# Arithmetic

# Arithmetic

Early development of arithmetic

Axioms of the operations of arithmetic

Arithmetic is a branch of mathematics concerned with the numerical manipulation of numbers using the operations of addition, subtraction, multiplication, division, and the extraction of roots. General arithmetic principles slowly developed over time from the principle of counting objects. Critical to the advancement of arithmetic was the development of a positional number system and a symbol to represent the quantity zero. All arithmetic knowledge is derived from the primary axioms of addition and multiplication. These axioms describe the rules which apply to all real numbers, including whole numbers, integers, and rational and irrational numbers.

## Early development of arithmetic

Arithmetic developed slowly over the course of human history, primarily evolving from the operation of counting. Prior to 4000 BC, few civilizations were even able to count up to ten. Over time however, people learned to associate objects with numbers. They also learned to think about numbers as abstract ideas. They recognized that four trees and four cows had a common quantity called four. The best evidence suggests that the ancient Sumerians of Mesopotamia were the first civilization to develop a respectable method of dealing with numbers. By far the most mathematically advanced of these ancient civilizations were the Egyptians, Babylonians, Indians, and Chinese. Each of these civilizations possessed whole numbers, fractions, and basic rules of arithmetic. They used arithmetic to solve specific problems in areas such as trade and commerce. As impressive as the knowledge that these civilizations developed was, they still did not develop a theoretical system of arithmetic.

The first significant advances in the subject of arithmetic were made by the ancient Greeks during the third century BC. Most importantly, they realized that a sequence of numbers could be extended infinitely. They also learned to develop theorems that could be generally applied to all numbers. At this time, arithmetic was transformed from a tool of commerce to a general theory of numbers.

## Numbering system

Our numbering system is of central importance to the subject of arithmetic. The system we use today, called the Hindu-Arabic system, was developed by the Hindu civilization of India some 1,500 years ago. It was brought to Europe during the Middle Ages by the Arabs and fully replaced the Roman numeral system during the seventeenth century.

The Hindu-Arabic system is called a decimal system because it is based on the number 10. This means that it uses 10 distinct symbols to represent numbers. The fact that 10 is used is not important, because it could have just as easily been based on another number of symbols, like 14. An important feature of our system is that it is a positional system. This means that the number 532 is different from the number 325 or 253. Critical to the invention of a positional system is perhaps the most significant feature of our system: a symbol for zero. Note that zero is a number just as any other, and we can perform arithmetic operations with it.

## Axioms of the operations of arithmetic

Arithmetic is the study of mathematics related to the manipulation of real numbers. The two fundamental properties of arithmetic are addition and multiplication. When two numbers are added together, the resulting number is called a sum. For example, 6 is the sum of 4 + 2. Similarly, when two numbers are multiplied, the resulting number is called the product. Both of these operations have a related inverse operation that reverses or “undoes” its action. The inverse operation of addition is subtraction. The result obtained by subtracting two numbers is known as the difference. Division is the inverse operation of multiplication and results in a quotient when two numbers are divided. The operations of arithmetic on real numbers are subject to a number of basic rules, called axioms. These include axioms of addition, multiplication, distributivity, and order. For simplicity, the letters a, b, and c denote real numbers in all of the following axioms.

There are three axioms related to the operation of addition. The first, called the commutative law, is denoted by the equation a + b = b + a. This means that the order in which you add two numbers does not change the end result. For example, 2 + 4 and 4 + 2 both mean the same thing. The next is the associative law, which is written a + (b + c) = (a + b) + c. This axiom suggests that grouping numbers also does not affect the sum. The third axiom of addition is the closure property, which states that the equation a + b is a real number.

From the axioms of addition, two other properties can be derived. One is the additive identity property, which says that for any real number a + 0 = a. The other is the additive inverse property, which suggests that for every number a, there is a number –a such that –a+ a = 0.

Like addition, the operation of multiplication has three axioms related to it. There is the commutative law of multiplication stated by the equation a× b=b× a.

There is also an associative law of multiplication denoted by a× (b× c) = (a× b)× c. And finally, there is the closure property of multiplication, which states that a× b is a real number. Another axiom related to both addition and multiplication is the axiom of distributivity represented by the equation (a + b)× c=(a× c) + (b× c).

