INTEGERS ARTHIMETIC OPERATIONS AND ABSOLUTE VALUES

Addition of integers - 
    As we already know addition of any two numbers a and b means moving b units to right of a on the line. The same is followed even in the addition of Integers.

Case-I                                     
Additing of two positive integers 
Ex - (+2) + (+3) = +5
  
We observe from the number line that (2) + (3) means 2 units from ‘0’ towards its right and 3 units from 2 towards its right gives 5 units from ‘0’.
Case-II  Addition of ‘2’ negative Integers
Example:-  (-2) + (-3) = -5

We observe from the number line that  (-2) + (-3) means 2 units to the left of ‘0’ and 3 units to the left (-2) gives ‘5’ units to the left of zero. i.e. (-2) + (-3) = 5
Case-III - Addition of a negative and a positive Integer
Ex-1 :     (-2) + (-3) = +1

We observe from the number line that (-2) +3 means 2 units to the left of zero and 3 units right of (-2) gives 1 unit to right of zero i.e. (-2)+3 =1
Properties of addition in Integers
1.Closure Property
Observe Property :-
    1. 25+75 = 100            2. 100+ (-200) = -100
    3. (-200) +450= 250        4. (-130) + (-150) = -280
We observe that the sum of Integers is always an integer.  In such a case  we say that Integers are closed under Addition.
Conclusion - for.  This is called closure property.
2. Commulative Property
Observe the following
    1. I) 25+75 =100            II) 75 + 25 =100
    2. I) (-100) + (-200) = -300    II) (-200)+(-100) = -300
We observe that the order of adding the Integers does not change the sum. In this case we say commutative propery holds good under Addition .
Conclusion:- 
For any \(a,b\in z\), we can see that a+b = b+a.  This is called commutative property under Addition.
3. Associative Property -
Observe the following
    1. I. 25+(75+100) = 25+175=200        II.(25+25)+100=100+100=200
    2. I. 25+(-75)+100) = 25+25=50        II. (25+(-75)+100=-50+100=5
For any three Integers
    1st Integers +(2nd Integers + 3rd Integers)= (1st Interger+2nd Integer) + 3rd Integer
 Conclusion for any three Integers 
    \(a,b,c\in z\)
    (a+b)+c = a+(b+c)
This property is called associative property for Integers over Addition.
4. Additive Identity:-
Observe the following 
    1. I) 2+0=2         II) 0+2=2
    2. I) -11+0=-11     II) 0+(-11) = -11
We can see that addition of zero and order of adding zero does not change the result.
 Conclusion - for any,\(a\in z\) ,a+0=0+a=a,
Here ‘0’ is called Additive  Identity. This property is called property of Indentity for Integers over Addition.
5.Additive Inverse :-
For any Integer 
We can say that a and -a are Additive
 Inverse of each other
Ex :- (2)+(-2)=0 and (-2)+(2)=0
     ‘-2’ is the additive Inverse of 2 and 2 is the Additive Inverse of (-2)
Note
 1. The sum of any two positive Integers is always positive
    i.e.\(5,6\in {{z}^{+}}\Rightarrow 5+6=11\in {{z}^{+}}\)
2. the sum of two negative Integers is always negative 
    (-2)+(-3)= -2-3= -5
3. The sum of an Integer and its Additive Inverse is always is zero
    Ex:-11+(-11)=+11-11 = 0
 4. the sum of any two Intergers of Opposite signs could be positive, negative or zero
    ex :- I.-5+7-2    II.  +5(-8)= -3    III. (-11)+(-4) = – 11 - 4 = -15
5. Successor of an Integer is the more than itself 
    ex :- 5,6(5+1)
6. Substraction of Integers
    Substraction of any two integers a and b is a+( – b) = a= b .
The substraction of Integers follows the rules of signs:-
    i)a – (+b) = a – b        ii) a – (-b) = a+b
    iii) a+(-b)= a – b        iv) a + (+b)= a+b
Case -I substraction of any two prositive Integers takes the sign of greatest Integers
E.X. :-7 – 2=5 here 5 
gets the sign of 7 and the number line it is represented as follows:
i.e 7 – 2 means 2 units to left of 7
Case -II Substraction of two  negative Integers is represented on the number line as follows 
example :-  – 7 – ( -2) = -7 +2 = – 5
    
