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Pythagoras
Pythagoras was a Greek philosopher, mathematician and founder of Pythagoreanism religious movement. He was known for the Pythagorean Theorem and made influential contributions in the field of mathematics and natural philosophy.
1,004 Questions
How do you do Pythagoras theorem in square based pyramids?
There are no right angles in a square based pyramid,
so Pythagoras and his theorem are not involved.
him
How can you prove the pythagorean theorem and use it to solve problems?
There are two ways to prove the theorem: one graphically (visually), one algebraically:
They both start in the same way (this is going to take some explaining as I can't draw a diagram very easily - try and draw the diagram on a piece of paper from the instructions):
Take a square ABCD of side length a + b.
Mark E at distance 'a' from A along AB (which is distance 'b' from B along AB)
Mark F at distance 'a' from B along BC (which is distance 'b' from C along BC)
Mark G at distance 'a' from C along CD (which is distance 'b' from D along CD)
Mark H at distance 'a' from D along DA (which is distance 'b' from A along DA)
Now draw in: EF, FG, GH, HE.
This forms four right angled triangles: AEH, BFE, CGF, DHG
And a quadrilateral EFGH.
Using the above diagram we can show:
The four right angled triangles are congruent using Side-Angle-Side on the sides with lengths 'a' and 'b' and the enclosed right angle. Thus the hypotenuses of the four triangles are the same and can be labelled distance 'c'.
Next, the sides of the quadrilateral EFGH are the hypotenuses of the triangles which are all distance 'c', so EFGH has four sides of equal length.
Now consider angle HEF:
Since the angles in a triangle sum to 180°
AEH + EHA + HAE = 180°, thus AEH + EHA = 180° - HAE = 180° - 90° = 90°
Since triangles AEH and BFE are congruent, angle EHA = angle FEB
Thus the sum of angles AEH + FEB = 90°
E is along the line AB so angle AEB is 180°.
Thus the sum of angles AEH + FEB + HEF = 180° but AEH + FEB = 90°, thus HEF = 180° - 90° = 90°.
Similarly for all the angles of the quadrilateral EFGH, thus EFGH is a square of side length 'c'.
(Starting with four congruent triangles and arranging them in the big square of the diagram is easier to start the proof, but harder to explain.)
Thus the diagram shows four congruent triangles of side lengths a, b, c arranged around a square of side length c forming a square of side length a + b.
Graphical proof:
Take triangle AEH and translate it so that HE coincides with BF; the new position of A is called A' and the rectangle formed is A'FCB
Take triangle EBF and translate it so that E coincides with A; where B ends up on AB call B' and the new position of F call F'
Finally take triangle DHB and translate it to that HB coincides with AF; the new position of D is called D' to form a rectangle AB'F'D'.
Length B'F' is 'a' so the distance from CD of F' is 'b'.
Length D'F' is 'b' so the distance from BC of F' is 'a'.
But GA' is length 'b' and FA' is length 'a' so F' and A' coincide
Thus the square ABCD is now divided up into two rectangles (each made up of two of the original four triangles) and two squares: B'BFA' (of side length 'a') and D'A'GD (of side length 'b')
As the area of the triangles has not changed, the area of the square EFGH (of side length 'c') must equal the sum of the areas of these two small squares.
Thus c² = a² + b²
[This is the image above]
Algebraic proof:
Consider the area of the big square. It can be calculated as the square of the side length or as the sum of the areas of the four triangles and the smaller square:
area = (a + b)² = 4 x ½ ab + c²
→ a² + 2ab + b² = 2ab + c²
→ a² + b² = c²
But as 'c is the length of the hypotenuse of one of the triangles, and 'b' & 'c' are the lengths of the other two sides, this proves Pythagoras' Theorem.
One example where Pythagoras' Theorem is used in reverse: if a² + b² = c² then the angle opposite 'c' is a right angle. Making a triangle with sides 3, 4, 5 ensures a right angle is opposite the 5 side and this is used in building work to ensure construction is square.
WHO DISAGREE THE PYTHAGORAS THEOREM?
