NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

Short Note:

• Rational Number: A number that can be written in the form p/q, where p and q are integers and q ≠ 0.

• Irrational Number: A number that cannot be written in the form p/q; examples include √2, √3, π.

• Every integer, whole number and natural number is a rational number. Zero is also a rational number (0 = 0/1).

• Between any two rational numbers, there exist infinitely many rational numbers as well as infinitely many irrational numbers.

Short Note:

• The collection of rational and irrational numbers together forms the set of Real Numbers.

• A rational number has either a terminating decimal expansion or a non-terminating repeating (recurring) decimal expansion.

• An irrational number has a non-terminating, non-repeating decimal expansion.

• Every real number can be represented by a unique point on the number line, and every point on the number line represents a unique real number.

Short Note:

• Irrational numbers such as √2, √3 can be located on the number line using the Pythagoras theorem construction.

• Process of Successive Magnification: visualising a real number with a non-terminating decimal expansion by zooming into smaller and smaller intervals on the number line.

Short Note:

• Sum or difference of a rational and an irrational number is always irrational.

• Product or quotient of a non-zero rational number with an irrational number is always irrational.

• For positive real numbers a, b: √a · √b = √(ab); √a / √b = √(a/b); (√a + √b)(√a − √b) = a − b.

• Laws of Exponents: a^p · a^q = a^p+q; (a^p)^q = a^pq; a^p/a^q = a^p−q; a^pb^p = (ab)^p, for a, b > 0 and p, q rational numbers.