Number System Class 9 Notes Maths - Chapter 1

Chapter: 1

Natural numbers are counting numbers, like 1, 2, 3,.......
Whole numbers are just like natural numbers but they start from O, like 0, 1, 2, 3,.......
If we put the whole numbers and their negative together, the new collection of numbers will look like 0, 1, 2, 3,....... –1, –2, –3....... and such numbers are called integers.
A number is called a rational number, if it can be written in the form (p/q) , where p and q are integers and q ≠ 0.
A number which cannot be expressed in the form (p/q) (where p and q are integers and q ≠ 0) is called an irrational number.
All rational numbers and all irrational numbers together make the collection of real numbers.

Decimal expansion of a rational number is either terminating or non-terminating recurring.
Decimal expansion of an irrational number is non-terminating non-recurring.

The sum, the difference and the product of two rational numbers is a rational number. Also, the quotient of two non-zero rational numbers is also a rational number.
If u is a rational number and v is an irrational number, then u + v and u – v are irrationals. Also, if u is non-zero rational, then uv and (u/v), are irrationals.
If u and v both are irrationals, then u + v, u - v, uv and (u/v) are either rational or irrational.

For any positive real numbers a and b:

(i) √ab = √a . √b

(ii) √(a/b)=(√a/√b),provided b ≠0

(iii) ( √a + √b) ( √a – √b) = a – b

(iv) (a + √b) (a – √b) = a² - b and ( √a + b) ( √a – b) = a – b²

(v) ( √a + √b)² = a + 2 √(ab) + b and ( √a – √b)² = a – 2 √(ab) + b

(vi) ( √a + √b) ( √c + √d) = √(ac) + √(ad) + √(bc) + √(bd)

To rationalise the denominator of the form a + √b, multiply the numerator and denominator by its conjugate, i.e. by a – √b.
To rationalise the denominator of the form √a + √b, multiply the numerator and denominator by its conjugate, i.e. √a – √b.
Conjugate of a – √b is a + √b and the conjugate of √a – √b is √a + √b.

If a and b are positive real numbers and p and q are rational numbers, then

(i) a^p × a^q = a^p+q

(ii) (a^p)^q = (a^q)^p = a^pq

(iii) (a^p/a^q)=a^(p-q) and (1/a^{p)=(a0/ap)= a0-p}

(iv) (ab)^p = a^pb^p

(v) (a/b)^p=(a^p/b^p) and (a/b)^-p=(b/a)^p