5 = 180 which are the three angles of the triangle.
(if your picture worked and you got this proof, you deserve an ice cream, well done)
I once saw a 3rd grade teacher take a straw and ask the class how many degrees are there in a straight line. The class yell out "180 !!" and then the teacher folded the staw up into a triangle and asked and how many degrees does this triangle have now ? and the class yell, "180 degrees !!"
Another proof is to cut a Big triangle out of a piece of paper. And label the angles 1,2,3. Then cut the triangle into three pieces. Line up the angles 1, 2, 3 into a straight line. WHALA there is your proof. Angles 1 + 2 + 3 = 180.
Tuesday, February 1, 2011
The Father of Geometry 300 BC
When ever i ask my students, "who is the father of geometry ?", they always yell back, "Mr. Suarez !!". (They know how to get a good grade on a test...lol)
Since we live in Tampa, we go driving south down Dale Mabry (in our heads)...we pass Kennedy Blvd, and then Bay to Bay...and El Prado...Gandy is too far.
Finally some students yells out EUCLID !! and sure enough most cities in America have a Euclid street or avenue (the books says MLK has surpassed Washington and Jefferson as the number one named street in America).
According to Trivial Pursuit, Euclid, Ohio had the first Stop Light in the USA.
So Euclid lived in Alexandria, Egypt. (At this point my students all say, why are these guys all GREEKS, but no one lives in Greece....good question...research that one)
As it turns out Alexander the Great conquered the known world (Western Civilization...Greece to India to Egypt)...and he established the University of Alexandria in Egypt. He brought ALL the known mathematics to Alexandria.
It just happened that Euclid was the math department chair. And he wrote the number TWO best seller of all time called "The Elements". It had 13 chapters and one of the chapters was on Pythagoras. This book today (2,500 years later)is till being taught in our high school geometry class.
So Euclid ranks in the Top 100 most influential men of western civilization.
(the sad part is that Archimedes who was the greatest mathematician before christ, is not mentioned in many lists)....and worse yet, many Geometry students do NOT even know who Euclid or Archimedes were...shame on you math teachers.
So Euclid was a textbook publisher, but more than that he organized formal proofs into a system that could be followed by other students of mathematics.
He also gave us the Parallel Postulate.
This postulate reads like a theorem, but Euclid could NOT proof it, so he had to accept it as postulate (something that we must accept without proof...or something so obvious we accept without proof).
("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
(you got to be kidding....this makes NO sense)...how about
parallel postulate ("through a point outside a line there is exactly one parallel")
that is much better !! (I think ?)
Just look around you, there are parallel lines every where. All the parallel definition states is that through a point NOT on the line, there exist only one line parallel to the original line. It is obvious, right ?
Interesting that even though it sounds like a theorem, Euclid could NOT prove it. (Even with his 13 chapters in his book The Elements.)
Many years later, Einstein would show that parallel lines do NOT exist...OUCH !!
but that is called non-Euclidean geometry and for another blog someday. Einstein was a little different....lol...probably one of the greatest minds of ALL time.
The problem in space is that the gravity of the sun bends the light coming from the stars...and this is an important concept if you are trying to land on the MOON !!
It would not have been "one step for mankind" if the space agency did not have Einstein to tell them that light bends in space....it would have been a BIG step for mankind...and they would have missed the Moon altogether !!! next stop MARS...lol
Since we live in Tampa, we go driving south down Dale Mabry (in our heads)...we pass Kennedy Blvd, and then Bay to Bay...and El Prado...Gandy is too far.
Finally some students yells out EUCLID !! and sure enough most cities in America have a Euclid street or avenue (the books says MLK has surpassed Washington and Jefferson as the number one named street in America).
According to Trivial Pursuit, Euclid, Ohio had the first Stop Light in the USA.
So Euclid lived in Alexandria, Egypt. (At this point my students all say, why are these guys all GREEKS, but no one lives in Greece....good question...research that one)
As it turns out Alexander the Great conquered the known world (Western Civilization...Greece to India to Egypt)...and he established the University of Alexandria in Egypt. He brought ALL the known mathematics to Alexandria.
It just happened that Euclid was the math department chair. And he wrote the number TWO best seller of all time called "The Elements". It had 13 chapters and one of the chapters was on Pythagoras. This book today (2,500 years later)is till being taught in our high school geometry class.
So Euclid ranks in the Top 100 most influential men of western civilization.
(the sad part is that Archimedes who was the greatest mathematician before christ, is not mentioned in many lists)....and worse yet, many Geometry students do NOT even know who Euclid or Archimedes were...shame on you math teachers.
So Euclid was a textbook publisher, but more than that he organized formal proofs into a system that could be followed by other students of mathematics.
He also gave us the Parallel Postulate.
This postulate reads like a theorem, but Euclid could NOT proof it, so he had to accept it as postulate (something that we must accept without proof...or something so obvious we accept without proof).
("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
(you got to be kidding....this makes NO sense)...how about
parallel postulate ("through a point outside a line there is exactly one parallel")
that is much better !! (I think ?)
Just look around you, there are parallel lines every where. All the parallel definition states is that through a point NOT on the line, there exist only one line parallel to the original line. It is obvious, right ?
Interesting that even though it sounds like a theorem, Euclid could NOT prove it. (Even with his 13 chapters in his book The Elements.)
Many years later, Einstein would show that parallel lines do NOT exist...OUCH !!
but that is called non-Euclidean geometry and for another blog someday. Einstein was a little different....lol...probably one of the greatest minds of ALL time.
The problem in space is that the gravity of the sun bends the light coming from the stars...and this is an important concept if you are trying to land on the MOON !!
It would not have been "one step for mankind" if the space agency did not have Einstein to tell them that light bends in space....it would have been a BIG step for mankind...and they would have missed the Moon altogether !!! next stop MARS...lol
Monday, January 31, 2011
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.
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!!)
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!!)
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