The axioms of multiplication also suggest two more properties. These include the multiplicative identity property, which says for any real number a, 1× a = a, and the multiplicative inverse property, which states for every real number there exists a unique number (1/a) such that (1/a)× a=1.

The axioms related to the operations of addition and multiplication indicate that real numbers form an algebraic field. Four additional axioms assert that within the set of real numbers there is an order. One states that for any two real numbers, one and only one of the following relations is true: either a < b, a > b or a = b. Another suggests that if a < b, and b < c, then a < c. The monotonic property of addition states that if a < b, then a + c < b + c. Finally, the monotonic property of multiplication states that if a < b and c >0, then a× c < b× c.

## Numbers and their properties

These axioms apply to all real numbers. It is important to note that the term real numbers describes the general class of all numbers that includes whole numbers, integers, rational numbers, and irrational numbers. For each of these number types only certain axioms apply.

Whole numbers, also called natural numbers, include only numbers that are positive integers and zero. These numbers are typically the first ones to which a person is introduced, and they are used extensively for counting objects. Addition of whole numbers involves combining them to get a sum. Whole number multiplication is just a method of repeated addition. For example, 2× 4 is the same as 2 + 2 + 2 + 2. Since whole numbers do not involve negative numbers or fractions, the two inverse properties do not apply. The smallest whole number is zero, but there is no limit to the size of the largest.

Integers are whole numbers that include negative numbers. For these numbers the inverse property of addition does apply. In addition, for these numbers zero is not the smallest number, but it is the middle number with an infinite number of positive and negative integers existing before and after it. Integers are used to measure values that can increase or decrease, such as the amount of money in a cash register. The

### KEY TERMS

**Associative law—** Axiom stating that grouping numbers during addition or multiplication does not change the final result.

**Axiom—** A basic statement of fact that is stipulated as true without being subject to proof.

**Closure property—** Axiom stating that the result of the addition or multiplication of two real numbers is a real number.

**Commutative law—** Axiom of addition and multiplication stating that the order in which numbers are added or multiplied does not change the final result.

**Hindu-Arabic number system—** A positional number system that uses 10 symbols to represent numbers and uses zero as a place holder. It is the number system that we use today.

**Inverse operation—** A mathematical operation that reverses the work of another operation. For example, subtraction is the inverse operation of addition.

standard rules for addition are followed when two positive or two negative numbers are added together and the sign stays the same. When a positive integer is added to a negative integer, the numbers are subtracted and the appropriate sign is applied. Using the axioms of multiplication it can be shown that when two negative integers are multiplied, the result is a positive number. Also, when a positive and negative are multiplied, a negative number is obtained.

Numbers to which both inverse properties apply are called rational numbers. Rational numbers are numbers that can be expressed as a ratio of two integers, for example, 1/2. In this example, the number 1 is called the numerator and the 2 is called the denominator. Though rational numbers represent more numbers than whole numbers or integers, they do not represent all numbers. Another type of number exists called an irrational number, which cannot be represented as the ratio of two integers. Examples of these types of numbers include square roots of numbers that are not perfect squares and cube roots of numbers that are not perfect cubes. Also, numbers such as the universal constants π and e are irrational numbers.

The principles of arithmetic create the foundations for all other branches of mathematics. They also represent the most practical application of mathematics in everyday life. From determining the change received from a purchase to calculating the amount of sugar in a batch of cookies, learning arithmetic skills is extremely important.

*See also* Algebra; Calculus; Function; Geometry; Trigonometry.

## Resources

### BOOKS

Immergut, Brita, and Jean Burr-Smith. *Arithmetic and Algebra Again*. New York: McGraw-Hill, 2005.

Perry Romanowski

# Arithmetic

# Arithmetic

Arithmetic is a branch of mathematics concerned with the addition, subtraction, multiplication, division, and extraction of roots of certain numbers known as real numbers. Real numbers are numbers with which you are familiar in everyday life: whole numbers, fractions, decimals, and roots, for example.

