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.

• 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.

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.

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.

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.

◦ 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.

• 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

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.

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

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