Identity element
Definition
In mathematics, an identity element, or neutral element, is a special type of element in a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic structures such as groups, rings, and fields.
Properties
The identity element's defining property is that when it is combined with any element in the set, the result is that element. More formally, if * is a binary operation and e is the identity element, then for any element a in the set: a * e = e * a = a
This property holds for various mathematical structures. In a group, the identity element with respect to the group operation is often denoted by e or 1. In a ring, there are two identity elements: one for addition (usually denoted as 0) and one for multiplication (usually denoted as 1).
Examples
Identity elements can be found in many areas of mathematics. Here are some examples:
- In the set of integers under addition, the identity element is 0, because adding 0 to any integer returns that integer. - In the set of real numbers under multiplication, the identity element is 1, because multiplying any real number by 1 gives that real number. - In the set of n-by-n matrices under matrix addition, the identity element is the zero matrix, because adding the zero matrix to any matrix gives that matrix. - In the set of invertible n-by-n matrices under matrix multiplication, the identity element is the identity matrix, because multiplying any invertible matrix by the identity matrix gives that matrix.
Identity Elements in Different Structures
Different algebraic structures have different requirements for identity elements. Here are some examples:
- In a group, there is a single identity element with respect to the group operation. This identity element is unique. - In a ring, there are two identity elements: one for addition and one for multiplication. The additive identity is unique, but the multiplicative identity is not necessarily unique. - In a field, there are also two identity elements: one for addition and one for multiplication. Both identity elements are unique in a field.
Identity Elements in Computer Science
In computer science, identity elements are used in various algorithms and data structures. For example, in the context of binary search trees, the identity element is often used as a sentinel value. In functional programming languages, the concept of an identity function, which is a function that always returns its input, is a common example of an identity element.
See Also
- Group (mathematics) - Ring (mathematics) - Field (mathematics) - Binary operation - Computer Science
