Introduction: The review of the development and the structure of the present number system will serve as the background to understand the underlying basic principle on how arithmetic and algebra solutions proceed. Man's growth is infinite. It must be qualified to measure its growth, hence the development of our new number system is a consequence. From its crude beginning, consisted of natural numbers ( counting numbers ) were used primary for counting, was denoted by N= {1,2,3,4,....), performing the operations: addition, subtraction, Multiplication and division. Further to explore was done, and found out that the need for a convenient representation was not yet perfect. The first problem was, when two identical numbers were subtracted, no solution in the set of natural number can be found. and so, this was enlarged to include zero, the set of the whole numbers denoted by W = ( 0,1,2,3,4,...). The next problem that arose was, when a subtrahend is greater than a minuend, that give rise to the concepts of signed numbers, called the set of integers, denoted by I = {.....,-3,-2,-1,0,1,2,3,....). Making the operation of division in all situation is possible, enlarged the number system to include fractions. this enlarged number system thus developed is the REAL NUMBER SYSTEM to distinguish it from the imaginary numbers of the complex number.DEFINITION OF TERMS: Natural numbers----- consist primarily of counting numbers. N= {1,2,3,4,...) Whole numbers------ consist of the counting numbers and zero. W= {0,1,2,3,4,.
..) Integers ---------------- consist of negative, zero, positive numbers. I= {,....,-3,-2,-1,0,1,2,3,..} Rational number----- a number which can be expressed as a quotient of two integers; and whose denominator is a non-zero integer, such that M/N where N is not equal to zero.----- Repeating decimal representation example: 1/3 = 0.333333.... 5/3 = 1.666666... 1/9 = 0.111111.... 4/9 = 0.44444...... 9/11 = 0.818181...----- Terminating decimal representation example: 1/4 = 0.25 7/8 = 0.875 3/5 = 0.6 5/2 = 2.50 1/2 = 0.50---- Periodic decimal representation example: 3/11 = 0.272727... 3/7 = 0.714285714285..... 12/11 = 1.090909....... 2/13 = 0.153846153846....IRRATIONAL NUMBER---- a number whose equivalent decimal representation is non-repeating, non-terminating, non-periodic example: pi = 3.1415826358..... 18/13 = 1.384615385.....