## Early development of arithmetic

Arithmetic grew out of the need that people have for counting objects. For example, Stone Age men or women probably needed to count the number of children they had. Later, one person might want to know the number of oxen to be given away in exchange for a wife or husband. For many centuries, however, counting probably never went beyond the 10 stage, the number of fingers on which one could note the number of objects.

At some time, people began to realize that numbers could stand for something other than real objects. They understood that four oxen, four stones, four stars, and four baskets all had something in common, a "fourness" that could be expressed by some symbol, such as 4. It appears that the ancient Sumerians of Mesopotamia (after 4000 b.c.) were the first to develop systematic ways of dealing with numbers in an abstract sense.

By far the most mathematically advanced of the ancient civilizations were the Egyptians, Babylonians, Indians, and Chinese. Each of these civilizations knew about and used whole numbers, fractions, and basic rules for dealing with such numbers. They used arithmetic to solve specific problems in areas such as trade and commerce, but they had not yet developed a theoretical system of arithmetic.

The establishment of such a theoretical arithmetic system occurred among the ancient Greeks in the third century b.c. The Greeks developed a set of theorems for dealing with numbers in the abstract sense, not just for the purpose of commerce.

## Numbering system

The numbering system we use today is called the Hindu-Arabic system. It was developed by the Hindu civilization of India about 1,500 years ago and then brought to Europe by the Arabs in the Middle Ages (400–1450). During the seventeenth century, the Hindu-Arabic system completely replaced the Roman numeral system that had been in use earlier.

The Hindu-Arabic system is also called a decimal system because it is based on the number 10. The ten symbols used in the decimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Other number systems are possible and, in fact, are also used today. Computers, for example, operate on a binary system that consists of only two numbers, 0 and 1. Our system of time uses the sexagesimal (pronounced sek-se-JES-em-el) system, consisting of the numbers 0 to 60.

A key feature of the decimal system is the concept of positional value. The value of a number depends not only on the specific digit (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) used, but also on the position of that digit in the number. For example, the number 532 is different from the number 325 or 253. The difference results from the fact that the 5, 3, and 2 appears in a different position in each case.

Another key feature of the decimal system is the digit zero (0). Ancient civilizations had no way of representing the concept of nothing. They apparently had little need to express the fact that they owned no oxen or had no children. Even the Roman numeral system has no way of representing the concept of zero (0). That concept is critical in the Hindu-Arabic number system, however, where 0 is treated in exactly the same way as any other number.

## Axioms in arithmetic

The things that one does with numbers in arithmetic are said to be operations. The two basic operations in arithmetic are addition and multiplication, and the rules used to carry out these operations are referred to as the axioms of arithmetic. Axioms are statements that we accept as being true without asking that they be proved.

## Words to Know

**Associative law:** An axiom that states that grouping numbers during addition or multiplication does not change the final result.

**Axiom:** A basic statement of fact that is stipulated as true without being subject to proof.

**Closure property:** An axiom that states that the result of the addition or multiplication of two real numbers is a real number.

**Commutative law:** An axiom of addition and multiplication that states that the order in which numbers are added or multiplied does not change the final result.

**Hindu-Arabic number system:** A positional number system that uses ten symbols to represent numbers and uses zero as a place holder. It is the number system that we use today.

**Inverse operation:** A mathematical operation that reverses the work of another operation; for example, subtraction is the inverse operation of addition.

You may wonder why subtraction, division, raising a number of an exponent, and other mathematical operations are not listed as basic operations of arithmetic. The reason is that all of these operations can be considered as the extensions or inverse (backward operations) of addition or multiplication. For example, subtracting 3 from 9 is the same operation as adding the negative value of 3 (−3) to 9. In other words: 9 − 3 is the same as 9 + (−3). Similarly, division is the inverse operation of multiplication.

Three axioms control all addition operations. The first of these is called the commutative law and can be expressed by the equation a + b = b + a. In other words, it doesn't make any difference in which sequence numbers are added. The result will be the same. That concept is probably common sense to you. It doesn't make any difference whether you have $3 and earn $6 more ($3 + $6) or have $6 and earn $3 more ($6 + $3). In either case, you end up with $9.

