Inverse element

In mathematics come inverse elements in the study of algebraic structures. Such a structure is simply put out a lot and one in her defined two -digit shortcut ( arithmetic operation ). In this context, ie: if you linked an arbitrary element of the set and its inverse with the arithmetic operation, you always get the so-called neutral element as a result.

Colloquially one could the inverse element and the "reverse " or " opposite " element name. But we must not forget the context in which it is located, because there are a lot of ways, a quantity or an operation, to define (which you usually do not knows from school mathematics ).

Definition

Be a lot of double-digit combination and a neutral element. Be.

Next is no commutativity given, that is, it is only so called rechtsinvertierbar with the right inverse element, and it's called linksinvertierbar with the left inverse element.

However, exists for an element with an element, it means only invertible or double-sided invertible with the inverse element.

A double inverse element is often used as written in additive notation of the link, often in multiplicative notation as.

Properties

The link is assumed to be associative, meaning that is a monoid.

  • If an item is both left - and rechtsinvertierbar, then match all the left -and right- inverse elements. In particular, both sides invertible, and the inverse of an invertible element on both sides element is uniquely determined.
  • The inverse of the inverse is the original item ( involution ):
  • If a product is rechtsinvertierbar, so is rechtsinvertierbar; is linksinvertierbar, so is linksinvertierbar. Are both sides invertible, then also, and it is
  • A monoid homomorphism is Inverse to inverse from, ie is invertible, so is invertible, and it is

Applies in an algebraic structure with neutral element the associative law does not generally, it may be that an item has multiple links and multiple inverse right inverse.

Examples

Additive inverse

In the known sets of numbers ( natural numbers including zero 0, rational numbers, etc.) has an addition with neutral element 0, the additive inverse of a number is the number that is added to yield 0, so their opposites or its additive inverse. Adding up to a term, one adds a so-called constructive or productive zero.

For example, the opposite of Figure 7, because. For the same reason, the opposite is of turn 7, so is. This is true in general for all numbers.

Therefore, the opposite of a number is not always a negative number, ie a number applies for negative numbers: that is the opposite of a negative number is a positive number. However, the opposite of a positive number is always a negative number.

The opposite is obtained in these cases always by multiplying by -1, ie.

Generally there is the additive inverse element regularly in additively written abelian groups. The main examples are:

  • Integers
  • Rational numbers
  • Real numbers
  • Complex numbers
  • P- adic numbers
  • Hyper- real numbers

There are also sets of numbers in which, although an addition is executable, but in which no additive inverse elements exist. Such are, for example

  • Natural numbers
  • Cardinal numbers
  • Ordinals

It is the integers from the natural numbers to construct, defined by formally the negatives ( and 0, if 0 is not defined as a natural number ) is added and takes appropriate calculation rules. In this sense, every natural number has an opposite, which is its negative simultaneously. However, because it is (except for 0, when 0 is defined as a natural number ) is not a natural number, the set of natural numbers is not completed at the opposition or subtraction ( addition with an opposite ).

Multiplicative inverse

In the above- mentioned sets of numbers it has also a multiplication with neutral element 1, the multiplicative inverse of a number a is the number that is multiplied with a 1 results. Thus, it is the inverse of A.

For example, the inverse value of the rational number 7 1/7; in the integers has 7 but no multiplicative inverse.

Is generally a ring R is given, then the names of the elements that have multiplicative inverses, units of the ring. In the theory of divisibility is usually different (that is, a = eb with a unit e) does not distinguish between ring members, the multiplicative differ by one unit.

In residue class rings, one can calculate the multiplicative inverse using the extended Euclidean algorithm, if it exists.

Inverse function

Consider the set of all functions of a quantity. This quantity has the composition ( sequential execution ), defined as linking through

The composition is associative and has the identity map as a neutral element.

Now if a function is bijective, then the inverse function is the inverse element of in.

We generalize this concept to bijective functions and receives an inverse function with and

If A is a body such as the real numbers, then we must not be confused with the inverse of the inverse of! The inverse function is defined only if is bijective, and the inverse is defined only if has no zeros. Even if a subset of bijective maps on to vote inverse and inverse in general not match.

For example, the function has an inverse function, and a return value, which, however, do not match. (This is the set of positive real numbers. )

Involution (mathematics) Hyperreal number Closure (mathematics) Unit (ring theory)
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