Division (mathematics)

The Division is one of the four basic operations of arithmetic. It is the inverse operation of multiplication. The Division is colloquially referred to as parts. The long division is the method of sharing with pen and paper. It is taught in the classroom of elementary school and once again represented in textbooks for 5th grade, but is rarely used after the introduction of electronic means of learners. Arithmetic operator for division are the Divided characters " :", " ÷ " or "/".

Definition

Parts or divide means: to find x to a given number b ( the first factor) an appropriate number ( the second factor ), so that the multiplication results in a desired product a:

• Find in due and such that.

One is limited to natural or to whole numbers, so this is not always possible (see divisibility ).

In articles, for example in the field of rational numbers or in the bodies of the real and complex numbers, however, the following applies:

For every number and every number different from zero, there is exactly one number that satisfies the equation.

Division is therefore the inverse of multiplication to determine this. We write

It read:

Properties

For the Division neither commutative nor associative law applies. However, they can be attributed to the multiplication, because it is

It can therefore be advantageous to leave the division as a multiplication with the reciprocal value, since the multiplication is both associative and commutative and therefore easier and less error-prone forming allowed. For the division, however, applies with the addition and the subtraction of the second distributive law, ie

And

This is also known by the Rechtsdistributivität Division. The first distributive law ( Linksdistributivität ) is not satisfied with the addition and subtraction in general.

Division by zero

Example

Example of a pastry:

It is not possible to answer the question, there is no one there who could get the cake. If we translate this question in the language of mathematics and abstracts of all sorts except mathematical meanings, is from the philosophical question " How do I distribute something to 0 places? " The purely mathematical problem, " How do I divide by 0? ".

Mathematical Proof

The quotient is solution of the equation, ie. If you wanted to divide by 0, b = 0 would be.

Since the quotient has either no ( for ) or more than one solution ( for ), they say in mathematics:

Is ¹ ​​/ ₀ = ∞?

Some people have the intuition that the solution of the division must be infinite by zero, since experience shows that the individual gets more and more, the less are there, with whom he has to share something. If we apply this idea of spreading also to positive sizes, which can be less than 1. For example, on the distribution of 1 liter of water in rectangular or cylindrical vessels with ever smaller footprint, the water column is the higher, the smaller the footprint is. Indeed, there are in the mathematics of the limit, with a meaningful result can be obtained for a non- directly computable task method. Applying this method to, for example, so the result sought actually to infinity. However, only when the approaches zero from the positive side. As one approaches the zero from the direction of negative numbers to, the exact opposite happens and the value of the function tends towards. Thus, the function seeks to the point work against you as well as against, so it has no clear limit. This also shows that it is not reasonably possible to define. However, be said caveat that this only applies as long as you are moving in the field of real numbers - for the complex numbers is only an infinity usually defined.

Division by zero in the computer

In electronic computing systems division by zero produces mostly (or NaN, in the case of 0/0), a runtime error or otherwise intercepted with exception handling, since a further calculations with an undefined intermediate result would not make sense. In careless programming divisions by zero can lead to misbehavior in the current program, and even bring in some rare cases (for example, occurrence in the kernel ) the entire computer to crash.

Division with remainder

In the field of integers is considered: A division is then entirely feasible if the dividend is an integer multiple of the divisor. In general, however, the division is not carried out completely, that is, it remains a residue.

Spellings

There are several spellings for the division:

The colon as a sign of division is common only since Leibniz (1646-1716), although he is also known in older writings. William Oughtred introduced the notation in his book Clavis Mathematicae one of 1631.

The next to the last -mentioned notation is also called fraction form or short ( real ) fraction. The fraction notation is unique only in commutative multiplication; playing in more general mathematical structures involved, as mentioned below under " generalization ".

Generalization

In abstract algebra, algebraic structures defined, the bodies are called. Bodies are characterized by the fact that the division (except by 0) is always possible in them. The division is done here by multiplying by the inverse element of the divisor.

In more general structures ( with noncommutative multiplication) must distinguish between left and right division Division. Also, the ( non-) validity of the associative law influence on the properties of quotients.