CBSE Class 9 Maths Chapter 1 Number Systems Revision Notes

Glossary of Key Terms

Natural Numbers (N)

The counting numbers starting from 1, 2, 3, 4, ... up to infinity.

Whole Numbers (W)

The collection of natural numbers along with the number zero: 0, 1, 2, 3, ... up to infinity.

Integers (Z)

The collection of all positive whole numbers, zero, and negative natural numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational Numbers (Q)

Numbers that can be expressed in the fraction form p/q, where p and q are integers, and q is not equal to 0.

Irrational Numbers

Numbers that cannot be written in the form p/q (where p and q are integers, q != 0). They feature non-terminating and non-recurring decimal expansions.

Real Numbers (R)

The complete collection containing all rational numbers and all irrational numbers combined.

Rationalization

The algebraic process used to convert a fraction with an irrational denominator into an equivalent fraction with a rational denominator.

Must Remember: Key Points & Laws

• Every natural number is a whole number, and every whole number is an integer, but the converse is not true.
• There are infinitely many rational numbers between any two given distinct rational numbers.
• The decimal expansion of a rational number is either terminating (e.g., 0.5) or non-terminating recurring (e.g., 0.333...).
• The decimal expansion of an irrational number is always non-terminating and non-recurring (e.g., 0.101101110...).
• The sum, difference, product, or quotient of a non-zero rational number and an irrational number is always irrational.
• The sum, difference, product, or quotient of two irrational numbers can be either rational or irrational.
• Law of Exponents 1: a^p * a^q = a^(p + q)
• Law of Exponents 2: (a^p)^q = a^(pq)
• Law of Exponents 3: a^p / a^q = a^(p - q)
• Law of Exponents 4: a^p * b^p = (ab)^p

Real Numbers and Their Decimal Expansions

In CBSE Class 9, understanding the real number line and decimal expansions is critical for scoring well. Real numbers are broadly divided into rational and irrational numbers. The decimal expansion of rational numbers is either terminating (meaning the division ends with a remainder of 0) or non-terminating recurring/repeating (meaning a digit or group of digits repeats infinitely after the decimal point). Conversely, irrational numbers are characterized by non-terminating non-recurring decimal expansions that never end and never show a repeating block pattern. You can geometrically locate irrational roots such as sqrt(2), sqrt(3), or sqrt(5) using a compass and ruler by applying the Pythagoras Theorem step-by-step on a number line.

Comparison: Rational vs. Irrational Numbers

How to Rationalize a Denominator of the Form 1 / (a + sqrt(b))

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Solved Exemplar Problems

• {"title":"Example 1: Express 0.666... in p/q form","description":"Let x = 0.666... (Equation 1). Since one digit is repeating, multiply both sides by 10 to get 10x = 6.666... (Equation 2). Subtract Equation 1 from Equation 2: 10x - x = (6.666...) - (0.666...), which gives 9x = 6. Therefore, x = 6/9 = 2/3. Thus, 0.666... in rational form is 2/3."}
• {"title":"Example 2: Rationalize the denominator of 1 / (7 + 3sqrt(2))","description":"Multiply both the numerator and the denominator by the conjugate (7 - 3sqrt(2)). Numerator becomes 1 (7 - 3sqrt(2)) = 7 - 3sqrt(2). Denominator becomes (7 + 3sqrt(2))(7 - 3sqrt(2)) = 7^2 - (3sqrt(2))^2 = 49 - (9 2) = 49 - 18 = 31. The rationalized expression is (7 - 3sqrt(2)) / 31."}
• {"title":"Example 3: Simplify (125)^(-1/3)","description":"First, write 125 as a power of 5: 125 = 5^3. Now apply the law of exponents (a^p)^q = a^(pq). (5^3)^(-1/3) = 5^(3 * -1/3) = 5^(-1) = 1/5."}

Board Exam Traps & Smart Marking Cues

Beware of the Square Root trap: Students frequently make calculations errors such as writing (sqrt(3))^2 = 9 instead of 3. Always apply the identity (sqrt(x))^2 = x carefully.

Conjugate Sign Errors: When rationalizing, remember to change the sign. If the denominator is 3 - sqrt(2), your rationalizing factor must be 3 + sqrt(2).

Representation on Number Line: In 2-mark questions requiring locating roots geometrically, label all points (like O, A, B) and write down the step of using Pythagoras theorem (OB^2 = OA^2 + AB^2) clearly to avoid step-mark deductions.

Self-Test Quick Revision Checks

• Is zero (0) a rational number? Support your answer. Yes, zero is a rational number because it can be written in the form p/q, such as 0/1, 0/2, or 0/-5, where the denominator is a non-zero integer.
• Give an example of two irrational numbers whose sum is a rational number. Let the two irrational numbers be (3 + sqrt(2)) and (3 - sqrt(2)). Their sum is (3 + sqrt(2)) + (3 - sqrt(2)) = 6, which is a rational number.
• Find the value of (64)^(1/2). Since 64 can be written as 8^2, we apply laws of exponents: (8^2)^(1/2) = 8^(2 * 1/2) = 8^1 = 8.
• What is the rationalizing factor of (sqrt(5) + sqrt(2))? The rationalizing factor is (sqrt(5) - sqrt(2)). When multiplied, they yield (sqrt(5))^2 - (sqrt(2))^2 = 5 - 2 = 3 (a rational number).

Frequently Asked Questions