Arithmetic

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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 ]



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