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.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Explore the basics of trigonometry, including angles, triangles, and the fundamental trigonometric ratios: sine, cosine, and tangent.
Explore real-world applications of trigonometry in measuring heights, distances, and angles in various fields such as astronomy, navigation, and architecture.
Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
Explore the concepts of calculating areas related to circles, including sectors, segments, and combinations with other geometric shapes.
