Properties of Rational Numbers
1. Closure Property of Rational Numbers
For two rational numbers a and b, a + 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.
a × 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 b, a + 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 b × a = 3/4 × 1/2 = 3/8
Thus, a × b = b × a
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 c, a + (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 × (b × c) = 1/4 × (1/2 × 3/4) = 1/4 × 3/8 = 3/32 and (a × b) × c = (1/4 × 1/2) × 3/4 = 1/8 × 3/4 = 3/32
Thus, a × (b × c) = (a × 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 a is a rational number, then a + 0 = 0 + a = a.
For example, if a = -1/5, then
a + 0 = -1/5 + 0 = -1/5 and 0 + a = 0 + (-1/5) = -1/5
Thus, a + 0 = 0 + a = a
Hence, the rational number 0 is called the additive identity of rational numbers.
5. Multiplicative Identity of Rational Numbers
If a is a rational number, then a × 1 = 1 × a = a.
For example, if a = -1/5, then
a × 1 = -1/5 × 1 = -1/5 and 1 × a = 1 × (-1/5) = -1/5
Thus, a × 1 = 1 × a = 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 a, b and c are rational number, then a(b + c) = a × b + a × c and a(b – c) = a × b – a × c.
For example, if a = 1/2, b = 2/5 and c = -1/3, then
a(b + c) = 1/2 (2/5 + -1/3) = 1/2 (1/15) = 1/30
a × b + a × c = 1/2 × 2/5 + 1/2 × -1/3 = 1/5 - 1/6 = 1/30
Thus, a(b + c) = a × b + a × c
Similarly, we can show that a(b – c) = a × b – a × c.
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