Proof by Mathematical Induction

The square root of 2 is an irrational number, meaning it cannot be expressed as a ratio of two integers \(p/q\). The classic proof, often attributed to the Pythagoreans, is a proof by contradiction: it assumes \(\sqrt{2} = p/q\) in lowest terms, which leads to the conclusion that both \(p\) and \(q\) must be even, contradicting the initial assumption.

Ptolemy's theorem provides an elegant geometric proof for the sum and difference formulas in trigonometry. By inscribing a quadrilateral in a circle with one side as the diameter, the side lengths can be expressed as sines and cosines of the inscribed angles. Applying the theorem \(AC \cdot BD = AB \cdot CD + BC \cdot DA\) directly yields identities like \(sin(alpha + beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\).

The Cartesian coordinate system provides an algebraic model for Euclidean geometry. It uses one or more numbers, or coordinates, to uniquely determine the position of a point in space. In a plane, two perpendicular lines (the x-axis and y-axis) are used, allowing geometric shapes to be described by algebraic equations. This fusion of algebra and geometry is known as analytic geometry.

A second-order linear hyperbolic partial differential equation that governs the propagation of various types of waves. In its simplest form, it is written as \(\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u\), where \(u(\vec{x},t)\) is the amplitude of the wave, \(c\) is the wave speed, and \(\nabla^2\) is the Laplace operator. It models phenomena like vibrating strings, sound waves, and light waves.

The Euler characteristic is a topological invariant, a number that describes a topological space's structure or shape regardless of how it is bent. For polyhedra, it is defined by the formula \(\chi = V - E + F\), where V, E, and F are the number of vertices, edges, and faces, respectively. For a sphere, \(\chi = 2\), while for a torus, \(\chi = 0\).

One of the earliest problems in geometric probability, it is considered a precursor to the Monte Carlo method. It involves dropping a needle of length \(l\) onto a floor with parallel lines a distance \(t\) apart. The probability that the needle will cross a line is \(P = \frac{2l}{\pi t}\) (for \(l \le t\)). This provides a physical experiment to estimate \(\pi\).

Proof by contradiction, or reductio ad absurdum, is a form of indirect proof. It establishes the truth of a proposition by showing that assuming the proposition to be false leads to a logical contradiction. To prove a proposition \(p\), one assumes its negation, \(\neg p\), and deduces a contradiction, such as \(q \land \neg q\), thereby concluding that \(p\) must be true.

The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right-angled triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The formula is expressed as \(a^2 + b^2 = c^2\).

Also known as the method of indivisibles, this principle states that if two solids lying between two parallel planes have the property that every plane parallel to the two given planes intersects them in cross-sections of equal area, then the two solids have equal volumes. It provides a powerful method for calculating volumes of complex shapes without calculus.

The Pythagorean theorem provides the basis for the distance formula in Cartesian coordinates. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a plane is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This formula is a direct application of the theorem to a right triangle whose legs are the differences in the x and y coordinates.

This is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The problem asked if the seven bridges of the city of Königsberg could all be traversed in a single trip without doubling back, with the trip ending on the same landmass it began.

A fundamental theorem in topology and geometry stating that for any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the formula \(V - E + F = 2\). This value, 2, is the Euler characteristic of a sphere, revealing a deep topological property independent of the polyhedron's specific shape.

Bayes' theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event. It is a fundamental concept in probability theory and statistics. Mathematically, it is stated as \(P(A|B) = \frac{P(B|A)P(A)}{P(B)}\), where A and B are events and \(P(B) \neq 0\). It relates the conditional and marginal probabilities of two random events.

A second-order linear elliptic partial differential equation that describes systems in a steady-state or equilibrium condition. It is written as \(nabla^2 u = 0\) or \(Delta u = 0\), where \(nabla^2\) (or \(Delta\)) is the Laplace operator. Solutions, called harmonic functions, are the smoothest possible functions and represent potentials in fields like electrostatics, gravitation, and fluid flow.