NCERT Solutions Class 9 Maths Chapter 1 — Number Systems

Solution:

Step 1: Real numbers are composed of both rational and irrational numbers. Therefore, every irrational number is a real number.

Step 2: Option B is false because rational numbers are also real numbers.

Step 3: Option C is false because fractions like 1/2 are rational but not integers.

Step 4: Option D is false because negative integers (e.g., -3) are integers but not whole numbers.

Solution:

Step 1: A rational number can be expressed as p/q, where p and q are integers and q ≠ 0. Its decimal expansion is either terminating or non-terminating repeating.

Step 2: The decimal expansion of 7/8 is 0.875, which is terminating. Thus, 7/8 is a rational number. Ravi's conclusion for 7/8 is correct.

Step 3: The decimal expansion of 1/3 is 0.333..., which is non-terminating repeating. According to the definition, this also represents a rational number. Ravi's conclusion that 1/3 is irrational is incorrect.

Step 4: Therefore, 7/8 is rational, and 1/3 is also rational.

Solution:

Step 1: In the right-angled triangle OAB, by Pythagoras theorem, OB² = OA² + AB².

Step 2: Given OA = 2 units and AB = 1 unit, OB² = 2² + 1² = 4 + 1 = 5. So, OB = √5 units.

Step 3: To represent √5 on the number line, we need to transfer the length OB from the origin (O) to the number line.

Step 4: Therefore, the next logical step is to draw an arc with center O (the origin) and radius OB (which is √5) to intersect the number line.

Solution:

Step 1: A. The sum of a rational number and an irrational number is always irrational. For example, 2 + √3 is irrational. This statement is ALWAYS TRUE.

Step 2: B. The product x × y is not always rational. If x is a non-zero rational number, x × y is irrational. If x = 0, then 0 × y = 0, which is rational. So, this statement is FALSE.

Step 3: C. The product y × y (or y²) is not always irrational. For example, √2 × √2 = 2, which is rational. So, this statement is FALSE.

Step 4: D. The quotient x / y is not always rational. If x is a non-zero rational number, x / y is irrational. If x = 0, then 0 / y = 0, which is rational. So, this statement is FALSE.

Solution:

Step 1: The property of rational numbers is that they are 'dense' on the number line.

Step 2: This means that between any two distinct rational numbers, no matter how close they are, there exists another rational number.

Step 3: This process can be repeated infinitely, leading to infinitely many rational numbers between any two distinct rational numbers.