INTEGERS

Integers and Absolute Values

    Till now we have learnt two types of number system. Those are natural numbers and whole numbers. Here we try to study whole numbers in depth
Observation :
                                                            
    both Samantha and Anushka walk in opposite direction from a  point count their steps with +ve sign when walking to wards right and with -ve sign when walking towards left.

    Anushka has walked 6 steps to wards right. This is represented as (+6) samantha has walked 6 steps towards left this is denoted as(-6)
    Observe the answers to the following questions.
    i) 11 steps from ‘0’ towards left (-11)
    ii) 5 steps from’0’ towards right  (+5)
    iii) x steps from ‘0’ towards left  (-x) (x > 0)
    iv) y steps from ‘0’ towards right   (+y) (y>0)
    Ismail has borrowed Rs 4 from his friend Divid, this is denoted by (-4). This type of numbers are called negative numbers those are -1, -2, -3, -4, -5, -6....etc.
    Let A person has got a lottery Rs 50,000 this is repersented by +50,000 as some money is added to his account. This type of numbers are called positive numbers these are represented by +1, +2, +3...... ect. 
    Let A person Y has no, balance in his account. i.e repersented by ‘0’ it is called zero.
    Earning is denoted by ‘+’ symbol 
    expenditure is denoted ‘-’ symbol.
    
   The height of the above sea level is  denoted by positive numbers.
    Level the depth below the sea is represented by negative numbers.

Note: 
    1) The collection of positive numbers is represented by N and those are called natural numbers i.e N = {1,2,3....}
    2) If ‘0’ is joined to the set of natural numbers, then the new set formed is called, whole numbers set and represented by ‘W’ 
     W = {0,1,2,3........}
Integers :The set of whole numbers together with negatives of natural numbers is called integers this set is repersented by I or Z

Number line : 
    The integers are represented by points at equal intervals on a line is called number line.

Representation of Integers on number line:

1.    Draw a straight line
    
2.    make a point on the middle of it
    
3.    make equal distances on the right as well as left
    
4.    On the right hand side of the middle point ‘O’ label the points of division as +1, +2, +3, +4, +5,+6,+7 etc. Which on the left hand side as -1, -2, -3, -4 -5 etc as show below

5.    Hence the number line is
   
Note : 1) Let us observe the integers on the number line. Numbers on the number line have a important property i.e integers occuring on the right are greater than the integers occuring on the left
Some examples 
    1) +5 >+3,since +5 is to the right of +3 on the number line.
    2) +1 > 0, since +1 is to the right of  zero on the number line.
    3) 0 > -1  since ‘0’ is the right of (-1) on the number line.
    4) -1 > -2 sicne -1 is to the right of -2 on the number line.
    5) All positive integers are greater than zero
    i.e 0 < 1 < 2 < 3 < 4 < .....etc.
    6) All negative numbers are lessthan 0
    0 > -1 > -2 > -3 > ........etc


Absolute value of an integer :
    The absolute value of an integer is the numerical value of the integer  irrespective of its sign the symbol ‘| |’ is used to represent the absolute value of an integer. Thus |+8| = +8 , |-8| = 8 
     This shows that the absolute value of an integer is either equal to or greater than the integer but never less than the integer
    Let ‘a’ be a real number
    |a| = +a, if a is +ve
    |a| = 0 if a= 0
    |a| = -a, if ‘a’ is negative      Ex : |5| = +5     |-5| = +5