Square number in the context of "Irrational number"

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3, which is the number 9.In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x.The adjective which corresponds to squaring is quadratic.

The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) = x + 2x + 1.

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:

1. Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
2. Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
3. Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
4. Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n + 1?

As of 2025, all four problems are unresolved.

In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term.An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. An extensional definition gives meaning to a term by specifying every object that falls under the definition of the term in question.

For example, in set theory one would extensionally define the set of square numbers as {0, 1, 4, 9, 16, ${\displaystyle \dots }$}, while an intensional definition of the set of the square numbers could be {${\displaystyle x\mid x}$ is the square of an integer}.

In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together: n = n × n × n × n.

Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.

In chemistry and atomic physics, an electron shell may be thought of as an orbit that electrons follow around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on further and further from the nucleus. The shells correspond to the principal quantum numbers (n = 1, 2, 3, 4 ...) or are labeled alphabetically with the letters used in X-ray notation (K, L, M, ...). Each period on the conventional periodic table of elements represents an electron shell.

Each shell can contain only a fixed number of electrons: the first shell can hold up to two electrons, the second shell can hold up to eight electrons, the third shell can hold up to 18, continuing as the general formula of the nth shell being able to hold up to 2(n) electrons. For an explanation of why electrons exist in these shells, see electron configuration.