Pythagorean Theorem
Statement
In a right triangle, the square of the hypotenuse equals the sum of squares of the legs. Converse: if a²+b²=c², the triangle is right-angled.
Examples
Example 1. Legs 3 and 4. Find hypotenuse.
Solution. c=√(9+16)=5.
In Depth
The Pythagorean theorem \(a^2+b^2=c^2\) is one of the oldest and most proved theorems in mathematics — over 370 distinct proofs are known. It is the foundation of Euclidean distance in any number of dimensions: in \(\mathbb{R}^n\), the distance between points \(\mathbf{u}\) and \(\mathbf{v}\) is \(\|\mathbf{u}-\mathbf{v}\|=\sqrt{\sum(u_i-v_i)^2}\).
Pythagorean triples are integer solutions: (3,4,5), (5,12,13), (8,15,17), (7,24,25). The general formula generates all primitive triples: \(a=m^2-n^2\), \(b=2mn\), \(c=m^2+n^2\) for integers \(m>n>0\) with \(\gcd(m,n)=1\) and \(m-n\) odd.
The converse is equally important: if \(a^2+b^2=c^2\), the triangle is right-angled. This is used in construction to verify right angles (the 3-4-5 check). In non-Euclidean geometry, the Pythagorean theorem fails: on a sphere, \(\cos c=\cos a\cos b\) (spherical law of cosines).
Fermat's Last Theorem, proved by Andrew Wiles in 1995, states that \(a^n+b^n=c^n\) has no positive integer solutions for \(n>2\) — a profound generalization of the Pythagorean equation.
Further Reading & Context
The study of pythagorean theorem connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.