Real Numbers and Arithmetic Theorems
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Questions and Answers

What is the set of numbers that includes all rational and irrational numbers?

  • Whole numbers
  • Integers
  • Real numbers (correct)
  • Natural numbers

    • What is the purpose of Euclid's Division Lemma?

  • To find the prime factorization of a number
  • To find the HCF of two numbers (correct)
  • To represent numbers on a number line
  • To express irrational numbers as a ratio of integers

    • What is a characteristic of irrational numbers?

  • They are not real numbers
  • They cannot be expressed as a ratio of integers (correct)
  • They can be represented as terminating decimals
  • They can be expressed as a ratio of integers

    • What is the decimal expansion of a rational number?

    <p>Either terminating or non-terminating, recurring</p> Signup and view all the answers

    What is a repeating decimal?

    <p>A decimal that has a block of digits that repeats infinitely</p> Signup and view all the answers

    What rule should be followed when performing operations on real numbers?

    <p>The order of operations (PEMDAS) should be followed</p> Signup and view all the answers

    What is the general form of a polynomial?

    <p>a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0</p> Signup and view all the answers

    What is the degree of the polynomial 3x^4 + 2x^2 + x?

    <p>4</p> Signup and view all the answers

    What is a monomial?

    <p>A polynomial with only one term</p> Signup and view all the answers

    What is the Remainder Theorem?

    <p>If p(x) is divided by (x - a) and the remainder is r, then p(a) = r</p> Signup and view all the answers

    What is an algebraic identity?

    <p>An expression that is always equal to another expression</p> Signup and view all the answers

    What is a zero of a polynomial?

    <p>A value of x that makes the polynomial equal to zero</p> Signup and view all the answers

    Study Notes

    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 &lt; 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 &lt; 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.

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    Description

    Explore the basics of real numbers, Euclid's Division Lemma, and the Fundamental Theorem of Arithmetic. Learn how to represent real numbers and apply theorems to find the HCF of two numbers.

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