CBSE Class 8 Maths Exemplar Chapter 1: Rational Numbers NCERT Solutions

The correct option is (d) rational number.

By definition, a number that can be written in the form \(\frac{p}{q}\), where 'p' and 'q' are integers and 'q' is not zero, is called a rational number. Natural numbers, whole numbers, and integers do not always satisfy this condition (e.g., integers like 5 can be written as \(\frac{5}{1}\), but the definition specifically defines the set of rational numbers).

The correct option is (b) p and q are integers and <math>\(q \neq 0\)</math>.

The definition of a rational number requires two conditions: first, that both the numerator (p) and the denominator (q) must be integers, and second, that the denominator (q) cannot be zero. The condition <math>\(p \neq 0\)</math> is not necessary for a number to be rational (e.g., 0 can be written as \(\frac{0}{1}\) and is a rational number).

The correct option is (a) rational numbers are closed under addition.

The closure property states that if you perform an operation on any two numbers from a set, the result is also a number within that same set. In this case, we are adding two rational numbers, \(\frac{3}{8}\) and \(\frac{-5}{7}\). To perform the addition, we find a common denominator, which is the Least Common Multiple (LCM) of 8 and 7. The LCM of 8 and 7 is 56.

Now, we express each fraction with the common denominator:

<math display="block">\(\frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56}\)</math>

<math display="block">\(\frac{-5}{7} = \frac{-5 \times 8}{7 \times 8} = \frac{-40}{56}\)</math>

Adding these fractions:

<math display="block">\(\frac{3}{8} + \frac{(-5)}{7} = \frac{21}{56} + \frac{-40}{56} = \frac{21 - 40}{56} = \frac{-19}{56}\)</math>

Since the result, \(\frac{-19}{56}\), is also a rational number, this demonstrates that rational numbers are closed under addition.

The statement that is not true is (d) rational numbers are closed under division.

Rational numbers are indeed closed under addition, subtraction, and multiplication. For example:

• Addition: \(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) (rational)
• Subtraction: \(\frac{1}{2} - \frac{1}{3} = \frac{1}{6}\) (rational)
• Multiplication: \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\) (rational)

However, rational numbers are not closed under division. This is because division by zero is undefined. For instance, if we try to divide 5 (a rational number) by 0 (which can be considered as \(\frac{0}{1}\)), the operation is not possible. Even if we consider division of two non-zero rational numbers, like \(\frac{1}{2} \div \frac{1}{3} = \frac{3}{2}\) (which is rational), the fundamental issue of division by zero prevents closure for the entire set.

The correct option is (a) Addition of rational numbers is commutative.

The given expression is in the form <math>\(a + b = b + a\)</math>, where <math>\(a = -\frac{3}{8}\)</math> and <math>\(b = \frac{1}{7}\)</math>. This form illustrates the commutative property of addition, which states that the order of operands does not change the outcome of the addition.

Let's verify:

Left Hand Side (LHS): <math>\(-\frac{3}{8} + \frac{1}{7}\)</math>

LCM of 8 and 7 is 56.

<math display="block">\(-\frac{3 \times 7}{8 \times 7} + \frac{1 \times 8}{7 \times 8} = \frac{-21}{56} + \frac{8}{56} = \frac{-21 + 8}{56} = \frac{-13}{56}\)</math>

Right Hand Side (RHS): <math>\(\frac{1}{7} + \left(-\frac{3}{8}\right)\)</math>

<math display="block">\(\frac{1 \times 8}{7 \times 8} + \frac{-3 \times 7}{8 \times 7} = \frac{8}{56} + \frac{-21}{56} = \frac{8 - 21}{56} = \frac{-13}{56}\)</math>

Since LHS = RHS, the commutative property holds true.

The expression (a) <math>\(\frac{2}{3} \times \left( \frac{-6}{7} \times \frac{3}{5} \right) = \left( \frac{2}{3} \times \frac{-6}{7} \right) \times \frac{3}{5}\)</math> shows that rational numbers are associative under multiplication.

The associative property of multiplication states that for any three rational numbers a, b, and c, the product remains the same regardless of how they are grouped. The property is expressed as: <math display="block">\(a \times (b \times c) = (a \times b) \times c\)</math>.

In the given expression, we have:

• <math>\(a = \frac{2}{3}\)</math>
• <math>\(b = \frac{-6}{7}\)</math>
• <math>\(c = \frac{3}{5}\)</math>

The expression groups 'b' and 'c' first on the left side and 'a' and 'b' first on the right side, demonstrating the associative property.