Basic arithmetic

Symbols of the four basic arithmetic operations: plus , minus , times and divided .

The basic arithmetic operations (also called basic arithmetic operations or simply 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 that students have to acquire during their school days .

Of the four basic arithmetic operations, addition and multiplication are regarded as basic operations and subtraction and division as derived operations. A number of calculation rules apply to the two basic operations , such as the commutative laws , the associative laws and the distributive laws . In algebra , these concepts are then abstracted so that they can be transferred to other mathematical objects .

The four basic arithmetic operations

addition

Example of an addition
${\ displaystyle 1 + 2 = 3}$

The addition is the process of co-counting two (or more) numbers. The operator for the addition is the plus sign +, the operands are called summands , the term sum and the result is called the sum value / value of the sum:

Summand + summand = sum value

The result of adding natural numbers is again a natural number. By memorizing and using elementary arithmetic techniques , small numbers can be added up in your head. The addition of large numbers can be done by hand with the help of written addition .

subtraction

Example of a subtraction
${\ displaystyle 5-1 = 4}$

Subtraction is the act of subtracting one number from another number. The operator for the subtraction is the minus sign -, the two operands are called the minuend and subtrahend , the term difference and the result is called the difference value / value of the difference.

Minuend - Subtrahend = difference value

However, the result of subtracting two natural numbers is only a natural number again if the minuend is greater than the subtrahend. If the minuend and subtrahend are the same, the result is the number zero , which is often also counted among the natural numbers. If the subtrahend is greater than the minuend, the result is a negative number . In order to be able to carry out the subtraction without restrictions, the number range is therefore extended to the whole numbers . The subtraction of large numbers can be done by hand with the help of written subtraction .

multiplication

Example of a multiplication
${\ displaystyle 3 \ cdot 5 = 15}$

Multiplication is the act of taking two (or more) numbers. The operator for multiplication is the mark of · (or x), the operands are multiplier called and multiplicand, the term product and the result is product value / value of the product:

Multiplicand · Multiplier = product value

If there is no need to distinguish between multiplier and multiplicand, both are often referred to collectively as factors .

If the factors are natural or whole numbers, the result of the multiplication is again a natural or whole number. By memorizing the multiplication tables , small numbers can be multiplied in your head. The multiplication of large numbers can be done by hand with the help of written multiplication .

division

Example of a division
${\ displaystyle 12: 3 = 4}$

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

Dividend: Divisor = quotient value

However, the result of dividing two natural or whole numbers is only a natural or whole number again if the dividend is a multiple of the divisor. Otherwise you get a fraction . In order to be able to carry out the division without restrictions, the number range is therefore expanded to include the rational numbers . The division by zero , however, can not be meaningfully defined. The division of large numbers can be done by hand using the written division .

Extract from the 2017/2018 curriculum .

Basic arithmetic in class

The basic arithmetic operations are dealt with in mathematics lessons during the first years of school . In elementary school ( primary level ) arithmetic is first taught with small natural numbers and later expanded to include larger numbers. Lessons also include the multiplication tables, division by remainder , solving simple equations and the rule of three . There are mental arithmetic , written arithmetic, rollover calculations and applications in the form of word problems practiced. Simple calculation laws are used for advantageous calculation. In the first years of a secondary school ( secondary level I ) negative numbers are also considered, fractions and thus rational numbers are introduced, and the laws relating to the connection of the four basic arithmetic operations are dealt with.

Calculation rules

Illustration of the commutative laws

The following are , and numbers from the underlying number range. The commutative laws apply to addition and multiplication ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$

{\ displaystyle a + b = b + a}

and ,

{\ displaystyle a \ cdot b = b \ cdot a}

that is, the result of a sum or a product is independent of the order of the summands or factors. Next apply associative laws

{\ displaystyle (a + b) + c = a + (b + c)}

and .

