Scott P. Raffone
Pythagoras was another Greek mathematician He lived around 500 BC, two hundred years prior to the work of Euclid Pythagoras was not only a great mathematician, he was a philosopher and astronomer. Pythagoras linked his mathematics to philosophy, music and astronomy He was interested in the concepts of number theory, mathematical figures and the abstract idea of a proof.
Pythagoras had six main contributions to mathematics and the sciences:
1.
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In astronomy, the Earth was a sphere at the center of the Universe. He also recognized that the orbit of the Moon was inclined to the equator of the Earth He was one of the first to realize that Venus as an evening star and as a morning star was the same planet.
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2.
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He discovered the five regular solids
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a.
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Tetrahedron
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b.
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Octahedron
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c.
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Icosahedron
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d.
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Hexahedron (cube)
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e.
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Dodecahedron
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3.
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The discovery of irrationals Because of his belief that all things are numbers, it would be logical to prove that the hypotenuse of an isosceles right triangle had a length corresponding to a number.
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4.
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Constructing figures of a given area and geometrical algebra. This is a skill that we find very important in the modern day teaching of geometry.
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5.
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The sum of the angles of a triangle is equal to two right angles. Also the sum of exterior angles is equal to four right angles. We use. numerical values of 180
0
and 360
0
for each of these statements
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6.
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The Pythagorean Theorem- For a right triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two legs (The square on the hypotenuse would not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side.)
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The Pythagorean Theorem was used by both the Babylonians and Egyptians for thousands of years but it wasn't until Pythagoras proved it did it get its due recognition in "the western world." He found the sum of the area of the two squares is equal to the area of the third square meant that the two squares could be cut up and reassembled to form a square identical to the third square.
Using the coordinate plane and adapting the teachings of Pythagoras, We can find the distance between any two coordinates (x
1
, y
1
) and (x
2
, y
2
). The distance formula:
d = ((x
2
- x
1
)
2
+ (y
2
- y
1
)
2
)
1/2
. We can show this in the diagram below showing the Pythagorean Theorem in the Cartesian plane. We want to find the distance between Point A and Point B.
In order to find the distance from point B to Point A we need to Place C on the graph such that it forms a right triangle with points A and B. According to the graph above Point B has a coordinate of (4, 3) and Point A has a coordinate of (8, 6) and point C has (8, 4)
The distance of a = (x
2
- x
1
) = (8 - 4) = 4
The distance of b = (y
2
- y
1
) = (6 - 3) = 3
c
2
= a
2
+ b
2
|
d = ((x
2
- x
1
)
2
+ (y
2
- y
1
)
2
)
1/2.
|
c
2
= 4
2
+ 3
2
|
d = ((8 - 4)
2
+ (6 - 3)
2
)
1/2.
|
c
2
= 16+ 9
|
d = (4
2
+ 3
2
)
1/2.
|
c
2
= 25
|
|
d = (16+ 9)
1/2.
|
c = 5
|
|
d = (25)
1/2
= 5
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Within this tool, Des Carte made simple rules for the Euclidean Transformations:
1.
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Translation is a movement or slide along the coordinate plane.
|
-
= (x + X, y + Y). Where X and Y are the distances of the shift.
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2.
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Reflection (x, y) is (-x, y) if reflected across the y-axis
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-
Reflection (x, y) is (x, -y) if reflected across the x-axis
-
3.
|
Rotation (x, y) around the origin by some angle A.
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-
X' = xcosA - ysinA
-
Y' = xsinA - y cosA
-
4.
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Dilation is to make object bigger by a scale factor (m)
|
-
(x', y') = (mx my)