Rational number in the context of "Arithmetic"

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation).

A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in language with number words or by dedicated symbols called numerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, which can display any non-negative integer using a combination of ten symbols, called numerical digits. Numerals can be used for counting (as with cardinal number of a collection or set), labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$, and π, and complex numbers which extend the real numbers with a square root of −1, and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio ⁠a/b⁠ is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory.

For example, the numbers 3 and 2 are commensurable because their ratio, ⁠3/2⁠, is a rational number. The numbers ${\displaystyle {\sqrt {3}}}$ and ${\displaystyle 2{\sqrt {3}}}$ are also commensurable because their ratio, ${\textstyle {\frac {\sqrt {3}}{2{\sqrt {3}}}}={\frac {1}{2}}}$, is a rational number. However, the numbers ${\textstyle {\sqrt {3}}}$ and 2 are incommensurable because their ratio, ${\textstyle {\frac {\sqrt {3}}{2}}}$, is an irrational number.

An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface Z or blackboard bold ${\displaystyle \mathbb {Z} }$.

The set of natural numbers ${\displaystyle \mathbb {N} }$ is a subset of ${\displaystyle \mathbb {Z} }$, which in turn is a subset of the set of all rational numbers ${\displaystyle \mathbb {Q} }$, itself a subset of the real numbers ⁠${\displaystyle \mathbb {R} }$⁠. Like the set of natural numbers, the set of integers ${\displaystyle \mathbb {Z} }$ is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, ⁠5+1/2⁠, 5/4, and the square root of 2 are not.

In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number p/q is a "good" approximation of a real number α if the absolute value of the difference between p/q and α may not decrease if p/q is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions.

In mathematics, an algebraic equation or polynomial equation is an equation of the form ${\displaystyle P=0}$, where P is a polynomial, usually with rational numbers for coefficients.

For example, ${\displaystyle x^{5}-3x+1=0}$ is an algebraic equation with integer coefficients and

A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically 0 (zero) and 1 (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.