**Rational Numbers and Commutative Property**

The commutative property is a fundamental property of addition and multiplication. It states that changing the order of the numbers being added or multiplied does not change the result. However, subtraction does not follow this property. In other words, the order of subtraction does matter, and therefore, rational numbers are not commutative under subtraction.

**Understanding Rational Numbers**

Rational numbers are numbers that can be expressed as fractions, where the numerator and denominator are both integers. These numbers can be positive, negative, or zero and can be represented on a number line. Rational numbers include integers, fractions, and terminating or repeating decimals.

**Commutative Property**

The commutative property states that for any two numbers a and b, the operation being performed (addition or multiplication) will yield the same result, regardless of the order in which the numbers are arranged. Mathematically, it can be expressed as follows:

- Addition: a + b = b + a
- Multiplication: a × b = b × a

**Non-Commutative Property of Subtraction**

Unlike addition and multiplication, subtraction does not follow the commutative property. The order of subtraction matters, and changing the positions of the numbers being subtracted will affect the result. Mathematically, it can be expressed as follows:

- Subtraction: a - b ≠ b - a

**Illustrating Non-Commutativity with an Example**

Let's consider the rational numbers 2/3 and 1/2. We will evaluate the subtraction in two different orders to demonstrate that the result varies based on the order.

Step 1: Subtracting 2/3 - 1/2
- We can simplify the fractions by finding a common denominator: 2/3 = 4/6 and 1/2 = 3/6.
- Subtracting 4/6 - 3/6 gives us 1/6.

Step 2: Subtracting 1/2 - 2/3
- Again, finding a common denominator: 1/2 = 3/6 and 2/3 = 4/6.
- Subtracting 3/6 - 4/6 gives us -1/6.

As we can see from this example, changing the order of the subtraction operation yields different results. This demonstrates that rational numbers are not commutative under subtraction.

In conclusion, rational numbers, which include fractions and terminating or repeating decimals, do not follow the commutative property under subtraction. Changing the order of subtraction changes the result, highlighting the non-commutative nature of this operation.