Number theory

Number Theory

Number theory (pronounced: /ˈnʌmbər ˈθɪəri/), also known as arithmetic or higher arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The etymology of the term "number theory" comes from the French "théorie des nombres", first used by Adrien-Marie Legendre in the 19th century.

Definition

Number theory studies properties of the integers, more broadly, properties of mathematical objects that can be defined using integers. This includes prime numbers, rational numbers, irrational numbers, and algebraic numbers, among others.

History

Number theory has a long and distinguished history. The ancient Greeks were the first to systematically study number theory, with Euclid's Elements being a notable early treatise on the subject. The field has since expanded to include the study of many different types of numbers and number systems.

Subfields

Number theory is divided into several subfields, each with its own unique focus. These include:

• Algebraic number theory: This field studies algebraic structures related to algebraic numbers, integers, and rational numbers.
• Analytic number theory: This field uses methods from mathematical analysis to solve problems about integers and prime numbers.
• Geometric number theory: This field is concerned with the relationships between numbers and geometry.
• Computational number theory: Also known as algorithmic number theory, this field focuses on algorithms for performing number theoretic computations.

Related Terms

• Prime number: A natural number greater than 1 that has no positive divisors other than 1 and itself.
• Composite number: A positive integer that has at least one positive divisor other than one or itself.
• Rational number: A number that can be expressed as the quotient or fraction of two integers.
• Irrational number: A real number that cannot be expressed as a simple fraction.
• Algebraic number: A number that is a root of a non-zero polynomial equation in one variable with integer coefficients.

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