Real Numbers and Arithmetic Theorems

Real Numbers

Introduction

• Real numbers are a set of numbers that include all rational and irrational numbers.
• They can be represented on a number line.

Euclid's Division Lemma

• If a and b are positive integers, then there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
• This lemma is used to find the HCF (Highest Common Factor) of two numbers.

The Fundamental Theorem of Arithmetic

• Every composite number can be expressed as a product of prime numbers in a unique way, except for the order in which they are written.
• This theorem is used to find the prime factorization of a number.

Irrational Numbers

• Irrational numbers are real numbers that cannot be expressed as a ratio of integers (e.g., √2, π, e).
• They can be represented as non-terminating, non-repeating decimals.

Decimal Expansion of Rational Numbers

• The decimal expansion of a rational number is either terminating or non-terminating, recurring.
• Examples: 1/2 = 0.5 (terminating), 1/3 = 0.333... (non-terminating, recurring)

Repeating Decimals

• A repeating decimal is a decimal that has a block of digits that repeats infinitely.
• Examples: 1/3 = 0.333... (repeating decimal), 1/7 = 0.142857... (repeating decimal)

Operations on Real Numbers

• Real numbers can be added, subtracted, multiplied, and divided, following the usual rules of arithmetic.
• The order of operations (PEMDAS) should be followed when performing operations on real numbers.

Properties of Real Numbers

• Commutative property: a + b = b + a, a × b = b × a
• Associative property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
• Distributive property: a × (b + c) = a × b + a × c
• Existence of additive and multiplicative identities (0 and 1, respectively) and inverse elements.

Real Numbers

• Real numbers include rational and irrational numbers and can be represented on a number line.

Euclid's Division Lemma

• The lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
• This lemma is used to find the Highest Common Factor (HCF) of two numbers.

The Fundamental Theorem of Arithmetic

• Every composite number can be expressed as a product of prime numbers in a unique way, except for the order in which they are written.
• This theorem is used to find the prime factorization of a number.

Irrational Numbers

• Irrational numbers are real numbers that cannot be expressed as a ratio of integers (e.g., √2, π, e).
• They can be represented as non-terminating, non-repeating decimals.

Decimal Expansion of Rational Numbers

• The decimal expansion of a rational number is either terminating or non-terminating, recurring.
• Examples: 1/2 = 0.5 (terminating), 1/3 = 0.333... (non-terminating, recurring)

Repeating Decimals

• A repeating decimal is a decimal that has a block of digits that repeats infinitely.
• Examples: 1/3 = 0.333... (repeating decimal), 1/7 = 0.142857... (repeating decimal)

Operations on Real Numbers

• Real numbers can be added, subtracted, multiplied, and divided, following the usual rules of arithmetic.
• The order of operations (PEMDAS) should be followed when performing operations on real numbers.

Properties of Real Numbers

• Commutative property: a + b = b + a, a × b = b × a.
• Associative property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
• Distributive property: a × (b + c) = a × b + a × c.
• Existence of additive and multiplicative identities (0 and 1, respectively) and inverse elements.

Polynomials

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• The general form of a polynomial is: a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0.
• The degree of a polynomial is the highest power of the variable (x) in the polynomial.
• Polynomials can be classified into three types: monomial, binomial, and trinomial.
• A monomial is a polynomial with only one term.
• A binomial is a polynomial with two terms.
• A trinomial is a polynomial with three terms.

Remainder Theorem

• If p(x) is a polynomial and p(a) = 0, then (x - a) is a factor of p(x).
• If p(x) is divided by (x - a) and the remainder is r, then p(a) = r.

Factor Theorem

• If (x - a) is a factor of p(x), then p(a) = 0.
• If p(a) = 0, then (x - a) is a factor of p(x).

Algebraic Identities

• (x + y)^2 = x^2 + 2xy + y^2.
• (x - y)^2 = x^2 - 2xy + y^2.
• x^2 + y^2 = (x + y)^2 - 2xy.
• (x + y)(x - y) = x^2 - y^2.

Zeroes of a Polynomial

• A zero of a polynomial p(x) is a value of x that makes the polynomial equal to zero.
• The number of zeroes of a polynomial is equal to its degree.