Real number Flashcards
What is the Fundamental Theorem of Arithmetic?
Every composite number can be expressed as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.
Also called the Unique Factorization Theorem.
What is the condition for a rational number x to have a terminating decimal expansion?
x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2^n * 5^m where n, m are non-negative integers.
What happens if the prime factorisation of q is not of the form 2^n * 5^m?
x has a decimal expansion which is non-terminating repeating.
What is the relationship between HCF and LCM of two integers a and b?
HCF(a, b) × LCM(a, b) = a × b.
What is the first step in Euclid’s Division Lemma?
Apply Euclid’s Division Lemma to c & d: c = dq + r.
What indicates that d is the HCF of c and d in Euclid’s Division process?
If r = 0.
What is the prime factorization of a natural number?
It is unique, except to the order of its factors.
What is the prime factorization of 12?
12 = 2 × 2 × 3 or 12 = 2^2 × 3.
What are rational numbers?
Numbers in the form p/q, where p and q are co-prime numbers and q ≠ 0.
What defines an irrational number?
A number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
What is a terminating decimal?
A decimal expansion of a rational number that comes to an end.
If p is a prime number and p divides a², what can be concluded about p and a?
p divides a, where a is a positive integer.
What is the method to find HCF using prime factorization?
HCF = Product of the smallest power of each common prime factor involved in the numbers.
What is the method to find LCM using prime factorization?
LCM = Product of the greatest power of each prime factor involved in the numbers.
What is the relationship between composite numbers and prime factors?
Every composite number can be expressed as a product of prime numbers.
Fill in the blank: A rational number x can be expressed in the form _____ where p and q are coprime.
p/q.
True or False: The prime factorization of a natural number is not unique.
False.
