,
,
, KST STUDY POINT, KK NAGAR, CHENNAI
CHAPTER 1
REAL NUMBERS
(A) Main Concepts and Results
• Euclid’s Division Lemma : Given two positive integers a and b, there exist unique
integers q and r satisfying a = bq + r, 0 ≤ r < b.
• Euclid’s Division Algorithm to obtain the HCF of two positive integers, say c and d,
c > d.
Step 1 : Apply Euclid’s division lemma to c and d, to find whole numbers q and r,
such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to
d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage
will be the required HCF.
• Fundamental Theorem of Arithmetic : Every composite number can be expressed
as a product of primes, and this expression (factorisation) is unique, apart from the
order in which the prime factors occur.
• Let p be a prime number. If p divides a2, then p divides a, where a is a positive
integer.
• 2, 3, 5 are irrational numbers.
• The sum or difference of a rational and an irrational number is irrational.
• The product or quotient of a non-zero rational number and an irrational number is
irrational.
• For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.
03/05/18
,
, KST STUDY POINT, KK NAGAR, CHENNAI
CHAPTER 1
REAL NUMBERS
(A) Main Concepts and Results
• Euclid’s Division Lemma : Given two positive integers a and b, there exist unique
integers q and r satisfying a = bq + r, 0 ≤ r < b.
• Euclid’s Division Algorithm to obtain the HCF of two positive integers, say c and d,
c > d.
Step 1 : Apply Euclid’s division lemma to c and d, to find whole numbers q and r,
such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to
d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage
will be the required HCF.
• Fundamental Theorem of Arithmetic : Every composite number can be expressed
as a product of primes, and this expression (factorisation) is unique, apart from the
order in which the prime factors occur.
• Let p be a prime number. If p divides a2, then p divides a, where a is a positive
integer.
• 2, 3, 5 are irrational numbers.
• The sum or difference of a rational and an irrational number is irrational.
• The product or quotient of a non-zero rational number and an irrational number is
irrational.
• For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.
03/05/18
