PROPERTIES OF SUBTRACTION OF RATIONAL NUMBERS -
Subtraction of rational numbers can be understood through various properties and principles derived from the properties of addition and the concept of additive inverses. Here are some key properties of subtraction of rational numbers:
1. Definition of Subtraction:-
Subtraction can be defined in terms of addition and the additive inverse. For any rational numbers a and b:−
a − b = a + (−b)
This definition helps to utilize the properties of addition when dealing with subtraction.
2. Closure Property:-
The set of rational numbers is closed under subtraction. This means that for any rational numbers a and b, a − b is also a rational number.
3. Non-Commutativity:-
Subtraction is not commutative. For rational numbers a and b:
a − b ≠ b − a (in general)
For example, 3 − 2 ≠ 2 − 3
4. Non-Associativity:-
Subtraction is not associative. For rational numbers a, b, and c:-
(a − b) − c ≠ a − (b − c)
For example, (5 − 2) − 1 ≠ 5 − (2 − 1).
5. Subtracting Zero:-
Subtracting zero from any rational number does not change its value. For any rational number a:−
a − 0 = a
6. Subtracting a Number from Itself:-
Subtracting any rational number from itself results in zero. For any rational number a:-
a − a = 0
7. Negative of a Difference:-
The negative of a difference is the same as the difference of the negatives. For any rational numbers a and b:−
− (a − b) = (−a) − (−b)
8. Distributive Property:-
Subtraction distributes over addition and subtraction in the following ways. For rational numbers a, b, and c:−
a − (b + c) = (a − b) − c
a − (b − c) = (a − b) + c
By understanding these properties, one can handle the subtraction of rational numbers more effectively and apply these principles to solve related mathematical problems.