MATHEMATICS- NUMBER THEORY 1

                                                    NUMBER THEORY
Number theory is that branch of mathematics devoted primarily to the study of numbers especially integers.

DEFINITION of a number: A number is an idea. It is something you cannot see but can only think about.You only think about a number when you affiliate it to something. You see. For example 2 pens, 3 pencils, 5 bags etc. Five therefore is an idea of collecting objects together.

 In our system of counting called Arabic or Decimal system of counting, we represent number base on the idea of groupings in tens. We use digits to represent numbers as below:

  Zero, one, two, three, four, five, six, seven, eight, nine.

    0  ,   1  ,  2   ,   3   ,   4  ,  5  ,   6 ,    7   ,     8  ,   9 .

All other numbers are obtained from the above digits by combining two or more of them and some times by comparing two of them. In each case the value (place value) of a digit in the number depends on its position in the number.

  CLASSIFICATION OF NUMBERS

1)     COUNTING NUMBERS. A counting number is a positive whole number. Example 1, 2, 3,4,The set of counting numbers is written as IN. The set of counting numbers is written as IN^*, Hence IN^*={ 1,2,3,4…} The three dots in the set indicate that the set is so large and all its members cannot be listed.

2)     NATURAL NUMBERS. A natural number is a counting number but also includes the number zero (0). It is a positive whole number starting with zero. The set of natural numbers is denoted by IN and written as IN= ﻉ0, 1, 2, 4…}

3)     INTEGER NUMBERS. An integer number is either a positive whole or negative whole number including zero. For example 0,3,5,6,... etc. The set of integers is denoted by Z and written as Z=} -2, -1, 0, 1, 2…}. So, Z the set of integers can be split into three parts namely

i-the set of positive integers denoted Z⁺ and written as Z⁺=ﻉ1,2,3,…}

ii-the set of negative integers denoted Z^- and written as Z^-=ﻉ…-3,-2,-1}

iii-the neutral set  or empty set .

Hence Z= Z⁺+Z^-+ neutral or empty set.

4)     RATIONAL NUMBERS. A number is said to be rational if it can be expressed as a fraction. That is , it can be expressed in the for ﻉp/q, q≠0 , p,q ﻉZ.}.A rational number is denoted by Q. From the here 4 is a rational number because it be written as 4/1.That means all integers are rational numbers. Hence, Zﮯ Q where the symbol, ﮯ represents subset of.

5)     IRRATIONAL NUMBERS. Any number which is not rational is irrational. An irrational number cannot be expressed as a fraction. For example π, √2, ³√3 etc. The set of irrational numbers is denoted by Q^c.

6)     REAL NUMBERS. A real number is any number which is not Complex. It is counting number, natural number, integer number, rational number or irrational number. The set of rational numbers is denoted by IR. So IR={…-3,-2,-1,-1/2,0,1/2,1,2,3,…}. Observe that here all the sets above are subsets of the set of IR.

7)     DIRECTED NUMBERS. There are two signs on any number which indicate the direction of the number on the number line. These signs are positive (+) and negative (-). Zero is neutral; it is neither positive nor negative. Any number without a sign is considered positive unless it is zero. On the number line, negative numbers are on the left of zero while positive numbers are on the right . 

  -4 -3 -2-1 0 1 2 3 4 

As you move from left to right of the number line ,numbers increase . That is to say -3>-4, -2>-3, 0>-1, 1>0, 2>1, 3>2, 4>3 and so on. As you move from right to left the numbers decrease that is 3<4, 2<3,1<2,0<1, -1<0, -2<-1,-3<-2,-4<-3 etc. The arrow heads pointing to the left and right shows that there are many more numbers in these directions.

8)      ABSOLUTE VALUES OF NUMBERS. The absolute value of a positive number remains positive while that of a negative number becomes positive. In general, the absolute value of a number,x, denoted by ↾x↾. It means ↾x↿ is the positive value ox. E.g. I5I=5, I-5I=5 
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