The second axiom of arithmetic is the associative law, which can be expressed as a + (b + c) = (a + b) + c. In other words, if you have more than two items to be added, it doesn't make any difference how you group them for adding. A delivery person might collect $2 from a newspaper customer at one building and $5 and $7 from two customers in a second building; that is, $2 + ($5 + $7), or $14. Or that same delivery person might collect $2 and $5 from two customers in the first building and $7 from one customer in the second building, or ($2 + $5) + $7, or $14. In either case, the total collected is the same.

Finally, the closure axiom says that if you add any two real numbers, a + b, the result you get is also a real number.

Three multiplication axioms similar to the addition axioms also exist. The commutative law says that a × b = b × a. The associative law says that a × (b × c) = (a × b) × c. And the closure law says that a × b = a real number.

Other laws and axioms can be derived from the three basic laws of addition and multiplication. Those derivations are not essential to this discussion of arithmetic and will not be included here.

## Kinds of numbers

The numbers used in arithmetic can be subdivided into various categories: whole numbers, integers, rational numbers, and irrational numbers. Whole numbers, also called natural numbers, include all of the positive integers plus zero. The numbers 3, 45, 189, and 498,992,353 are whole numbers. Integers are all whole positive and negative numbers. A list of integers would include 27, −14, 203, and −398,350.

Rational numbers are numbers that can be expressed as the ratio of two integers. Some examples include ½, ¾, 801/57, and 19/3,985. These numbers are also examples of fractions in which the first number (the number above the division sign) is the numerator, and the second number (the number below the division sign) is the denominator.

Finally, irrational numbers are numbers that cannot be expressed as the ratio of two integers. The ratio of the circumference of a circle (total length around) to its diameter is known by the name pi (π). The value of π can be calculated, but has no determinate (final) result. Depending on how long you calculate, the value of π can be expressed as 3.14 or 3.1416 or 3.14159265 or 3.141592653589793. The point is that no matter how long you look, you cannot find two integers that can be divided such that the answer will be the same as the value of π. Pi is, therefore, an irrational number.

The principles of mathematics provide the foundations for all other branches of mathematics. They also represent the most practical application of mathematics to everyday life. From determining the change received from a purchase to calculating the amount of sugar needed to make a batch of cookies, arithmetic skills are extremely important.

[*See also* **Algebra; Calculus; Function; Geometry; Trigonometry** ]

# Arithmetic

# Arithmetic

Arithmetic is a branch of **mathematics** concerned with the numerical manipulation of numbers using the operations of **addition** , **subtraction** , **multiplication** , **division** , and the extraction of roots. General arithmetic principles slowly developed over **time** from the principle of counting objects. Critical to the advancement of arithmetic was the development of a positional number system and a symbol to represent the quantity **zero** . All arithmetic knowledge is derived from the primary axioms of addition and multiplication. These axioms describe the rules which apply to all **real numbers** , including whole numbers, **integers** , rational, and irrational numbers.

## Early development of arithmetic

Arithmetic developed slowly over the course of human history, primarily evolving from the operation of counting. Prior to 4000 b.c., few civilizations were even able to count up to ten. Over time however, people learned to associate objects with numbers. They also learned to think about numbers as abstract ideas. They recognized that four trees and four cows had a common quantity called four. The best evidence suggests that the ancient Sumerians of Mesopotamia were the first civilization to develop a respectable method of dealing with numbers. By far the most mathematically advanced of these ancient civilizations were the Egyptians, Babylonians, Indians, and Chinese. Each of these civilizations possessed whole numbers, fractions, and basic rules of arithmetic. They used arithmetic to solve specific problems in areas such as trade and commerce. As impressive as the knowledge that these civilizations developed was, they still did not develop a theoretical system of arithmetic.

The first significant advances in the subject of arithmetic were made by the ancient Greeks during the third century b.c. Most importantly, they realized that a sequence of numbers could be extended infinitely. They also learned to develop theorems which could be generally applied to all numbers. At this time, arithmetic was transformed from a tool of commerce to a general theory of numbers.

## Numbering system

Our numbering system is of central importance in the subject of arithmetic. The system we use today called the Hindu-Arabic system, was developed by the Hindu civilization of India some 1,500 years ago. It was brought to **Europe** during the middle ages by the Arabs and fully replaced the Roman numeral system during the seventeenth century.

