Pythagoras and the Pythagorean Theorem

1. Pythagoras and the Pythagorean Theorem Made By Christine Gerwig

2. Pythagoras Pythagoras, a Greek philosopher, was born around 570 BCE in Samos, near modern Turkey. When he was around 18, he left to Phoenicia, Egypt, and possibly Babylonia (to study), and Croton (or Kroton) in Italy to escape the rule of the tyrant, Polycrates. He died around 490 BCE in another southern Italian city named Metapontion.

3. Pythagoras Pythagoras, a Greek philosopher, was born around 570 BCE in Samos, near modern Turkey. When he was around 18, he left to Phoenicia, Egypt, and possibly Babylonia (to study), and Croton (or Kroton) in Italy to escape the rule of the tyrant, Polycrates. He died around 490 BCE in another southern Italian city named Metapontion.

4. Pythagoras While in Croton, he established a society made of ‘Pythagoreans.’ The main idea of the Pythagorean philosophy is that there are three kinds of men: (from lowest to highest) those who buy and sell (lovers of wisdom), those who compete (lovers of honor), and those who simply watch (lovers of gain). This expresses the belief of the tripartite soul, the belief that every soul has three parts.

5. Pythagoras Pythagoras also taught about Rebirth, if the soul, a divine and immortal being, was purified from being in contact with the material body by the individual’s conduct and observance of rules.

6. Pythagoras Although not much is known about Pythagoras, it is fairly certain that he experimented the relationships between mathematics and music, as he also wanted to further explore the relationship between the physical world and mathematics. For example, he attached different weights to strings or used different string lengths, and examined the weights on the strings or the string lengths and the note they produced. He discovered that a string and another string twice its length produced harmonious tones, which led to musical scales and octaves. This also began the science of mathematical physics, where a physical law is mathematically expressed.

7. Pythagoras Pythagoras and his followers were also some of the first to imagine the world as a sphere, for it created a ‘perfect’ mathematical interrelation between a globe moving in circles and the stars’ behavior in a spherical universe. This was more pleasing that Anaximander’s cylindrical earth or a flat one, and later caused Greek scholars, such as Aristotle, to seek and find evidence to support the idea of a spherical Earth.

8. Pythagoras Pythagoras also believed that the Sun, Moon, and planets all move independently. His successors developed the idea of the Earth revolving around a central fire. This eventually led to the Copernican theory of the universe.

9. Pythagoras Some other beliefs of the Pythagoreans were: • To abstain from beans • Not to pick up what has fallen • Not to touch a white cock • Not to stir a fire with iron • Not to look in a mirror beside a light • Not to pick bread • Not to step over a crossbar • Not to eat from the whole loaf • Not to walk on highways • To regard men and women equally • To enjoy a common way of life • To live communally • Discoveries were communal and all were attributed to Pythagoras • The number is the essence of all things (virtues, colors)

10. Pythagoras Some other disciplines of the Pythagoreans were: • Silence • Music • Incenses • Physical and moral purifications • Rigid cleanliness • A mild asceticism • Utter loyalty • Common possessions • Secrecy • Daily self-examinations • Pure linen clothes

11. Pythagoras The Pythagoreans were finally destroyed in the 400’s BCE by the people of Croton. The people were suspicious of the Pythagoreans because they were aristocrats, so they killed the Pythagoreans in a political uprising.

12. The Pythagorean Theorem The Pythagorean Theorem was one of the theorems known earliest to ancient civilizations. 1000 years before Pythagoras, clay tablets from Babylonia had rules for creating Pythagorean Triplets, and showed that the Babylonians had some knowledge on the relationships between the sides of a right triangle, and had even come up with an estimation for the square root of 2.

13. The Pythagorean Theorem • A Chinese treatise from the Han Dynasty (209 BC – AD 202) called the Chao Pei Suan also mentions the Pythagorean Theorem. It explains the Theorem, and give an explanation of a 3, 4, 5 right triangle. Mathematicians are unsure whether it was meant to explain the Theorem as a whole, or just the 3, 4, 5 triangle.

14. The Pythagorean Theorem • The Sulbasutras, a group of texts written by the Vedic people, also show understanding of the Pythagorean Theorem. The Verdic people lived in India from 1500 BCE – 200 BCE.

15. The Pythagorean Theorem • An important part of the Vedics’ religion were sacrificial ceremonies at intricate altars. The Sulbasutras were a set of rules for how to make altars, and gave rules for creating geometric shapes, such as directions to make squares with the same area as rectangles, and circles with the same area as squares, because they believed that better ceremonies were created by accurate measurements. The directions include the diagonals of squares and rectangles, which provide knowledge of the Theorem.

16. The Pythagorean Theorem • The Pythagorean Theorem is most likely only named after Pythagoras because he was probably the first to offer a proof of the theorem. However, since he was credited for the Pythagoreans’ discoveries, it might have been one of the Pythagoreans who discovered the proof.

17. The Pythagorean Theorem • The Pythagorean Theorem, simply stated, says that ‘the sum of the squares of the 2 legs of a right triangle equals the square of the third side, the hypotenuse,’ or a²+b²=c². Using this formula, one can find the missing side length of a right triangle if presented with the lengths of the other 2 sides.

18. The Pythagorean Theorem • The Pythagorean Theorem, simply stated, says that ‘the sum of the squares of the 2 legs of a right triangle equals the square of the third side, the hypotenuse,’ or a²+b²=c². Using this formula, one can find the missing side length of a right triangle if presented with the lengths of the other 2 sides.

19. The Pythagorean Theorem To prove this theorem, we are going to use the figure to the left. It has one large square with side length c, 4 congruent right triangles with legs a and b and hypotenuse c, and a smaller square in the middle with side length b-a.

20. The Pythagorean Theorem Since the larger square has a side length of c, we know that the area of the larger square (A) equals c², or A=c².

21. The Pythagorean Theorem Since the smaller square has a side length of b-a, the area of the smaller square is (b-a)². Also, since the area of a triangle is (bh)/2, the area of each triangle is (ab)/2. But, we have four triangles, so the total area of the triangles is (4bh)/2.

22. The Pythagorean Theorem So, since the triangles have an area of (4ab)/2, and the smaller square has an area of (b-a)², we know that the area of the larger square is (4ab)/2 plus (a-b)², or A=(4ab)/2 + (a-b)².

23. The Pythagorean Theorem • Using the formulas we created for the area of the larger square (A), we can solve for a, b, and c as follows: • A=c² and A= + (b-a)² • c²= + (b-a)² • c²= 2ab + b² -2ab +a² • c²= a² + b² 4ab 2 4ab 2

24. The Pythagorean Theorem The Pythagorean Theorem can be used to find the distance between 2 points. By making a right triangle between 2 points you can solve for c to find the distance between the points. In the example, by using a^2+b^2=c^2, you would do (2-(-1))^2+(6.5-0)^2=c^2, or 3^2+6.5^2=c^2, or 51.25=c^2. So c, or the distance between the points, is about 7.1589 units.

25. The Pythagorean Theorem • This idea can be shown in the distance formula, D=√(x₂-x₁)²+(y₂-y₁)². By knowing the coordinates of 2 points, you can substitute the x and y values into the equation and solve for the distance between them.

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