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A ladder is leaning against a wall. The base of the ladder is 6 feet from the wall, and the ladder reaches a height of 8 feet. How long is the ladder?

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A ladder is leaning against a wall. The base of the ladder is 6 feet from the wall, and the ladder reaches a height of 8 feet. How long is the ladder?

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1. How did Pythagoras influence later philosophical and mathematical thought?

2. How did Pythagoras connect mathematics with the structure of reality?

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1)Pythagoras was born on the Greek island of Samos around 570 BCE.

2)The Pythagorean Theorem is named after Pythagoras. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides

4)Pythagoras founded the Pythagorean Brotherhood (or School of Pythagoras), a philosophical and religious group that studied mathematics, music, and philosophy.

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2. Theano was a philosopher and mathematician who is believed to have been one of the most prominent students — and possibly the wife — of Pythagoras. She played a crucial role in preserving and expanding Pythagorean teachings after his death. While historical records about her are sparse and sometimes mixed with legend, she is credited with writing works on mathematics, cosmology, medicine, and ethics, often from the perspective of Pythagorean philosophy. Theano also helped manage the Pythagorean school, especially after Pythagoras died, ensuring that his ideas lived on. In Pythagorean thought, the combination of mathematics, mysticism, and morality was central, and Theano contributed to preserving that holistic vision.

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Mathematical Harmony and Philosophy!

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DALL·E 2024-12-19 14.03.30 - Logo przedstawiające stylizowaną postać Pitagorasa z elementami matematycznymi, takimi jak trójkąt prostokątny i wzór pitagorejski. Postać jest uprosz

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DALL·E 2024-12-20 09.40.20 - A lively scene of Pythagoras actively teaching a class of students. Pythagoras, depicted as a middle-aged man with a neatly kept beard, wearing a clas

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DALL·E 2024-12-20 09.38.02 - A vibrant scene of Pythagoras, depicted as a middle-aged man with a neatly kept beard and lively expression, wearing a classical Greek robe, actively 

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DALL·E 2024-12-23 00.50.29 - A detailed modern classroom setting featuring Pythagoras teaching his students. Pythagoras, depicted as an ancient Greek philosopher with a flowing ro

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Pythagorean Wisdom

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Wanna become one of the Pythagoreans?

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1. In what ways did Archytas of Tarentum apply Pythagorean principles to his work in mathematics, music theory, and political philosophy?

2. Who was the female philosopher and mathematician who was one of Pythagoras' most prominent students and later played a key role in preserving his teachings?

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February is the second month of the year, and the number 2 is significant in Pythagorean numerology. It represents duality, balance, and relationships, which are core ideas in Pythagorean philosophy.

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DALL·E 2025-02-27 14.38.53 - An artistic depiction of the number 2 alongside Pythagoras, the ancient Greek mathematician. Pythagoras is shown as an elderly bearded man wearing a w

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1)Pythagoras influenced mathematics, philosophy, and science in many ways:

Mathematics: His Pythagorean Theorem is fundamental in geometry. He also contributed to number theory and the study of irrational numbers.

Philosophy: His ideas about numbers and reality influenced Plato’s Theory of Forms and metaphysics.

Science & Astronomy: He believed the universe followed mathematical harmony, influencing later astronomers like Kepler.

Music: He discovered numerical ratios in music, contributing to acoustics.

His belief that mathematics explains the universe shaped Western philosophy, logic, and science.

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1) a. Mathematics: Pythagoras' Pythagorean Theorem became a fundamental principle in geometry. He also contributed to number theory, the study of irrational numbers, and the idea that numbers underlie all reality.

b. Philosophy: He introduced the idea that numbers and mathematical relationships govern the universe, influencing later philosophers like Plato and Aristotle.

c. Science & Music: He discovered mathematical ratios in music, laying the foundation for harmonics.

2)Pythagoras believed that everything in the universe could be explained through numbers and mathematical relationships. His philosophy suggested tahtreality itself was structured according to mathematical principles, an idea that influenced both Platonism and later scientific thought.

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1. Pythagoras influenced later philosophical and mathematical thought by introducing the idea that numbers and mathematical relationships are essential for understanding the universe. His teachings inspired philosophers like Plato and helped establish a mathematical approach to science.

2. Pythagoras connected mathematics with the structure of reality by claiming that the universe is governed by numerical principles. Discoveries regarding musical harmony in numbers demonstrated that mathematics underlies both physical phenomena and aesthetic experiences.

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3. Pythagoras studied mathematics and philosophy in Ancient Egypt. He is believed to have traveled there around the 6th century BCE, where he learned from Egyptian priests, who were known for their advanced knowledge of mathematics, geometry, and religious philosophy. Some accounts also suggest that he later studied in Babylon, where he was exposed to Mesopotamian mathematical conceps and astronomical knowledge.

