In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory and calculus.
What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, and some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places.
In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16.
Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by √, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted √ = 3, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.
A root name server is a name server for the Domain Name System's root zone. It directly answers requests for records in the root zone and answers other requests by returning a list of the authoritative name servers for the appropriate top-level domain (TLD). The public root name servers are a critical part of the Internet infrastructure because they are the first step in translating (resolving) human readable host names into IP addresses that are used in communication between Internet hosts.
A combination of limits in the DNS and certain protocols, namely the practical size of unfragmented User Datagram Protocol (UDP) packets, resulted in a decision to limit the number of root servers to 13 logical servers. To serve the needs of the public Internet worldwide, the number of root server instances is 376 as of 22 August 2013[update].
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients). Numbers such as π that are not algebraic are said to be transcendental; almost all real and complex numbers are transcendental. (Here "almost all" has the sense "all but a countable set"; see Properties below.)
The sum, difference, product and quotient of two algebraic numbers is again algebraic (this fact can be demonstrated using the resultant), and the algebraic numbers therefore form a field, sometimes denoted by A (which may also denote the adele ring) or Q. Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. This can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.