mathematics

The decimal numeral system, also called base-10, is the most widely used number system in the world.

Binary, base-2, is the foundational number system for digital computing.

The Roman numerals system uses combinations of letters: I, V, X, L, C, D, and M.

The Millennium Prize Problems are seven mathematical problems whose solutions carry a one million dollar prize each, established in 2000.

Integer factorization is computationally hard for large numbers, a property underlying RSA cryptography.

The central limit theorem states that the sum of a large number of independent identically distributed variables is approximately normally distributed.

Topology is the branch of mathematics that studies properties of space preserved under continuous deformations.

Differential equations relate functions to their derivatives and are used to model many physical systems.

Number theory is the branch of mathematics devoted to the study of integers and integer-valued functions.

The four color theorem states that any planar map can be colored with at most four colors so that no two adjacent regions share a color.

Hexadecimal, base-16, is commonly used in computing for compact representation of binary data.

Negative numbers were not widely accepted in Western mathematics until the 17th century.

P versus NP is a major unsolved problem in computer science and mathematics asking whether every problem whose solution can be verified quickly can also be solved quickly.

The Collatz conjecture, also known as the 3n+1 problem, remains unsolved despite extensive computational verification.

The square root of 2 is irrational, a fact known to ancient Greek mathematicians.

Pi has been computed to over 100 trillion decimal places by 2022 with the help of distributed computing.

The normal distribution, also called Gaussian, is a continuous probability distribution characterized by its bell-shaped curve.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

Group theory studies algebraic structures known as groups, foundational for modern algebra and physics.

The Riemann zeta function, denoted by the Greek letter zeta, is central to analytic number theory.

Fermat's Last Theorem was proven by Andrew Wiles in 1994.

There are infinitely many prime numbers, as Euclid proved around 300 BC.

The golden ratio, denoted by the Greek letter phi, is approximately 1.618033988 and is irrational.

The natural logarithm has the base e, an irrational number approximately equal to 2.71828.

The Fibonacci sequence is defined by each term being the sum of the two preceding terms, starting from 0 and 1.

The area of a circle is pi times the radius squared.

The Riemann hypothesis, proposed by Bernhard Riemann in 1859, concerns the non-trivial zeros of the Riemann zeta function and remains unproven.

The Pythagorean theorem can be expressed as a^2 + b^2 = c^2 for the legs and hypotenuse of a right triangle.

Calculus was independently developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.

The fundamental theorem of calculus links differentiation and integration as inverse operations under suitable conditions.

Pi is an irrational number approximately equal to 3.14159 and represents the ratio of a circle's circumference to its diameter.

Euler's identity, e^(i*pi) + 1 = 0, links five fundamental mathematical constants.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

A prime number is a natural number greater than 1 with no positive divisors other than 1 and itself.

In Euclidean geometry, the sum of interior angles of a triangle equals 180 degrees.

Goedel's incompleteness theorems, published in 1931, show that any sufficiently rich consistent formal system contains true statements that cannot be proven within the system.

A vector space is a mathematical structure formed by a collection of vectors closed under addition and scalar multiplication.

Pierre de Fermat conjectured Fermat's Last Theorem around 1637, claiming a margin proof that was never written down.

Carl Friedrich Gauss is widely regarded as one of the greatest mathematicians in history, with major contributions to number theory, algebra, and statistics.

Georg Cantor founded modern set theory and proved that there are different sizes of infinity.

Alan Turing is considered the father of theoretical computer science and artificial intelligence.

The Poincare conjecture, posed in 1904, was proven by Grigori Perelman in 2003.

The Banach-Tarski paradox shows that a solid ball in three-dimensional space can be decomposed and reassembled into two identical copies of the original.

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between them.

A derivative measures the instantaneous rate of change of a function with respect to one of its variables.

An integral represents the accumulated total of a function's values over an interval, geometrically corresponding to area under the curve.

Probability theory is the branch of mathematics concerning the analysis of random phenomena.

Bayes' theorem describes the probability of an event based on prior knowledge of conditions related to the event.