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Pythagoras was born in a Greek island called Samos,Pythagorean theory is named after him,he studied in ancient Egypt and the name of the school he found is Pythagorean Brotherhood.Pythagoras believed that numbers are fundamental essence of the universe ,

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1)Pythagoras was born on the island of Samos, Greece

2)The Pythagorean Theorem,which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides

3)He studied in ancient Egypt and possibly in Babylon.

4)He founded the Pythagorean School or Pythagorean Brotherhood in Croton, a Greek colony in southern Italy.

5)Pythagoras believed that nymbers are the fundamental principle of reality. He thought the universe was structured according to mathematical relationships and that numbers had mystical and philosophical significance.

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1)◦ Mathematics: Archytas contributed to geometry and was known for solving the problem of doubling the cube using geometric means.

◦ Music Theory: He advanced Pythagorean harmonic theory, studying how mathematical ratios determine musical notes and scales. ◦ Political Philosophy: He applied mathematical reasoning to politics advocating for a structured and just society influenced by Pythagorean ideals of order and harmony.

2) ◦ Theano, who was believed to be either Pythagoras' wife or close disciple, played a crucial role in continuing and preserving Pythagorean teachings after his death She contributed to mathematics, ethics, and mysticism, and helped maintain the Pythagorean Brotherhood.

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March is a month devoted to honoring the Pythagoreans and the field of Mathematics

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March is a month devoted to honoring the Pythagoreans and the field of Mathematics

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March is a month devoted to honoring the Pythagoreans and the field of mathematics

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1. Where was Pythagoras born?

2. What famous mathematical theorem is named after Pythagoras?

3. In which ancient civilization did Pythagoras study mathematics and philosophy?

4. What was the name of the school Pythagoras founded?

5. What did Pythagoras believe about the relationship between numbers and the universe?

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We can use the Pythagorean theorem for this right triangle. Given that one leg is 6 feet (base) and the other leg is 8 feet (height) the length of the ladder is 10 feet long.

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2) Pythagoras connected math and reality by showing that:

Numbers and ratios underpin music, geometry, and physical laws

Geometry reveals cosmic order

Mathematical harmony reflects the deeper structure of the universe

Studying math brings us closer to understanding existence itself

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2) Theano was a prominent female philosopher and mathematician of the 6th century BCE, believed to be either the wife or student of Pythagoras. After Pythagoras’s death, she is said to have preserved and continued his teachings, possibly even leading the Pythagorean school. She is known for her work on mathematical ratios—particularly the golden ratio—as well as for her writings on cosmology, ethics, and women's education. With her contributions to both science and philosophy, Theano stands out as a pioneering figure who demonstrated that women could play a significant role in the intellectual world of antiquity.

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Explain how the Pythagorean Theorem can be used to measure the height of a building without directly measuring it.

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Explain how the Pythagorean Theorem can be used to measure the height of a building without directly measuring it.

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Pythagoras had a profound influence on both philosophical and mathematical thought, and he is considered one of the foundational figures in Western intellectual history.

Philosophically, Pythagoras introduced the idea that reality is deeply connected to numbers and mathematical relationships. He believed that everything in the universe could be understood through mathematics, asserting that “all is number.” This belief led to a more abstract way of thinking about the world, where invisible principles like harmony, proportion, and order governed the physical and spiritual realms.

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The Archites of Tarentum (or Archytas of Tarentum) played a key role in expanding and integrating Pythagorean realities into both mathematical and political philosophy. Archytas was a philosopher, mathematician, and statesman, and is often considered the last great representative of the early Pythagorean school.As for the female philosopher and mathematician, Theano is traditionally considered one of the earliest and most significant female figures in the Pythagorean tradition. She was believed to be either the wife or student of Pythagoras and continued his work after his death. Theano contributed to mathematics (possibly in areas like the Golden Ratio), ethics, and philosophy, and played a vital role in preserving and teaching Pythagorean doctrines. Her writings, though surviving only in fragments or later attributions, suggest a commitment to the integration of reason, ethics, and scientific thought—core Pythagorean values.

Together, Archytas and Theano represent key figures who helped expand, protect, and apply Pythagorean teachings in both abstract and practical realms.

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a and b are the lengths of the two shorter sides (legs) of the triangle,c is the length of the hypotenuse (the ladder, in this case)

a = 6 feet (distance from the wall)

b = 8 feet (height the ladder reaches on the wall)

6² + 8² = c²

36 + 64 = c²

100 = c²

c = √100 = 10