Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign". Thus there are three kinds of equality, which are formalized in different ways.

These may be thought of as the logical, set-theoretic and algebraic concepts of equality respectively.

Etymology

The etymology of the word is from the Latin aequālis (“equal”, “like”, “comparable”, “similar”) from aequus (“equal”, “level”, “fair”, “just”).

Equality in mathematical logic

Logical formulations

Equality is defined so that things which have the same properties are equal. If some form of Leibniz's law is added as an axiom, the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally:

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,

The value of 0! is 1, according to the convention for an empty product.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects). This fact was known at least as early as the 12th century, to Indian scholars.Fabian Stedman in 1677 described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):

Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers, ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body;

Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Math (disambiguation)

Math (or Maths in most English-speaking countries) is used colloquially to refer to mathematics.

Math may also refer to:

See also