Real Number Flashcards
What does Euclid’s division algorithm state?
Any positive integer a can be divided by another positive integer b, leaving a remainder r that is smaller than b.
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 of the prime factors.
What are the two main applications of the Fundamental Theorem of Arithmetic?
- Proving the irrationality of numbers such as √2, √3, and √5
- Exploring when the decimal expansion of a rational number is terminating or non-terminating repeating.
What is the unique factorisation of a composite number according to the Fundamental Theorem of Arithmetic?
A composite number can be factorised as x = p1^e1 × p2^e2 × … × pn^en, where p1, p2, …, pn are prime factors.
Who provided the first correct proof of the Fundamental Theorem of Arithmetic?
Carl Friedrich Gauss.
True or False: The prime factorisation of a natural number is unique except for the order of its factors.
True.
How can you express the number 32760 as a product of primes?
32760 = 2^3 × 3^2 × 5 × 7 × 13.
What is the HCF of 6 and 20 using the prime factorisation method?
HCF(6, 20) = 2.
What is the LCM of 6 and 20 using the prime factorisation method?
LCM(6, 20) = 60.
Fill in the blank: The product of any two positive integers a and b is equal to _______.
HCF(a, b) × LCM(a, b).
What is the relationship between HCF and LCM of two integers?
HCF(a, b) × LCM(a, b) = a × b.
What is an irrational number?
A number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
What theorem is used to prove the irrationality of numbers like √2?
The Fundamental Theorem of Arithmetic.
State Theorem 1.2 regarding prime numbers.
If p is a prime number and p divides a^2, then p divides a, where a is a positive integer.
What is an example of an irrational number?
√2.
True or False: There is a natural number n for which 4^n ends with the digit zero.
False.
What is the prime factorisation of 96?
96 = 2^5 × 3.
What is the prime factorisation of 404?
404 = 2^2 × 101.
Find the HCF of 96 and 404.
HCF(96, 404) = 4.
What is the LCM of 96 and 404?
LCM(96, 404) = 9696.
Fill in the blank: The product of three numbers is not equal to the product of their _______.
HCF and LCM.
List the first few prime numbers.
- 2
- 3
- 5
- 7
- 11
- 13.
What is the significance of the Fundamental Theorem of Arithmetic?
It establishes the uniqueness of prime factorisation for composite numbers.
If p is a prime and p divides a², what can be concluded about a?
p divides a.
What technique is used in the proof that √2 is irrational?
Proof by contradiction.
Assuming √2 is rational, what can be expressed as a fraction?
r/2 = s for integers r and s (s ≠ 0).
What conclusion is reached if a and b have a common factor other than 1 in the proof of √2?
a and b must be coprime.
What contradiction arises in the proof that √2 is irrational?
a and b cannot have a common factor other than 1.
What is the conclusion reached about √2?
√2 is irrational.
What assumption is made to prove that √3 is irrational?
Assume √3 is rational.
What results from assuming that a and b are coprime in the proof of √3?
b³ = a.
What contradiction arises in the proof that √3 is irrational?
a and b cannot have a common factor of 3.
What is concluded about √3?
√3 is irrational.
What is the sum or difference of a rational and an irrational number?
Irrational.
What is the product and quotient of a non-zero rational and irrational number?
Irrational.
Prove that 5 - 3 is irrational. What assumption leads to a contradiction?
Assume 5 - 3 is rational.
What is the outcome of the assumption that 5 - 3 is rational?
3 is rational, which contradicts its irrationality.
What is concluded about 5 - 3?
5 - 3 is irrational.
Prove that 3√2 is irrational. What assumption leads to a contradiction?
Assume 3√2 is rational.
What contradiction arises in the proof that 3√2 is irrational?
2 is irrational.
What is concluded about 3√2?
3√2 is irrational.
What is the relationship between HCF and LCM for three numbers p, q, and r?
LCM(p, q, r) = (p × q × r) / HCF(p, q, r).
What is the relationship between HCF and LCM for three numbers p, q, and r in another form?
HCF(p, q, r) = (p × q × r) / LCM(p, q, r).
