Properties of Rational Numbers

# Properties of Rational Numbers

## Properties of Rational Numbers

1. Closure Property of Rational Numbers

For two rational numbers a and ba + b, a – b and a × b are also rational numbers.
For example, if a = 4/5 and b = -1/5, then
a + b = 4/5 + (-1/5) = 3/5, which is a rational number.
a - b = 4/5 - (-1/5) = 4/5 + 1/5 = 5/5 = 1, which is a rational number.
× b = 4/5 × -1/5 = -4/25, which is a rational number.

Thus, rational numbers are closed under addition, subtraction and multiplication.
Since 45 ÷ 0 = not defined, rational numbers are not closed under division.

### 2. Commutative Property of Rational Numbers

For two rational numbers a and ba + b = b + a and a × b = b × a.
For example, if a = 1/2 and b = 3/4, then
a + b = 1/2 + 3/4 = 5/4 and b + a = 3/4 + 1/2 = 5/4
Thus, a + b = b + a
Similarly, a × b = 1/2 × 3/4 = 3/8 and × a = 3/4 × 1/2 = 3/8
Thus, × b = ×
Hence, rational numbers are commutative under addition and multiplication.
But rational numbers are not commutative under subtraction and division.

### 3. Associative Property of Rational Numbers

For three rational numbers a, b and ca + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c
For example, if a = 1/4, b = 1/2 and c = 3/4, then
a + (b + c) = 1/4 + (1/2 + 3/4) = 1/4 + 5/4 = 6/4 and (a + b) + c = (1/4 + 1/2) + 3/4 = 3/4 + 3/4 = 6/4
Thus, a + (b + c) = (a + b) + c
Similarly, a × (× c) = 1/4 × (1/2 × 3/4) = 1/4 × 3/8 = 3/32 and (× b) × c = (1/4 × 1/2) × 3/4 = 1/8 × 3/4 = 3/32
Thus, × (× c) = (× b) × c
Hence, rational numbers are associative under addition and multiplication.
But rational numbers are not associative under subtraction and division.

### 4. Additive Identity of Rational Numbers

If is a rational number, then + 0 = 0 + a.
For example, if a = -1/5, then
a + 0 = -1/5 + 0 = -1/5 and 0 + a = 0 + (-1/5) = -1/5
Thus, + 0 = 0 + a
Hence, the rational number 0 is called the additive identity of rational numbers.

### 5. Multiplicative Identity of Rational Numbers

If is a rational number, then × 1 = 1 × a.
For example, if a = -1/5, then
× 1 = -1/5 × 1 = -1/5 and 1 × a = 1 × (-1/5) = -1/5
Thus, × 1 = 1 × a
Hence, the rational number 1 is called the multiplicative identity of rational numbers.

### 6. Additive Inverse of Rational Numbers

If p/q is a rational number, then – p/q is its additive inverse, i.e., p/q + (– p/q) = 0.
For example, the additive inverse of 3/5 is -3/5 and the additive inverse of -3/5 is 3/5.
Similarly, the additive inverse of -10 is 10 and the additive inverse of 10 is -10.
Generally, the additive inverse of a rational number can be obtained by changing its sign.

### 7. Multiplicative Inverse of Rational Numbers

The multiplicative inverse of p/q is q/p, since p/q × q/p = 1.
For example, the multiplicative inverse of 4/7 is 7/4 and the multiplicative inverse of -2/5 is -5/2. The multiplicative inverse of a positive rational number is positive and the multiplicative inverse of a negative rational number is negative.
To find the multiplicative inverse, we interchange the numbers given in the numerator and the denominator of the rational number.

### 8. Distributive Property of Rational Numbers

If aand are rational number, then a(c) = ×  × c and a(– c) = × – × c.
For example, if a = 1/2, b = 2/5 and c = -1/3, then
a(c) = 1/2 (2/5 + -1/3) = 1/2 (1/15) = 1/30
×  × c = 1/2 × 2/5 + 1/2 × -1/3 = 1/5 - 1/6 = 1/30
Thus, a(c) = ×  × c
Similarly, we can show that a(– c) = × – × c.

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