Elementary arithmetic

The basic arithmetic operations are the four mathematical operations of addition, subtraction, multiplication and division. The mastery of basic arithmetic is one of the basic skills of reading, writing and arithmetic, which are to be acquired by students during school hours.

Of the four basic arithmetic operations addition and multiplication as basic operations and subtraction and division are considered to be derived operations in arithmetic. For the two basic operations of a set of calculation rules, such as the Kommutativgesetze, the associative law and the distributive laws apply. In algebra, these concepts are then abstracted in order to transfer them to other mathematical objects.

The four basic arithmetic


The addition is the operation of the combined count of two ( or more ) numbers. The operator for the addition is the plus sign , the operands are called addends and the result is the sum of:

The result of addition of natural numbers is a natural number again. Through memorization and fundamental computing techniques small numbers can be added in the head. The addition of large numbers can be done by hand with the help of the written addition.


The subtraction is the process of extracting a number from another number. The operator for subtraction is the minus sign -, the two operands are called minuend and subtrahend and the result is the difference:

The result of the subtraction of two natural numbers is only then is a natural number, when the minuend is greater than the subtrahend. Are minuend and subtrahend equal, the result obtained is the number zero, which is often counted among the natural numbers. The subtrahend is greater than the minuend, obtained as a result of a negative number. In order to perform the subtraction fully, hence the speed range is extended to the integers. The subtraction of large numbers can be done by hand with the help of written subtraction.


Multiplication is the operation of Malnehmens two (or more ) numbers. The operator for multiplication is the mark * ( or x ), the operands are called factors, and the result is the product:

Are the factors of a natural or an integer, the result of the multiplication is also another natural or integer. By memorizing the multiplication tables small numbers can be multiplied in the head. The multiplication of large numbers can be done by hand with the help of the written multiplication.


The Division is the process of dividing one number by another number. The division operator is the Shared characters: (or / ), the two operands are called the dividend and the divisor, and the result is the quotient:

The result of division of two natural or integer numbers is only then a natural or integer, if the dividend is a multiple of the divisor. Otherwise, we obtain a fractional number. In order to carry out the division fully, hence the speed range is extended to the rational numbers. However, the division by zero can not be meaningfully defined. The division of large numbers can be manually performed using the long division.

Basic arithmetic in the classroom

The basic arithmetic operations are treated during the first years of school in mathematics education. In elementary school ( primary level ), the arithmetic is first taught with small natural numbers and later expanded to larger numbers. Teaching content are also the basics, division with remainder, solving simple equations and the rule of three. It will be practiced in the form of word problems mental arithmetic, written arithmetic, rollover Computing and Applications. For advantageous computing simple arithmetic laws are applied. In the first years of secondary school ( secondary education) then negative numbers are considered, the break statement, and thus introduced the rational numbers, and addresses the laws in connecting the four basic arithmetic operations.

Calculation rules

The following are, and figures from the underlying number field. For addition and multiplication, the Kommutativgesetze apply

That is the result of a sum or a product, regardless of the order of summands and factors. Furthermore, the associative laws apply

In the addition or multiplication of several figures it does not matter in which order the partial sums or partial products are formed. Therefore, the parentheses may be omitted in sums and products. In addition, the distributive laws apply

Which by multiplying out the product can be converted into a sum and vice versa by factoring out a sum into a product. Furthermore, the number itself is neutral with respect to addition and the number neutral with respect to multiplication, ie

For subtraction and division, these laws apply, with the exception of the distributive laws do not generally, the first distributive law is valid only for the subtraction in combination with the multiplication. Further calculation rules, such as point before line, the clamp rules and laws of the fractions can be found in the formulary arithmetic.

Unit Operations and derived operations

In arithmetic we consider addition and multiplication as basic operations. The addition of natural numbers as the repeated calculation of the successor a summand and the multiplication of natural numbers is viewed as repeated addition of a factor with itself. This view is then transferred to other number ranges, such as integer or rational numbers.

Subtraction and division to lead a derived mathematical operations as the basic operations. For subtraction and division can be reached on the question of the solution of basic equations of the form

Where and are given numbers from the underlying number field are and the number is searched. In order to solve these equations, a reverse operation is required for addition, namely the subtraction, as well as an inverse operation of the multiplication, namely, the division:

Subtracting a number will now be defined as the addition of the counter and the number of division by a number as multiplication by the inverse of:

The counter number, and the inverse of a number are used as the inverse number with respect to addition and multiplication indicated. In this way, the calculation rules for addition and multiplication can also be transferred to the subtraction and division.

Algebraic Structures

In algebra, these concepts initially created for arithmetic are abstracted in order to transfer them to other mathematical objects. An algebraic structure then consists of a carrier set ( here a set of numbers ), and one or more links on this quantity ( here the arithmetic operations ) that do not lead out of her. The various algebraic structures then differ only on the properties of the links ( the calculation rules), which are defined as axioms, but not with regard to the specific elements of the carrier set. For the basic operations we obtain the following algebraic structures:

  • The set of natural numbers together with the addition of a commutative semigroup in which the associative law and the commutative law governing the relationship.
  • The set of natural numbers together with the multiplication also a commutative semigroup.
  • The set of all integers is the addition of a commutative group in which additionally there is an identity element and an inverse element for each element.
  • The set of integers forms, with the addition and multiplication a commutative ring, in addition, the distributive laws in force in the for the links.
  • The set of rational numbers forming a body in which, in addition, each element except for the zero has an inverse element with respect to multiplication with addition and multiplication.

After the permanence doing all calculation rules ( a simple number range with the basic operations here ) ( an extended range of numbers with the same operations here ) are a fundamental structure in accordance with a specific structure. This structuring and axiomatization now allows to transfer knowledge gained from numbers to other mathematical objects. For example, corresponding operations on vectors, the vector addition and with matrices are the matrix addition. Special structures arise in the consideration of finite sets, for example, residue class rings as a mathematical abstraction of a division with remainder

Other arithmetic operations

  • Duplation, the doubling of numbers; was once one of the fundamental operations of arithmetic
  • Exponentiation, the repeated multiplication of numbers
  • Square roots and logarithms, the inverse operations of exponentiation