Addition

( " Add " Latin addere ) The addition, known colloquially as plus or numeracy And Convert is one of the four basic operations in arithmetic. The addition is based on the process of counting. Therefore one used for the operation to execute an addition, besides adding also the term summing. The arithmetic operators for addition is the plus sign " ". It was introduced in 1489 by Johannes Widmann.

Example: 2 3 = 5 is read as " two plus three ( is ) equal to five " or colloquially as " two and three results in five ."

• 4.1 Traditional method
• 4.2 Written adding decimals

Language rules

The elements of an addition are called addends and sum the result.

Coming from the English, the first term is sometimes called Augend. The second term is then called addend.

Principles and characteristics

The addition can be carried out in all speed ranges.

Commutative

The value of the sum is independent of the order of summands. Both arise as a result. These are called the commutative property of addition or Vertauschungsgesetz. For all the numbers and thus formally applies:

Associative law

In addition Parentheses may be implemented or omitted without changing the value of the sum. We call this property the associative law or law of compound addition. For all the numbers, and the following applies:

Therefore, since it 's not available for the addition of several numbers on the brackets, it omits many times and writes a little shorter

Neutrality of zero

The number zero with the symbol is the neutral element of addition. For all numbers applies:

Zero is the only number with this property.

Counter number

The counter number to a number is the number for which it holds. For example, the counter number is too. One writes for the opposite number of and it is then:

With the construction of the integers from the natural numbers to the smallest number range is defined, where each number has a counter number. The additive inverse of a number is given here clearly.

Distributive

In conjunction with the addition of the multiplication, the distributive laws apply. For all the numbers, and the following applies:

Accordingly, it can be converted into a sum and vice versa by factoring out a sum into a product by multiplying out the product.

Reduction rules

By addition of a number to both sides of an equation or inequality, the truth of an equation does not change. For all the numbers, and the following applies:

Solution of equations

The inverse operation of addition is subtraction. Subtracts one arrives on the question of the solution of elementary equations of the form

Where and are given numbers, and the number is searched. Because of the reduction rule, the solution is unique, if it exists. Thus, can serve as a definition for subtraction. It is then

In the natural numbers, the equation is solvable if it is. However, for the inverse equation

Solvable. In the integers former equation is always solvable and it is

Which can be verified by inserting and applying the calculation rules as a solution.

Definition of addition of the Peano axioms

Based on the Peano axioms can be the addition on the natural numbers defined as follows:

Denotes the successor of which is uniquely determined from the Peano axioms. Since 1 is the successor of 0, then

The successor of so not accord with.

Written addition

The written addition is one of the basic culture techniques that are learned in the first years of primary school education. The mastery of the written addition is also a prerequisite for learning the written multiplication.

Traditional method

Wherein said method i.a. is taught in primary schools in the German-speaking countries, the numbers to be added in the representation of the decimal system are written one above the other, that such bodies are to each other ( one on Einern, tens on tens, etc.). The numbers are then - added digit by digit - from right to left; the intermediate result is listed below, but only the last digit. If the intermediate result with more digits so arise transfers that need to be taken into account during the execution of the next column. For carrying out the process, it is necessary, the sums of numbers between 0 and 9 to know by heart.

Example:

5 7 = 12 The 1 is added to the transfer of the next (left -adjacent) digits column.

1 6 4 = 11

Written adding decimals

Here you can write the numbers so among themselves that the decimal point is exactly between them. One can think away the comma and to later write the result in the same place again. If the addends have different number of decimal places, so many zeros are added until all the summands have the same number of decimal places after the decimal point.

More notation possibility

Sums can be recorded by means of the summation symbol ( after the great Greek letter sigma ):

Under the Sigma the count variable is written ( in this case). You can be a start value (in this case ) are assigned by the connection with an equal sign. If no such assignment, so that means a summation over all possible. About the sigma of the final value (here: ). Between the start value and the end value, the counter variable is increased by one. In order to calculate the sum, and must be integers. In the case of n = m is the sum of an addend, in the case of n < m, it is defined as 0.

If one forms a sum of infinitely many terms, so this infinite series is called. We write this as a lower or upper limit, the symbol for minus and plus infinity: or.

Dealing with this symbol as well as some sums frequently occurring are described in the article sum.