Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Real Numbers – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Real Numbers chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Euclid’s Division Lemma: a = bq + r, 0 ≤ r < b
a is the dividend, b is the divisor, q is the quotient, and r is the remainder. This lemma is foundational for understanding divisibility and finding HCF. Example: For a = 20, b = 6, q = 3, r = 2.
Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes uniquely.
This theorem states that prime factorisation is unique for every composite number, crucial for understanding numbers' structure. Example: 28 = 2² × 7.
HCF(a, b) × LCM(a, b) = a × b
HCF is the highest common factor, and LCM is the least common multiple of two numbers a and b. This relation is useful for finding LCM when HCF is known and vice versa.
√2 is irrational
This formula states that the square root of 2 cannot be expressed as a fraction of integers, fundamental for understanding irrational numbers.
√p is irrational, where p is a prime
Generalizes the concept that the square root of any prime number is irrational, extending the understanding of irrational numbers.
Decimal expansion of p/q terminates if q = 2ⁿ5ᵐ
p/q is a rational number in its simplest form. The decimal terminates if the prime factors of q are only 2 and/or 5. Example: 1/8 = 0.125.
Decimal expansion of p/q is non-terminating repeating if q has prime factors other than 2 or 5.
Indicates that the decimal expansion of a rational number repeats infinitely if the denominator has prime factors beyond 2 or 5. Example: 1/3 = 0.333...
a² = 2b² implies a is even
Used in proofs by contradiction to show the irrationality of √2, demonstrating that if a² is divisible by 2, then a is divisible by 2.
Product of a non-zero rational and an irrational number is irrational.
Highlights a property of irrational numbers, useful in proofs and understanding number systems. Example: 2 × √3 is irrational.
Sum or difference of a rational and an irrational number is irrational.
Another property of irrational numbers, important for algebraic manipulations. Example: 1 + √5 is irrational.
Equations
Finding HCF using Euclid’s algorithm: HCF(a, b) = HCF(b, r), where a = bq + r
This equation is used iteratively to find the HCF of two numbers by replacing the larger number with the remainder until the remainder is zero.
LCM(a, b) = (a × b) / HCF(a, b)
A direct method to find the LCM of two numbers using their HCF, simplifying calculations.
a = bq + r, 0 ≤ r < b
Represents the division of a by b, yielding quotient q and remainder r, foundational for divisibility tests.
2b² = a² leads to contradiction in proving √2 is irrational
Central to the proof by contradiction that √2 cannot be expressed as a fraction of integers.
p divides a² implies p divides a
A theorem used in proofs, especially in demonstrating the irrationality of square roots of primes.
n² is even implies n is even
A logical step in proofs by contradiction, showing properties of even numbers.
For primes p, √p is not rational
An equation stating the irrationality of the square root of any prime number, a key concept in number theory.
HCF of three numbers: HCF(a, b, c) = HCF(HCF(a, b), c)
Extends the concept of HCF to three numbers, useful for solving more complex problems.
LCM of three numbers: LCM(a, b, c) = LCM(LCM(a, b), c)
Extends the concept of LCM to three numbers, facilitating the calculation for multiple numbers.
a × b = LCM(a, b) × HCF(a, b)
Relates the product of two numbers to their LCM and HCF, a fundamental relation in number theory.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
