This chapter explores real numbers, focusing on key properties such as the Fundamental Theorem of Arithmetic and the concept of irrational numbers, which are crucial for understanding the number system.
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.
This chapter discusses polynomials, their degrees, and classifications such as linear, quadratic, and cubic. Understanding polynomials is essential for solving various mathematical problems.
This chapter focuses on solving pairs of linear equations with two variables and their real-life applications.
This chapter explores quadratic equations, highlighting their forms and significance in real-world applications.
This chapter introduces arithmetic progressions, which are sequences of numbers generated by adding a fixed value to the previous term. Understanding these patterns is crucial for solving real-life mathematical problems.
This chapter focuses on the properties of triangles, specifically their similarity and how it can be applied in various real-world contexts.
This chapter covers the concepts of coordinate geometry, including finding distances between points and dividing line segments. Understanding these concepts is essential for solving geometry problems using algebra.
This chapter focuses on the foundational concepts of trigonometry, particularly the relationships between the angles and sides of right triangles.
This chapter explores how trigonometry is applied in real-life situations, particularly in measuring heights and distances.
This chapter explores the properties of circles, particularly focusing on tangents and their relationship with radii and secants.
This chapter focuses on sectors and segments of circles, essential concepts in geometry. Understanding these helps in solving real-life problems related to areas and measurements.