{\ displaystyle (a \ cdot b) \ cdot c = a \ cdot (b \ cdot c)}

When adding or multiplying several numbers, it does not matter in which order the partial sums or partial products are formed. Therefore the brackets can be omitted from totals and products. In addition, the distributive laws apply

{\ displaystyle a \ cdot (b + c) = a \ cdot b + a \ cdot c}

and ,

{\ displaystyle (a + b) \ cdot c = a \ cdot c + b \ cdot c}

with which a product can be converted into a sum by multiplying and vice versa by excluding a sum into a product. Furthermore, the number behaves neutrally with regard to addition and the number neutrally with regard to multiplication, that is ${\ displaystyle 0}$ ${\ displaystyle 1}$

{\ displaystyle a + 0 = 0 + a = a}

and .

{\ displaystyle a \ cdot 1 = 1 \ cdot a = a}

These laws do not apply to subtraction and division, or only to a limited extent. Further calculation rules, such as dot before line , the rules in brackets and the laws of fractions , can be found in the arithmetic formula collection .

Basic operations and derived operations

Subtraction as addition
${\ displaystyle 5-1 = 5 + ({-} 1)}$

Division as multiplication
${\ displaystyle 12: 3 = 12 \ cdot {\ tfrac {1} {3}}}$

In arithmetic , addition and multiplication are considered basic operations. The addition of natural numbers is seen as repeated determination of the successor of a summand and the multiplication of natural numbers as repeated addition of a factor to itself. This view is then transferred to other number ranges, such as whole or rational numbers.

Subtraction and division are introduced as derived mathematical operations of the basic operations. One arrives at subtraction and division by asking about the solution of elementary equations of the form

{\ displaystyle a + x = b}

or ,

{\ displaystyle a \ cdot x = b}

where and are given numbers from the underlying number range and the number is sought. To solve these equations, an inverse operation of addition is required, namely subtraction, and also an inverse operation of multiplication, namely division: ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle x}$

{\ displaystyle x = ba}

or .

{\ displaystyle x = b \, / \, a}

The subtraction of a number is now defined as addition with the opposite number and division by a number as multiplication with the reciprocal value : ${\ displaystyle a}$ ${\ displaystyle -a}$ ${\ displaystyle a}$ ${\ displaystyle {\ tfrac {1} {a}}}$

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

or .

{\ displaystyle x = b \ cdot {\ tfrac {1} {a}}}

The opposite number and the reciprocal of a number are called the inverse numbers with respect to addition and multiplication. In this way, the calculation rules for addition and multiplication can also be applied to subtraction and division.

Algebraic structures

In algebra , these concepts that were initially created for arithmetic are abstracted in order to be able to transfer them to other mathematical objects . An algebraic structure then consists of a set of carriers (here a set of numbers) and one or more links on this set (here the arithmetic operations) that do not lead out of it . The various algebraic structures then differ only in terms of the properties of the links (the rules of calculation), which are defined as axioms , but not in terms of the concrete elements of the set of carriers. The following algebraic structures are obtained for the basic operations:

The set of natural numbers forms a commutative semigroup with the addition , in which the associative law and the commutative law apply to the link. ${\ displaystyle (\ mathbb {N}, +)}$
The set of natural numbers also forms a commutative semigroup with multiplication . ${\ displaystyle (\ mathbb {N}, \ cdot)}$
Together with the addition, the set of whole numbers forms a commutative group in which there is also a neutral element and an inverse element for each element. ${\ displaystyle (\ mathbb {Z}, +)}$
The set of whole numbers forms a commutative ring with addition and multiplication , in which the distributive laws also apply to the links. ${\ displaystyle (\ mathbb {Z}, +, \ cdot)}$
With addition and multiplication, the set of rational numbers forms a field in which every element apart from zero has an inverse element with regard to multiplication. ${\ displaystyle (\ mathbb {Q}, +, \ cdot)}$