     i.e.  – 7 – ( – 2)  follows the rule of sign and becomes -7+2 which is equal to 2 units to the right of (-7) on the number line.
Case-III :- substraction of a positive Integers from a negative Integer is represented as follows :-
eg :-   – 7 – ( + 2) = – 7 – 2 = – 9

i,e. 7 – 2 means 2 units to the left of  – 7 and gives  – 9
Properties of substraction of Integers
Observe the following
    1. 25 – 75=  – 50
    2. (– 125) – ( – 225) = – 125 +225 = 100
We observe that the difference of two Integers is always an Integer. So, we can say teh Integers are closed under substaraction.
Conclusion :- for any ‘2’ Integers (a-b) is also an Integer.
This property is called clousere property for Integer over substarction
2. Commentative Property :- 
    Observe the following 
    1. i) 75-25=50            ii) 25-75=-50
    2. i) 200-100=100            ii) 100-200= -100
    We deserve that the order of substraction changes the difference. Therefore we can  communative  property does not hold good over substaction.
    conclusion of a and b are any two Integers, then a-b = b-a 
3.Associate Property
    Observe the following :
    1. i)6 – (3 – 2) = 5 – 1 =4         ii)  (5 - 1) – 2 = 2 – 2 =0
    2. i)  16 – (8 – 4) = 16 – 4 =12       ii) (16 – 8 ) – 4 = 8 – 4 = 4 
    1st Integer = (2nd Integer-3rd Integer)(1st Integer-2nd Integer)-3rd Integer
    In this case associative law does not hold good over substraction \(\forall a,b\in z\)