The Hindu-Arabic system is called a decimal system because it is based on the number 10. This means that it uses 10 distinct symbols to represent numbers. The fact that 10 is used is not important because it could have just as easily been based on another number of symbols like 14. An important feature of our system is that it is a positional system. This means that the number 532 is different from the number 325 or 253. Critical to the invention of a positional system is perhaps the most significant feature of our system: a symbol for zero. Note that zero is a number just as any other and we can perform arithmetic operations with it.

## Axioms of the operations of arithmetic

Arithmetic is the study of mathematics related to the manipulation of real numbers. The two fundamental properties of arithmetic are addition and multiplication. When two numbers are added together, the resulting number is called a sum. For example, 6 is the sum of 4 + 2. Similarly, when two numbers are multiplied, the resulting number is called the product. Both of these operations have a related inverse operation which reverses or "undoes" its action. The inverse operation of addition is subtraction. The result obtained by subtracting two numbers is known as the difference. Division is the inverse operation of multiplication and results in a quotient when two numbers are divided. The operations of arithmetic on real numbers are subject to a number of basic rules, called axioms. These include axioms of addition, multiplication, distributivity, and order. For simplicity, the letters a, b, and c, denote real numbers in all of the following axioms.

There are three axioms related to the operation of addition. The first, called the commutative law, is denoted by the equation a + b = b + a. This means that the order in which you add two numbers does not change the end result. For example, 2 + 4 and 4 + 2 both mean the same thing. The next is the associative law which is written a + (b + c) = (a + b) + c. This axiom suggests that grouping numbers also does not effect the sum. The third axiom of addition is the **closure property** which states that the equation a + b is a real number.

From the axioms of addition, two other properties can be derived. One is the additive **identity property** which says that for any real number a + 0 = a. The other is the additive inverse property which suggests that for every number a, there is a number −a such that −a + a = 0.

Like addition, the operation of multiplication has three axioms related to it. There is the commutative law of multiplication stated by the equation a × b = b × a. There is also an associative law of multiplication denoted by a × (b × c) = (a × b) × c. And finally, there is the closure property of multiplication which states that a × b is a real number. Another axiom related to both addition and multiplication is the axiom of distributivity represented by the equation (a + b) × c = (a × c) + (b × c).

The axioms of multiplication also suggest two more properties. These include the multiplicative identity property which says for any real number a, 1 × a = a, and the multiplicative inverse property that states for every real number there exists a unique number (1/a) such that (1/a) × a = 1.

The axioms related to the operations of addition and multiplication indicate that real numbers form an algebraic **field** . Four additional axioms assert that within the set of real numbers there is an order. One states that for any two real numbers, one and only one of the following relations is true: either a < b, a > b or a = b. Another suggests that if a < b, and b < c, then a < c. The monotonic property of addition states that if a < b, then a + c < b + c. Finally, the monotonic property of multiplication states that if a < b and c > 0, then a × c < b × c.

## Numbers and their properties

These axioms apply to all real numbers. It is important to note that real numbers is the general class of all numbers that includes whole numbers, integers, rational numbers, and irrational numbers. For each of these number types only certain axioms apply.

Whole numbers, also called **natural numbers** , include only numbers that are positive integers and zero. These numbers are typically the first ones to which a person is introduced, and they are used extensively for counting objects. Addition of whole numbers involves combining them to get a sum. Whole number multiplication is just a method of repeated addition. For example, 2 × 4 is the same as 2 + 2 + 2 + 2. Since whole numbers do not involve **negative** numbers or fractions, the two inverse properties do not apply. The smallest whole number is zero but there is no limit to the size of the largest.

Integers are whole numbers that include negative numbers. For these numbers the inverse property of addition does apply. For these numbers, zero is not the smallest number but it is the middle number with an infinite number of positive and negative integers existing before and after it. Integers are used to measure values which can increase or decrease such as the amount of money in a cash register. The standard rules for addition are followed when two positive or two negative numbers are added together and the sign stays the same. When a positive integer is added to a negative integer, the numbers are subtracted and the appropriate sign is applied. Using the axioms of multiplication it can be shown that when two negative integers are multiplied, the result is a **positive number** . Also, when a positive and negative are multiplied, a negative number is obtained.

