Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.

Arithmetic systems can be distinguished based on the type of number they operate on. Integer arithmetic restricts itself to calculations with positive and negative whole numbers. Rational number arithmetic involves operations on fractions that lie between integers. Real number arithmetic includes calculations with both rational and irrational numbers and covers the complete number line.

Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Some arithmetic systems operate on mathematical objects other than numbers, such as interval arithmetic and matrix arithmetic.

Arithmetic operations form the basis of many branches of mathematics, such as algebra, calculus, and statistics. They play a similar role in the sciences, like physics and economics. Arithmetic is present in many aspects of daily life, for example, to calculate change while shopping or to manage personal finances. It is one of the earliest forms of mathematics education that students encounter. Its cognitive and conceptual foundations are studied by psychology and philosophy.

The practice of arithmetic is at least thousands and possibly tens of thousands of years old. Ancient civilizations like the Egyptians and the Sumerians invented numeral systems to solve practical arithmetic problems in about 3000 BCE. Starting in the 7th and 6th centuries BCE, the ancient Greeks initiated a more abstract study of numbers and introduced the method of rigorous mathematical proofs. The ancient Indians developed the concept of zero and the decimal system, which Arab mathematicians further refined and spread to the Western world during the medieval period. The first mechanical calculators were invented in the 17th century. The 18th and 19th centuries saw the development of modern number theory and the formulation of axiomatic foundations of arithmetic. In the 20th century, the emergence of electronic calculators and computers revolutionized the accuracy and speed with which arithmetic calculations could be performed.

Definition, etymology, and related fields

Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition, subtraction, multiplication, and division.^[1] In a wider sense, it also includes exponentiation, extraction of roots, and logarithm.^[2] The term "arithmetic" has its root in the Latin term "arithmetica" which derives from the Ancient Greek words ἀριθμός (arithmos), meaning "number", and ἀριθμητική τέχνη (arithmetike tekhne), meaning "the art of counting".^[3]

There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with natural numbers.^[4] However, the more common view is to include operations on integers, rational numbers, real numbers, and sometimes also complex numbers in its scope.^[5] Some definitions restrict arithmetic to the field of numerical calculations.^[6] When understood in a wider sense, it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations.^[7]

Arithmetic is closely related to number theory and some authors use the terms as synonyms.^[8] However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality.^[9] Traditionally, it is known as higher arithmetic.^[10]

Numbers

Numbers are mathematical objects used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different numeral systems to represent them.^[11]

Kinds

The main kinds of numbers employed in arithmetic are natural numbers, whole numbers, integers, rational numbers, and real numbers.^[12] The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as ${\displaystyle \{1,2,3,4,...\}}$. The symbol of the natural numbers is ${\displaystyle \mathbb {N} }$.^[a] The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as ${\displaystyle \{0,1,2,3,4,...\}}$ and have the symbol ${\displaystyle \mathbb {N} _{0}}$.^[14][b] Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers.^[16] The set of integers encompasses both positive and negative whole numbers. It has the symbol ${\displaystyle \mathbb {Z} }$ and can be expressed as ${\displaystyle \{...,-2,-1,0,1,2,...\}}$.^[17]

Based on how natural and whole numbers are used, they can be distinguished into cardinal and ordinal numbers. Cardinal numbers, like one, two, and three, are numbers that express the quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?".^[18]

A number is rational if it can be represented as the ratio of two integers. For instance, the rational number ${\displaystyle {\tfrac {1}{2}}}$ is formed by dividing the integer 1, called the numerator, by the integer 2, called the denominator. Other examples are ${\displaystyle {\tfrac {3}{4}}}$ and ${\displaystyle {\tfrac {281}{3}}}$. The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is ${\displaystyle \mathbb {Q} }$.^[19] Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to ${\displaystyle {\tfrac {3}{10}}}$, and 25.12 is equal to ${\displaystyle {\tfrac {2512}{100}}}$.^[20] Every rational number corresponds to a finite or a repeating decimal.^[21][c]

Irrational numbers are numbers that cannot be expressed through the ratio of two integers. They are often required to describe geometric magnitudes. For example, if a right triangle has legs of the length 1 then the length of its hypotenuse is given by the irrational number ${\displaystyle {\sqrt {2}}}$. π is another irrational number and describes the ratio of a circle's circumference to its diameter.^[22] The decimal representation of an irrational number is infinite without repeating decimals.^[23] The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is ${\displaystyle \mathbb {R} }$.^[24] Even wider classes of numbers include complex numbers and quaternions.^[25]

Numeral systems

A numeral is a symbol to represent a number and numeral systems are representational frameworks.^[26] They usually have a limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number.^[27] Numeral systems are either positional or non-positional. All early numeral systems were non-positional.^[28] For non-positional numeral systems, the value of a digit does not depend on its position in the numeral.^[29]

Tally marks and some tally sticks use the non-positional unary numeral system.

The simplest non-positional system is the unary numeral system. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers.^[30] Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks.^[31]

Egyptian hieroglyphics had a more complex non-positional numeral system. They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the Roman numeral system. It has the symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000.^[33]

A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the Hindu–Arabic numeral system, the radix is 10. This means that the first digit is multiplied by ${\displaystyle 10^{0}}$, the next digit is multiplied by ${\displaystyle 10^{1}}$, and so on. For example, the decimal numeral 532 stands for ${\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}}$. Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits.^[34]

Another positional numeral system used extensively in computer arithmetic is the binary system, which has a radix of 2. This means that the first digit is multiplied by ${\displaystyle 2^{0}}$, the next digit by ${\displaystyle 2^{1}}$, and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for ${\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}}$. In computing, each digit in the binary notation corresponds to one bit.^[35] The earliest positional system was developed by ancient Babylonians and had a radix of 60.^[36]

Arithmetic operations

Arithmetic operations underly many everyday occurrences, like when putting four apples from one bag together with three apples from another bag (top image) or when distributing nine apples equally among three children (bottom image).

Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output.^[37] The most important operations in arithmetic are addition, subtraction, multiplication, and division.^[38] Further operations include exponentiation, extraction of roots, and logarithm.^[39] If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations.^[40]

Two important concepts in relation to arithmetic operations are identity elements and inverse elements. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0.^[41]

There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in ${\displaystyle 13+4-4=13}$. Defined more formally, the operation "${\displaystyle \star }$" is an inverse of the operation "${\displaystyle \circ }$" if it fulfills the following condition: ${\displaystyle t\star s=r}$ if and only if ${\displaystyle r\circ s=t}$.^[42]

Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, ${\displaystyle 7+9}$ is the same as ${\displaystyle 9+7}$. Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since ${\displaystyle (5\times 4)\times 2}$ is the same as ${\displaystyle 5\times (4\times 2)}$.^[43]

Addition and subtraction

Addition and subtraction

Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is ${\displaystyle +}$. Examples are ${\displaystyle 2+2=4}$ and ${\displaystyle 6.3+1.26=7.56}$.^[44] The term summation is used if several additions are performed in a row.^[45] Counting is a type of repeated addition in which the number 1 is continuously added.^[46]

Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is ${\displaystyle -}$.^[47] Examples are $$Zdroj:https://en.wikipedia.org?pojem=Arithmetic
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