    we can say \(a(b-c) \ne (a-b)-c\)
    if a.b and c are Integers and  a-c>b-c.
Multiplication of Integers :- Let a and b are any two integers then 
  \(\begin{align} & i)(+a)\times (+b)=+ab,ex:(+7)\times (+6)=+42 \\ & ii)(+a)\times (-b)=-ab,ex:(+8)\times (-2)=-16 \\ & iii)(-a)\times (-b)=+ab,ex:(-11)\times (-3)=33 \\ \end{align} \)
Note : \(\begin{align} & 1)+\times +=+ \\ & 2)-\times +=- \\ & 3)-\times -=+ \\ & 4)+\times -=- \\ \end{align} \)    
Properties of Multiplication of Integers
1. Closure property :- Observe the following 
    1) \(25\times 75=1875\)        2) \(10 \times (-20)=-200\)        3) \((-25)+(-25)=+625\)
We can see that product of integers is always an integer.  So, the integers are closed under mulatiplication
Conclusion:- If a and b are any two integers then ‘ab’ is also an integer.
Multiplication is closed over integer set.
This is called closure property
Commutative property:-   Observe the following
   \(\begin{align} & \text{1})\text{ i})25\times 75=1875\text{ }~~\text{ }~\text{ii})~75\times 25=1875~~\text{ }~ \\ & \text{2})\text{ i})-10\times -20=200\text{ }~~\text{ }~\text{ii})~-20\times -10=200 \\ \end{align} \)
For any two integers a and b we can see that \(a\times b=b\times a\)
This property is called commutative property for integer over multiplication for integer.
Distributive property :-
Observe the following
\(\begin{align} & 1)2\times \left( 3+4 \right)=\left( 2\times 3 \right)+(2\times 4) \\ & 2\times 7=6+8 \\ & 14=14 \\ & 2)i)2\times [(-3)+4]=2\times (+1)=+2 \\ & ii)[2\times (-1)]+(2\times 4)=-6+8=+2 \\ & \therefore 2\times [(-3)+4]=[2\times (-3)]+(2\times 4) \\ \end{align} \)
1st  Integer (2\(^{nd}\) integer +3\(^{rd}\) integer) = (1st integer \(\times\)  2\(^{nd}\) integer)+ (1st integer \(\times\) 3\(^{rd}\) integer)
     We can say that distributive property hold good multiplication over addition
Division of Integers:- let us see the rule given below which helps us in performing the division operation over integers.
Rule I :- The quotient of two integers whth the same sign is positive integer obtained by dividing the numerical value of the divided with the numerical.
5.Associative property :- 
\(\text{1})\text{ i})2\times \left( 3\times 4 \right)\text{=2}\times \text{12=24 }~~\text{ }~~~\text{ ii})~\left( 2\times 3)\times 4 \right)=6\times 4=24\)
Conclusion :- For any three integers a, b, c we can see that 
  \(\left( a\times b)\times c \right)=a\times \left( b\times c \right)\)  
    This is called associative property for multiplication over integer
4.    Multiplicative Identity :-
    Observe the following
    \(\text{1})\text{ i})1\times 2=2~~\text{ }~\text{ii})2\times 1=2\text{ }~~\text{ 2})\text{ i})10\times 1=10\text{ }~\text{ }~~\text{ }~\text{ii})~1\times 10=10\)
    We observe that multiplication by ‘1’ and order of multiplying by ‘1’ doesn’t change the result
    Conclusion :- For any integer a, ax1=1xa=a.  Here ‘1’ is called multiplicative identity .  This property is  called property of identity .
Value of Divisor
Illustration:- \(\text{i})\left( +12 \right)\div \text{(+3)=+}\frac{12}{3}\text{=+4 }~~\text{ }~~~\text{ ii})~\left( -25 \right)\div \left( -5 \right)=+\frac{25}{5}=+5\)
Rule 2:- The quotient of ’2’ integers with different signs is the negative integer obtained by dividing the numerical value of the dividend with the numerical value of the divisor 
Illustration:-\(\text{i})\left( +36 \right)\div \text{(-6)=-}\frac{36}{6}\text{=- 6 }~~\text{ }~~~\text{ ii})~\left( -32 \right)\div \left( -4 \right)=\frac{-32}{4}=-8\)
    Now we shall study the properties of operations over integers.
Properties of Division of Integers
1) Closure property:-
    Observe the following 
   \(\text{i})75\div 25=3~~\text{ }~~~\text{ ii})~\left( 10 \right)\div \left( -20 \right)=\frac{10}{-20}=\frac{-1}{2}\)  is not an integer 
We observe that when an integer is divided by another integer then the result is not always an integer.  We can say that the integers are not closed under division.
The closure property doesn’t hold good for integers over division
Associative property:- 
Observe the following
    \(\begin{align} & \text{1})\text{ i})\left( 8\div 4 \right)\div \text{2=2}\div \text{2=1 }~~\text{ }~\text{ii})~8\div \left( 4\div 2 \right)\text{=8}\div \text{2=4}~\text{ }~ \\ & \text{2})\text{ i})\left( 36\div 6 \right)\div 3\text{=6}\div 3\text{=2 }~~\text{ }~\text{ii})~36\div \left( 6\div 3 \right)\text{=36}\div 3\text{=18} \\ \end{align} \) 
    I\(^{st}\) Integer :- (2\(^{nd}\) integer \(\div\) 3\(^{rd}\) integer) \(\ne\) (1\(^{st}\) integer \(\div\) 2nd integer) \(\div\) 3rd integer
Conclusion :-
     Associative law doesnot hold good under division
     Associative law doesnot hold good for division of integers
\(\therefore \forall a,b,c\in z,(a\div b)\div c\ne a\div \left( b\div c \right)\)