Numbers to which both inverse properties apply are called rational numbers. Rational numbers are numbers that can be expressed as a **ratio** of two integers, for example, 1⁄2. In this example, the number 1 is called the numerator and the 2 is called the denominator. Though rational numbers represent more numbers than whole numbers or integers, they do not represent all numbers. Another type of number exists called an **irrational number** which cannot be represented as the ratio of two integers. Examples of these types of numbers include square roots of numbers which are not perfect squares and cube roots of numbers which are not perfect cubes. Also, numbers such as the universal constants π and e are irrational numbers.

The principles of arithmetic create the foundations for all other branches of mathematics. They also represent the most practical application of mathematics in everyday life. From determining the change received from a purchase to calculating the amount of sugar in a batch of cookies, **learning** arithmetic skills is extremely important.

See also Algebra; Calculus; Function; Geometry; Trigonometry.

## Resources

### books

Paulos, John Allen. *Beyond Numeracy.* New York: Alfred A. Knopf, Inc., 1991.

Perry Romanowski

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Associative law**—Axiom stating that grouping numbers during addition or multiplication does not change the final result.

**Axiom**—A basic statement of fact that is stipulated as true without being subject to proof.

**Closure property**—Axiom stating that the result of the addition or multiplication of two real numbers is a real number.

**Commutative law**—Axiom of addition and multiplication stating that the order in which numbers are added or multiplied does not change the final result.

**Hindu-Arabic number system**—A positional number system that uses 10 symbols to represent numbers and uses zero as a place holder. It is the number system that we use today.

**Inverse operation**—A mathematical operation that reverses the work of another operation. For example, subtraction is the inverse operation of addition.

# arithmetic

a·rith·me·tic •
n. / əˈri[unvoicedth]məˌtik/ the branch of mathematics dealing with the properties and manipulation of numbers: *the laws of arithmetic.* ∎ the use of numbers in counting and calculation: *he could do arithmetic in his head.* ∎ fig. those aspects of a particular situation that can be expressed in numerical terms: *some unsettling parliamentary arithmetic.*•
adj. (ar·ith·met·ic) / ˌari[unvoicedth]ˈmetik/ (also ar·ith·met·i·cal) of or relating to arithmetic.DERIVATIVES: a·rith·me·ti·cian / əˌri[unvoicedth]məˈtishən/ n.

# arithmetic

**arithmetic** the branch of mathematics dealing with the properties and manipulation of numbers. The term comes (in Middle English, via Old French and Latin) from Greek *arithmētikē (tekhnē)* ‘(art) of counting’, from *arithmos* ‘number’. Early forms such as *arsmetrike* were influenced by Latin *ars metrica* ‘measuring art’.

In the Middle Ages, arithmetic was counted as one of the seven liberal arts, and was one of the subjects of the quadrivium.

# arithmetic

**arithmetic** Calculations and reckoning using numbers and operations such as addition, subtraction, multiplication and division. The study of arithmetic traditionally involved learning procedures for operations such as long division and extraction of square roots. The procedures of arithmetic were put on a formal axiomatic basis by Guiseppe Peano in the late 19th century. Using certain postulates, including that there is a unique natural number, 1, it is possible to give formal definition of the set of natural numbers and the arithmetical operations. Thus, addition is interpretable in terms of combining sets: in 2 + 7 = 9, 9 is the cardinal number of a set produced by combining sets of 2 and 7.

# arithmetic

**arithmetic** XIII. Earliest forms *arsmetrik(e)*, *-metike* — OF. *arismetique* — Rom. **arismetica*, for L. *arithmētica* — Gr. *arithmētikḗ* (sc. *tékhnē* art) ‘art of counting’, f. *arithmeîn*, f. *arithmós* number; assoc. with L. *ars metrica* ‘measuring art’ led to forms of the type *arsmetrik(e)* which were later (XV) conformed to the orig. L. and Gr.

So **arithmetic** XVI. — L. and Gr. **arithmetical** XVI. **arithmetician** XVI. — F.

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