Real Numbers and Irrational Numbers
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Questions and Answers

What is the characteristic of an irrational number?

  • Rational number
  • Finite decimal or fraction
  • Repeating and terminating decimal
  • Non-repeating and non-terminating decimal (correct)

    • The Fundamental Theorem of Arithmetic states that every prime number can be expressed as a product of composite numbers.

    False

    What is the purpose of the Fundamental Theorem of Arithmetic?

    To find the prime factorization of a number, simplify fractions, find the greatest common divisor (GCD), and solve Diophantine equations.

    A decimal expansion that has an infinite number of digits after the decimal point is called a ______________________ decimal.

    <p>non-terminating</p> Signup and view all the answers

    What is a real number?

    <p>A combination of rational and irrational numbers</p> Signup and view all the answers

    Irrational numbers can be expressed as a finite decimal or fraction.

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

    What is an example of an irrational number?

    <p>π, e, √2, etc.</p> Signup and view all the answers

    Euclid's Lemma states that if a prime number divides a product of two numbers, it must divide one of the ______________________ numbers.

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

    Match the following with their characteristics:

    <p>Terminating decimal = Has a finite number of digits after the decimal point Non-terminating decimal = Has an infinite number of digits after the decimal point Recurring decimal = Non-terminating decimal that repeats in a predictable pattern Non-recurring decimal = Non-terminating decimal that does not repeat in a predictable pattern</p> Signup and view all the answers

    What is the purpose of the Unique Factorization Theorem?

    <p>All of the above</p> Signup and view all the answers

    Study Notes

    Real Numbers

    • Real numbers are a combination of rational and irrational numbers.
    • They can be represented on the number line.
    • Real numbers can be either rational or irrational.

    Irrational Numbers

    • Irrational numbers are non-repeating, non-terminating decimals.
    • Examples: π, e, √2, etc.
    • