CHAPTER 1.2 - THE NATURAL OPERATIONS WITH THE NATURAL NUMBERS

Abstract

This chapter describes natural operations with the natural numbers such as addition, multiplication, exponentiation, and binary operations. Observant cave men noticed that whenever a number was added to another number, the procedure was invariably the same and the resulting number of elements in the union was likewise invariably same. The symbol “+” represents the operation of addition and is spoken as plus. An especially simple problem of a sort sometimes occurs in which all the numbers to be added are the same. For example, if a, b, c, and d each dug five clams, then the total number of clams dug by all of them would be obtained as 5 + 5 + 5 + 5, which could be evaluated as 5 + 5 + 5 + 5 = (5 + 5) + 5 + 5 = 10 + 5 + 5 = (10 + 5) + 5 = 15 + 5= 20. This sum of four fives is an example of what is called a multiple of five. The process of identifying the value of a multiple that is an extended application of addition has come to be called multiplication. It sometimes happens and rather frequently, that there are extended products in which all the factors are the same number. Such a special extended product is called a power. It is represented by the shorthand notation 54 in which the numeral 5 indicates the number that occurs repeatedly as a factor and the smaller numeral 4 indicates the number of times which the number five occurs as a factor. The number five is called the base of the power, and the number four is called the exponent of the power. The process of forming a power is called exponentiation. Any process that is used to combine two numbers to obtain a third number is called a binary operation. As each of the three operations of addition, multiplication, and exponentiation is of this type, these three processes are binary operations.