Integers are numbers both positive , negative and zero. Lets see them on number line below.
We need to understand that negative numbers before zero and including zero are less than positive numbers i.e -5 ,-4,-3,-2,-1,0 are less than 1,2,3,4,5 ..etc.
Adding and Multiplication of Integers
Lets try the following addition of positive integers.
1 + 2 =3
3 + 4 = 7
From the above the answers 3 and 7 are greater number than the two added positive integers.
Similarly lets try adding negative integers
-1 + -2
The above addition is written as -1 (+)(-) 2 ::only for understanding the signs are kept under brackets
Remember the following considering “a” and “b” as two integers.
- -a (+) -a = – 2a
- -a (x) -a = (minus X minus = plus)
- (-big number) + (+small number ) = -number ( i.e -5 +3 = -2)
- (+big number) + (-small number ) = + number (5+ (-3) =2)
- (+big number) – (-small number) = +big number (5 -(-3) = 8)
What is Closure property under addition?
When two integers are added, the answer is also an integer. This is called closure property under addition.
What is closure property under subtraction?
Similar to closure property under addition, when two integers are subtracted, answer is also an integer. This is called as closure property under subtraction.
From the above we can conclude that two integers are closed under addition and subtraction.
Examples for above theory
1 +2 = 3
5+3 = 8
-4 + – 4 = – 8 (because plus X minus = minus, so equation becomes -4 -4 = – 8 )
1-2 = -1 (because (-big number) + (+small number) = – number
Note that if no sign is mentioned before the number, then its assumed as positive i.e + .
What is commutative property for addition
When two integers are added ,irrespective of position the answer is the same, such addition is called commutative property for addition of integers
1+2 = 2+1 = 3
-1+2 =2-1 =1
-8+5 =5-8 =-3
5-9 = -9+5 = -4
What is commutative property for subtraction
1-2 ≠ -2+1
5-9 ≠ -9-5
For all above equations commutative property doesn’t hold good for subtraction.
As we have studied previously associative property holds good for addition of integers and subtraction doesn’t hold good similar to commutative property.
The additive identity and multiplicative identity are 0 and 1 respectively. To recall additive identity is number to which if any number is added we get the same number. Multiplicative identity is a number to which any integer is multiplied we get same number.
Division of integers
Suppose we have 2 integers “a” and “b”. The fractions are shown as below
Remember that denominator shall not be 0.
Division of integers is also not commutative. i.e a/b
On number line – 5 is less than -4 i.e -5 is smaller than -4 &
5 is greater than 0 , the more we proceed towards negative side of number line the numbers becomes smaller and smaller.