Irrational numbers

Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. In other words, they cannot be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not zero. This means that their decimal expansions are non-repeating and non-terminating.

Diophantine approximation is a branch of number theory that deals with the approximation of real numbers by rational numbers. It specifically studies the extent to which real numbers can be closely approximated by rational numbers, with a focus on the quality of these approximations. The name "Diophantine" comes from the ancient Greek mathematician Diophantus, who is known for his work in algebra, particularly in solving polynomial equations.

Real transcendental numbers are a subset of real numbers that are not algebraic. An algebraic number is defined as any number that is a root of a non-zero polynomial equation with integer coefficients. In contrast, transcendental numbers are not solutions to any such polynomial equation. For example, both rational numbers (like \( \frac{1}{2} \)) and irrational numbers (like \(\sqrt{2}\)) are algebraic, as they can be roots of polynomial equations with integer coefficients.

Apéry's constant is a mathematical constant denoted by \( \zeta(3) \), and it is defined as the value of the Riemann zeta function at \( s = 3 \): \[ \zeta(3) = \sum_{n=1}^\infty \frac{1}{n^3} \] This series converges to a specific numerical value, approximately \( 1.2020569 \).

The Copeland–Erdős constant is a real number that is constructed by concatenating the prime numbers in sequence. It is named after mathematicians Arthur Copeland and Paul Erdős. The constant is expressed as follows: \[ C = 0.

The Erdős–Borwein constant, often denoted as \( C_{E,B} \), is a mathematical constant that arises in the context of number theory, particularly in relation to certain infinite series and products.

Exact trigonometric values refer to the precise values of the sine, cosine, tangent, and other trigonometric functions for specific angles, typically in degrees or radians. These values are often expressed as fractions, whole numbers, or square roots, rather than decimal approximations.

Hippasus is a name associated with a few different contexts, primarily related to ancient Greece and mathematics. The most notable figure named Hippasus is a philosopher and mathematician from the Pythagorean school, who is traditionally credited with the discovery of irrational numbers, particularly in relation to the square root of 2.

An irrational number is a type of real number that cannot be expressed as a simple fraction or ratio of two integers. This means that if a number is irrational, it cannot be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Irrational numbers have non-repeating, non-terminating decimal expansions. This means their decimal representations go on forever without repeating a pattern.

Irrational rotation refers to a concept in mathematics, particularly in the field of dynamical systems and geometry. It typically involves rotations by an angle that is an irrational multiple of \( \pi \), meaning that the rotation angle cannot be expressed as a simple fraction. ### Key aspects of irrational rotation: 1. **Continuous Rotation**: When an object (like a point on a circle or a plane) is rotated continuously by an irrational angle, it never returns to its original position.

A Liouville number is a type of real number that is particularly significant in the field of number theory, especially in the study of transcendental numbers.

A **normal number** is a real number whose individual digits, and in broader terms, digits of any base, are uniformly distributed. More formally, a number is said to be normal in base \( b \) if, in its expansion in that base, all digits from \( 0 \) to \( b-1 \) appear with equal frequency in the limit as you consider more and more digits.

The Riemann zeta function, denoted as \(\zeta(s)\), is a complex function defined for complex numbers \(s\) with \(\text{Re}(s) > 1\) through the series: \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \] It can also be analytically continued to other values of \(s\) (with the exception of \(s

The prime constant, denoted as \( C_\pi \), is a mathematical constant related to the distribution of prime numbers. It is defined as the limit of the ratio of the number of prime numbers less than or equal to a given integer \( n \) and the logarithm of \( n \) as \( n \) approaches infinity.

The number \( e \), known as Euler's number and approximately equal to 2.71828, can be proven to be irrational using a proof by contradiction. Here’s the outline of the proof: ### Proof by Contradiction 1. **Assumption**: Assume \( e \) is rational.

The proof that π is irrational was first established by Johann Lambert in 1768. His proof is somewhat complex and relies on properties of continued fractions, but I can provide a high-level overview of the concepts involved in proving the irrationality of π. ### Overview of Lambert's Proof 1. **Definitions**: A number is irrational if it cannot be expressed as a fraction of two integers.

The Reciprocal Fibonacci Constant, denoted by \( R \), is defined as the sum of the reciprocals of the Fibonacci numbers.

A "schizophrenic number" is not a widely recognized term in mathematics or any scientific discipline, and there doesn't appear to be a standard definition or concept associated with it in the literature. It may be a colloquial or niche term that does not have broad use or acceptance.