REAL NUMBERS
Frequently Asked Questions
Real numbers include all rational and irrational numbers, representing all points on the number line.
Rational numbers can be expressed asp/qp/qp/qwherepppandqqqare integers andq?0q \neq 0q?=0.
Irrational numbers cannot be expressed asp/qp/qp/q; their decimal expansion is non-terminating and non-repeating.
Euclid, an ancient Greek mathematician, proposed the division lemma used for finding HCF.
For any two positive integersaaaandbbb, there exist unique integersqqqandrrrsuch thata=bq+ra = bq + ra=bq+r, where0=r<b0 \leq r < b0=r<b.
It helps find the Highest Common Factor (HCF) of two numbers using repeated division.
It is the process of applying Euclid’s Lemma repeatedly to find the HCF of two numbers.
HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly.
LCM (Least Common Multiple) is the smallest number divisible by the given numbers.
HCF×LCM=Product of the two numbers\text{HCF} \times \text{LCM} = \text{Product of the two numbers}HCF×LCM=Product of the two numbers.
Prime numbers are natural numbers greater than 1 that have only two factors: 1 and itself.
Composite numbers have more than two factors. Examples: 4, 6, 8, 9.
Every composite number can be expressed as a product of primes in a unique way, except for order of factors.
Expressing a number as a product of prime numbers.
List prime factors of each number and multiply common factors with least power.
Multiply all prime factors taking the highest power of each factor.
Two numbers having HCF = 1 are called co-prime numbers.
Yes, because their HCF is 1.
No, 1 is neither prime nor composite.
Decimals that end after a finite number of digits are terminating decimals.
Decimals that go on without ending are non-terminating decimals.
Non-terminating decimals that show repeating patterns, e.g., 0.333…, are repeating decimals.
Non-terminating decimals with no pattern, e.g.,p\pip, are non-repeating.
A rational numberp/qp/qp/qhas a terminating decimal expansion if its denominator has only 2 or 5 as prime factors.
1/4=0.251/4 = 0.251/4=0.25is a terminating decimal.
1/3=0.333…1/3 = 0.333…1/3=0.333…is a non-terminating repeating decimal.
2\sqrt{2}2is an irrational number.
22/7=3.142857…22/7 = 3.142857…22/7=3.142857…is non-terminating and repeating.
It simplifies finding the HCF of two positive integers quickly.
Prime factorization of any integer above 1 remains unique except for the order of factors.
The smallest prime number is 2.
The smallest composite number is 4.
An even number is divisible by 2.
A number not divisible by 2 is called an odd number.
Integers that follow one another in sequence, differing by 1.
The HCF of two consecutive integers is always 1.
The HCF is 2.
The HCF is 1.
The HCF of any number and 1 is 1.
The LCM of two co-prime numbers is equal to their product.
The prime factors are only 2 and 5.
60 = 2² × 3 × 5.
It ensures every integer has a unique prime factorization.
A number that cannot be written asp/qp/qp/qdue to non-terminating, non-repeating decimal expansion.
It’s a rational number because it’s a repeating decimal.
No, square roots of prime numbers are always irrational.
Yes, all rational numbers belong to the set of real numbers.
No, because irrational numbers are also real but not rational.
The product of two rational numbers is always rational.
The product is always irrational, provided the rational is not zero.
