Distributive property

The distributive (Latin distribuere " distribute " ) are mathematical rules that specify how two two-digit shortcuts behave at the resolution of brackets to each other, namely that a link in a certain way is compatible with the other link.

Especially in school mathematics refers to the use of the distributive law to convert a sum into a product as factoring out or lifting. The dissolution of parentheses by applying the distributive law is referred to as multiplying out.

The distributive forms with the associative law and the commutative basic rules of algebra.

Formal definition

On a lot had two double-digit links and defined. The link is called

• Linksdistributiv applies over if for all
• Rechtsdistributiv applies over if for all
• Distributive over when it is over the left and rechtsdistributiv.

If the link is commutative, then these three conditions are equivalent.

Importance

An example is the two -digit combinations of the addition and multiplication of numbers can be used.

A distinction is made between linksdistributiven and rechtsdistributiven links:

In words:

If the " parent " link, in this case, the multiplication is commutative, we can infer from the Linksdistributivität the Rechtsdistributivität and vice versa.

An example of " only " Rechtsdistributivität is the division, which is not even commutative:

Here in the rule does not apply:

In school mathematics usually only the two-sided ( commutative ) distributive laws are referred to as such and circumvented the law division. It is then only calculated:

The distributive laws are among the axioms for rings and fields. Examples of structures in which two functions are mutually related to each other distributive Boolean algebras are, as the algebra of quantities or the switching algebra. There are also combinations of shortcuts that do not distributive behavior; For example, the addition is not distributively on the multiplication.

Multiplying sums can also take the following into words: A sum is multiplied by a sum by each summand of a sum with each summand of the other sum - multiplied and the resulting products added - in compliance with the sign.

Examples

Real Numbers

In the following examples the use of the distributive law is illustrated in the set of real numbers. In school mathematics are usually meant in these examples of multiplying out. From the point of view of algebra, the real numbers form a field, which ensures the validity of the distributive law.

When mental math distributive is often used unconsciously:

One wants to calculate 6:16 in the head. To this end, we multiply 6:10 and 6.6, and adds the intermediate results. Also the written Multiply is based on the distributive law.

Matrices

For the matrix multiplication the distributive is valid. More precisely

For all matrices and matrices and

For all matrices and matrices. Since the commutative law does not apply to the matrix multiplication, it follows from the first law not the second. It is in this case to two different laws.