The addition ( Latin additio from addere "Add"), colloquially Plus arithmetic or And-computing called, is one of the four basic arithmetic operations in the arithmetic . The addition is based on the process of counting . Therefore one used for the operation to perform an addition, besides adding also the expression adding up . The arithmetic symbol for the addition is the plus sign "+". It was introduced by Johannes Widmann in 1489 .

Example: 2 + 3 = 5 is read as “two plus three (is) equal to five” or colloquially “two and three equals five”.

Language regulations

The elements of an addition are called summands and the result is called the sum :

first summand + second summand = sum

Until well into the 20th century , the terms Augend could be used for the first and Addend for the second summand, which are now very rare:

Basic rules and properties

The addition can be carried out in all number ranges .

Commutative law

The value of a sum is independent of the order of the summands. Both as a result . This property is called the commutative law or commutation law of addition. For all numbers and thus formally applies: ${\ displaystyle 6 + 7}$ ${\ displaystyle 7 + 6}$ ${\ displaystyle 13}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a + b = b + a}$ Associative law

When adding, brackets may be implemented or omitted without changing the value of the sum. This property is called the associative law or connection law of addition. For all numbers , and the following applies: ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle (a + b) + c = a + (b + c)}$ Since the brackets are not important when adding several numbers , they are often left out and written a little shorter

${\ displaystyle a + b + c}$ Neutrality of zero

The number zero with the symbol is the neutral element of addition. The following applies to all numbers : ${\ displaystyle 0}$ ${\ displaystyle a}$ ${\ displaystyle a + 0 = 0 + a = a}$ Zero is the only number with this property.

Opposite number

The opposite number (or the additive inverse ) to a number is the number for which applies. For example, the opposite number is too . One writes for the opposite number of and it then applies: ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a + b = 0}$ ${\ displaystyle -2}$ ${\ displaystyle 2}$ ${\ displaystyle -a}$ ${\ displaystyle a}$ ${\ displaystyle a + (- a) = (- a) + a = 0}$ Distributive laws

In the interplay of addition with multiplication , the distributive laws apply . For all numbers , and the following applies: ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle a \ cdot (b + c) = (a \ cdot b) + (a \ cdot c)}$ ${\ displaystyle (a + b) \ cdot c = (a \ cdot c) + (b \ cdot c)}$ Accordingly, a product can be converted into a sum by multiplying and, conversely, by factoring out a sum into a product.

Reduction rules

Adding a number to either side of an equation or inequality does not change the truth of an equation. For all numbers , and the following applies: ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle a = b \; \ Leftrightarrow \; a + c = b + c}$ ${\ displaystyle a ${\ displaystyle a> b \; \ Leftrightarrow \; a + c> b + c}$ This addition is a special case of equivalent conversion .

Solving equations

The reverse operation of addition is subtraction . One arrives at the subtraction by asking about the solution of elementary equations of the form

${\ displaystyle a + x = b}$ ,

where and are given numbers and the number is sought. Because of the truncation rule, the solution is unique if it exists. Thus can serve as a definition for the subtraction . It then applies ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle ba}$ ${\ displaystyle a + (ba) = b}$ In the natural numbers the equation is solvable if and only if is. However, for is the reverse equation ${\ displaystyle a + x = b}$ ${\ displaystyle a \ leq b}$ ${\ displaystyle a \ geq b}$ ${\ displaystyle b + x = a}$ solvable. The first equation is always solvable in whole numbers and it holds

${\ displaystyle x = ba = b + (- a)}$ ,

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

Definition of addition from the Peano axioms

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

• ${\ displaystyle n + 0 = n}$ • ${\ displaystyle n + m '= (n + m)'}$ ${\ displaystyle n '}$ denotes the successor of , who is uniquely determined on the basis of the Peano axioms. Since 1 is the successor of 0, the following applies ${\ displaystyle n}$ • ${\ displaystyle n + 1 = n + 0 '= (n + 0)' = n '.}$ So the successor to agrees with . ${\ displaystyle n}$ ${\ displaystyle n + 1}$ Written addition is one of the basic cultural techniques that is learned in the first years of primary school. Mastering written addition is also a prerequisite for learning written multiplication.

In the process that u. a. is taught in primary schools in German-speaking countries, the numbers to be added are written on top of each other in the representation of the decimal system so that the corresponding digits are below each other (one over ones, tens over tens, etc.). The digits are then added - from right to left - digit by digit; the intermediate result is noted below, but only the ones place. If the interim result has multiple digits, there are carry-overs that must be taken into account when processing the next column. To carry out the procedure it is necessary to be able to add numbers between 0 and 9 together.

Example:

Here you write the numbers one below the other so that the decimal point is exactly below one another. You can ignore the comma and later write it back in the same place in the result. If the summands have different numbers of decimal places, zeros are added to the decimal places until all the summands have the same number of decimal places.

Further notation options

Sums can also be noted using the sums symbol (after the capital Greek letter Sigma ): ${\ displaystyle \ Sigma}$ ${\ displaystyle \ sum _ {i = m} ^ {n} x_ {i} = x_ {m} + x_ {m + 1} + x_ {m + 2} + \ dotsb + x_ {n-1} + x_ {n}}$ The counting variable (in this case ) is written under the sigma . It can be assigned a starting value (here:) by connecting it with an equal sign . If this assignment is not made, this means a summation over all possible . Above the sigma is the final value (here:) . The counter variable is increased by one between the start value and the end value. To be able to calculate the sum, and must be whole numbers. In the case , the sum consists of a summand, in the case it is defined as 0. ${\ displaystyle i}$ ${\ displaystyle m}$ ${\ displaystyle i}$ ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle m}$ ${\ displaystyle n = m}$ ${\ displaystyle n If one forms a sum from an infinite number of expressions, then this infinite series is called. One example is the Leibniz series :

${\ displaystyle \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {2k + 1}} = 1 - {\ frac {1} {3}} + { \ frac {1} {5}} - {\ frac {1} {7}} + {\ frac {1} {9}} - \ dots = {\ frac {\ pi} {4}}}$ .

The symbol stands for infinite . How to use the sum symbol and some common sums are described in the article Sum . ${\ displaystyle \ infty}$ 