Key Notes
Chapter 01
Real Numbers
• For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying
the relation a = bq + r , 0 ≤ r < b. .
• Euclid’s division algorithms: HCF of any two positive integers a and b. With a > b is
obtained as follows:
Step 1: Apply Euclid’s division lemma to a and b to find q and r such that
a = bq + r , 0 ≤ r < b.
a= Dividend
b=Divisor
q=quotient
r=remainder
Step II: If r = 0, HCF ( a, b ) = b if r ≠ 0, apply Euclid’s lemma to b and r.
Step III: Continue the process till the remainder is zero. The divisor at this stage will be the
required HCF
• The Fundamental Theorem of Arithmetic: Every composite number can be expressed
(factorized) as a product of primes and this factorization is unique, apart from the order in
which the prime factors occur. Ex : 24 = 2 × 2 × 2 × 3 = 3 × 2 × 2 × 2
p
• Let x = , q ' ≠ 0 to be a rational number, such that the prime factorization of ‘q’ is of the
q
form 2m 5n, where m, n are non-negative integers. Then x has a decimal expansion which is
terminating.
p
• Let x = , q ≠ 0 be a rational number, such that the prime factorization of q is not of the
q
form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is
non-terminating repeating.
• p is irrational, which p is a prime. A number is called irrational if it cannot be written in the
P
form where p and q are integers and q ≠ 0.
q
Chapter 01
Real Numbers
• For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying
the relation a = bq + r , 0 ≤ r < b. .
• Euclid’s division algorithms: HCF of any two positive integers a and b. With a > b is
obtained as follows:
Step 1: Apply Euclid’s division lemma to a and b to find q and r such that
a = bq + r , 0 ≤ r < b.
a= Dividend
b=Divisor
q=quotient
r=remainder
Step II: If r = 0, HCF ( a, b ) = b if r ≠ 0, apply Euclid’s lemma to b and r.
Step III: Continue the process till the remainder is zero. The divisor at this stage will be the
required HCF
• The Fundamental Theorem of Arithmetic: Every composite number can be expressed
(factorized) as a product of primes and this factorization is unique, apart from the order in
which the prime factors occur. Ex : 24 = 2 × 2 × 2 × 3 = 3 × 2 × 2 × 2
p
• Let x = , q ' ≠ 0 to be a rational number, such that the prime factorization of ‘q’ is of the
q
form 2m 5n, where m, n are non-negative integers. Then x has a decimal expansion which is
terminating.
p
• Let x = , q ≠ 0 be a rational number, such that the prime factorization of q is not of the
q
form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is
non-terminating repeating.
• p is irrational, which p is a prime. A number is called irrational if it cannot be written in the
P
form where p and q are integers and q ≠ 0.
q
