COMMUTATIVE PROPERTY OF ADDITION OF RATIONAL NUMBERS -
Two rational numbers can be added in any order.
Thus for any two rational numbers a/b and c/d,
we have (a/b + c/d) = (c/d + a/b)
The commutative property of addition states that changing the order of the addends does not change the sum. In the context of rational numbers, this means that when adding rational numbers, the order in which you add them does not affect the result.
Mathematically, for any rational numbers a and b, the commutative property of addition can be expressed as:
a + b = b + a
This property holds true for all rational numbers, whether they are fractions, decimals, or integers.
1 2
Example.1) Let's consider the rational numbers -------- and --------
3 5
1 2 (1 X 5) + (2 X 3) (5 + 6) 11
-------- + -------- = ------------------------------ = ---------------- = -----------
3 5 (3 X 5) 15 15
And if we reverse the order of addition: -
2 1 (2 X 3) + (1 X 5) (6 + 5) 11
--------- + --------- = ------------------------- = -------------- = ----------
5 3 (5 X 3) 15 15
1 2 2 1 11
So, ---------- + ----------- = ---------- + ---------- = -----------
3 5 5 3 15
2 5
Example.2) Let's consider the rational numbers ----------- and -----------
3 7
2 5 (2 X 7) + (5 X 3) (14 + 15) 29
---------- + ---------- = --------------------------- = ----------------- = ------------
3 7 (3 X 7) 21 21
And if we reverse the order of addition: -
5 2 (3 X 5) + (2 X 7) (15 + 14) 29
---------- + ----------- = ------------------------------- = ------------------- = ------------
7 3 (7 X 3) 21 21
2 5 5 2 29
So, ----------- + ------------ = ----------- + ------------- = --------------
3 7 7 3 21
3 - 5
Example.3) Let's consider the rational numbers ------------ and ------------
7 11
3 - 5 (3 X 11) + {(-5) X 7} {33 + (-35)}
----------- + ----------- = -------------------------------- = ---------------------
7 11 (7 X 11) 77
(33 - 35) - 2
= ------------------ = -------------
77 77
And if we reverse the order of addition: -
- 5 3 {(- 5) X 7) + (3 X 11) {(-35) + 33)}
----------- + ------------ = -------------------------------- = ----------------------
11 7 (11 X 7) 77
(- 35 + 33) - 2
= -------------------- = -------------
77 77
3 - 5 - 5 3 - 2
So, ----------- + ----------- = ----------- + ----------- = -----------
7 11 11 7 77
As you can see, regardless of the order in which we add the rational numbers, the sum remains the same. This illustrates the commutative property of addition for rational numbers.
This property is fundamental in mathematics and has wide-ranging applications, ensuring that the result of addition remains consistent and predictable across different mathematical operations and contexts.