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 1500 years ago. It was brought to Europe during the middle ages by the Arabs and fully replaced the Roman numeral system during the 17th 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, note that the letters a, b, 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 x b = b x a. There is also an associative law of multiplication denoted by a x (b x c) = (a x b) x c. And finally, there is the closure property of multiplication which states that a x b is a real number. Another axiom related to both addition and multiplication is the axiom of distributivity represented by the equation (a + b) x c = a x c + b x c.
The axioms of multiplication also suggest two more properties. These include the multiplicative identity property which says for any real number a, 1 x a = a, and the multiplicative inverse property that states for every real number there exits a unique number (1/a) such that (1/a) x 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, the a + c < b + c. Finally, the monotonic property of multiplication states that if a < b and c > 0, then a x c < b x c.
Numbers and their properties
These axioms apply to all real numbers. It is important to note that a real numbers is the general class of all numbers which include 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 x 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 which also 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 which 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 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.
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