According to the principle of permanence , all calculation rules of a basic structure (here a simple number range with the basic operations) also apply in a correspondingly more specific structure (here an extended number range with the same operations). This structuring and axiomatization now makes it possible to transfer knowledge gained from numbers to other mathematical objects. For example, corresponding operations at are vectors the vector addition and matrices , the matrix addition . Special structures arise when considering finite sets , for example remainder class rings as a mathematical abstraction of a division with remainder .

history

All four basic arithmetic operations were already known in ancient Egyptian mathematics and in Babylonian mathematics . However, multiplication and division were not arithmetic operations in their own right. The multiplication of natural numbers was traced back to the continued doubling ( duplication ) of a factor and subsequent addition of the partial results. In the case of non-integer quotients, the division was carried out approximately by means of continued halving ( mediation ). Multiplication and division can only be found as independent operations in ancient Greek mathematics , for example in Euclid and Pappos .

Which arithmetic operations are included in the basic arithmetic operations has changed significantly over time. With Heron and Diophantos , squaring and taking square roots were added as further basic arithmetic operations to the four known arithmetic operations . In Indian mathematics , these operations have been replaced by the more general exponentiation and extraction of the roots , and in more recent times the logarithm has been added as the seventh basic arithmetic function. In Islamic mathematics , starting with Al-Chwarizmi , the duplatio and the mediatio were also viewed as separate arithmetic operations.

In the arithmetic books of the Middle Ages there were further additions to the basic arithmetic operations, which were called "species" there. Around 1225, Johannes de Sacrobosco found a total of nine of these species: Numeratio , Additio , Subtractio , Duplatio , Multiplicatio , Mediatio , Divisio , Progressio and Radicum extractio . The Numeratio dealt with counting, reading and writing numbers, the Progressio was the summation of successive natural numbers and the Extractio only included the taking of square roots. It was only in 1494 rejected Luca Pacioli the Duplatio and Mediatio as special cases of multiplication and division again. This was followed by further reductions until Gemma Frisius in 1540 was one of the first authors to limit the basic arithmetic operations to the four known.

literature

Walter Gellert (Hrsg.): Fachlexikon ABC Mathematik . Verlag Harri Deutsch , Thun and Frankfurt / Main 1978, ISBN 3-87144-336-0 .

Individual evidence

↑ Basic calculation type . In: Duden online dictionary . Bibliographical Institute GmbH.
↑ Calculation method . In: PONS online dictionary - spelling and foreign words . PONS GmbH.
^ Mullis, IVS, Martin, MO, Minnich, CA, Stanco, GM, Arora, A., Centurino, VAS, & Castle, CE (Eds.): TIMSS 2011 Encyclopedia: Education Policy and Curriculum in Mathematics and Science . Volumes 1 and 2. TIMSS & PIRLS International Study Center, Boston College, 2012, ISBN 978-1-889938-59-2 ( online ).
↑ ^a ^b ^c Johannes Tropfke : History of elementary mathematics in a systematic representation . First volume. Veit, Leipzig 1902, p. 29-31 .

Web links

Wikibooks: ${\ displaystyle {\ begin {smallmatrix} {\ mathbf {MATH} \ mu \ alpha T \ mathbb {R} ix} \ end {smallmatrix}}}$ - Mathematics for school

Wiktionary: Basic arithmetic - explanations of meanings, word origins, synonyms, translations

[1] Basic calculation type . In: Duden online dictionary . Bibliographical Institute GmbH.

[2] Calculation method . In: PONS online dictionary - spelling and foreign words . PONS GmbH.

[3] Mullis, IVS, Martin, MO, Minnich, CA, Stanco, GM, Arora, A., Centurino, VAS, & Castle, CE (Eds.): TIMSS 2011 Encyclopedia: Education Policy and Curriculum in Mathematics and Science . Volumes 1 and 2. TIMSS & PIRLS International Study Center, Boston College, 2012, ISBN 978-1-889938-59-2 ( online ).

[tropfke-4] Johannes Tropfke : History of elementary mathematics in a systematic representation . First volume. Veit, Leipzig 1902, p. 29-31 .