**Introduction**

**Pythagoras**, an eminent name in the realm of mathematics and geometry, propounded the renowned representation – **Pythagorean theorem**. This theorem encapsulates the profound essence of triangular science, continuing to cast a long shadow over mathematical progression nearly two and a half millennia after its inception.

**Pythagoras: The Man Behind the Mathematics**

**Pythagoras** was an ancient Ionian Greek philosopher hailing from the island of Samos. Born in 570 BC, he is often revered as the first true mathematician. Pythagoras’s work and philosophy laid the groundwork for modern mathematics, particularly within the field of geometry.

**The Epitome of Mathematics – Pythagorean Theorem**

The **Pythagorean Theorem** is a guiding beacon in mathematics, underpinning discrete concepts that ascertain the structural fabric of geometry. Its principle is encapsulated succinctly and elegantly in ‘a² + b² = c²’, acting as the bedrock of right-angled triangle calculations.

**Delving Deeper into the Pythagorean Theorem**

In any right triangle, the theorem postulates that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This proposition has been, and continues to be, instrumental in myriad aspects of applied and theoretical mathematics, including Euclidean geometry, trigonometry, and calculus.

**Explaining Pythagorean Theorem through Compelling Concepts**

To elaborate the theorem, envision a right-angled triangle comprising three sides: ‘a’, ‘b’, and ‘c’. Here, ‘c’ is the hypotenuse while ‘a’ and ‘b’ are the other two sides. The principle of the **Pythagorean theorem** says that if you square the length of ‘a’ and ‘b’ (i.e., a² + b²), the result will equate to the square of the length of ‘c’ (c²).

**The Application of Pythagorean Theorem in Real Life**

Contrary to popular belief, the theorem isn’t confined to the nebulous realm of theoretical mathematics, it permeates into daily life, used by construction workers juxtaposing walls, astronomers calculating distances, computer science professionals optimizing algorithms, and so much more.

**Making Sense of Pythagorean Theorem in Three Dimensions**

Delving further, when we venture into three-dimensional space geometry, the theorem evolves into the Pythagorean triple where it underpins the integral concepts of distance calculation between various spatial coordinates.

**Deeper Insights into Pythagorean Triples**

The Pythagorean triples carry the theorem’s essence, providing essential insights into number theory. A Pythagorean triple consists of three positive integers ‘a’, ‘b’, and ‘c’, such that a² + b² = c². These triples are the algebraic constituents of the geometric theorem.

**Conclusion: The Perennial Impact of Pythagoras and the Theorem**

Truly, the influence of **Pythagoras** and his influential **Pythagorean theorem** traverses across centuries, acting as a cornerstone for many mathematical advancements. It stands as a testimony to the power of human cognition, representing an enduring mathematical truth that navigates us through the intricate labyrinth of numbers, angles, and dimensions.

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