Integers And Absolute Values
Properties of addition in Integers
1. Closure Property
Observe the following
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 \(
a,b \in z,\,\,a + b \in z
\). This is called closure property.
2. Commutative 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=50
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 ‘o’ is called Additive Identity. This property is called property of Indentity for Integers over Addition.
5. Additive Inverse :-
For any Integer \(
a,a + ( - a) + ( - a) + a = 0
\)
We can say that a and -a are Additive
Inverse of each other
Ex :- \(% MathType!MTEF!2!1!+-
% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
%
(2) + ( - 2) = 0\,and\,( - 2) + 2 = 0
\)
\(
\therefore
\)‘-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.\(
5,6 \in \rlap{--} z^ + \Rightarrow 5 + 6 = 11 \in \rlap{--} z^ +
\)
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 :- \(
5,6\,(5 + 1)
\)