Pythagoras' theorem is only a theory because when the sides of a right angle triangle are equal then the length of its hypotenuse will be an irrational number which can't be finally determined.
Why does Pythagoras theorem work?
Pythagoras' theorem states that for any right angle triangle the square of its hypotenuse is equal to the sum of its square sides
If you mean Pythagoras' theorem then it states that for any right angle triangle when its hypotenuse is squared it is equal to the sum of its 2 squared sides.
How old was Pythagoras when he discovered Pythagoras theorem?
Pythagoras lived from 571 BCE to 495 BCE, so he was about 76 when he died.
Pythagoras and his students are often said to have constructed the first proof of this useful theorem. Pythagoras had a school with many students. but the school was secretive, and the students gave their teacher for what they discovered. Because of this, we don't know exactly who did what and when.
In any case, it is clear from their clay tablets that the Babylonians knew and used the theorem we call "Pythagoras' Theorem", about a thousand years earlier. They also knew some kind of proof.
Perfect
Pythagoras determined that the length of any side of a(n) _____ is based on the length of its other two sides.
Probably his theorem a2+b2=c2, which is used to calculate the hypotenuse (longest side) of a right-angled. triangle. You find the length of the two shorter sides, square them, add them together and then square root your answer. 3,4,5 is a Pythagorean triple, because 9(32)+16(42)=25(52), and a Pythagorean triple is where the length of each of the three sides is an integer(Whole number).
Pi (3.14 etc.) was actually just named for the first letter of the greek word for perimeter, and had nothing to do with Pythagoras, although many people think that it does.
He was the founder of a religious movement called Pythagoreanism.
Many people see him as a great mathematician, mystic and scientist, although most people just see the first part.
I'd say he was probably most famous for the first two though, despite that one of them was , as I said, nothing to do with him.
How old was Pythagoras when he died?
Pythagoras was believed to die at 99 years old because of all the dates leading up to his death
Towards the end of his life he fled to Metapontum because of a plot against him and his followers by a noble of Croton named Cylon. He died in Metapontum around 75 years old from unknown causes.
How did Pythagoras dress in samos?
He dressed kinda weird for his time. He wore a white robe and trousers.
Pythagoras is often described as what?
"The father of numbers."
Pythagoras was an ancient Greek mathematician famous for his theorem for a right angle triangle.
Who invented the pythagorean theorem?
Pythagoras of Samos was an Ionian Greek mathematician who founded a religious movement and came up with Pythagorean Theorum. Other theories include arguments that the theory predates him, or that it was rediscovered and popularized by one of his students.
Pythagoras (c. 580-500 B.C.) was an ancient Greek philosopher who was interested in numbers and their meanings. He discovered the relationships between mathematics and music, proposing that sounds and their relationships with other sounds can be measured using numbers. He also proposed that the Earth is a sphere, that the Earth, Moon, and stars revolve around the Sun, and that astronomy (the study of stars, planets, and heavenly bodies) could be written as mathematical sentences called equations. Pythagoras and his followers used lines, triangles, and squares made out of pebbles to represent numbers.
Today Pythagoras is best remembered for the Pythagorean theorem: The square of the length of the hypotenuse (the side of a right triangle opposite the right angle) of a right triangle equals the sum of the squares of the lengths of the triangle's other two sides.
What awards did Pythagoras receive?
He has not won any awards that we know of, being so old. However, he does have a prize named after him called the "Pythagoras Prize", given to mathematicians that were outstanding in their field.
Was there only one Pythagoras or where there a group of people called the Pythagoreans?
Both. There was a man named Pythagoras and his followers were called the Pythagoreans. Yes, there was someone called Pythagoras, and those loyal to him were Pythagoreans, though Pythagoreans sounds more like a tribe of natives than a bunch of people who followed a man.
What are some Italian contributions to mathematics?
Google: Peano, Fibonacci edit: The above are contributors, not contributions so it does not answer the question. Italian geometry of about a century ago was the start of what is considered "modern geometry." Italy is still a strong contributor to geometry as well as to practically all fields of mathematics.
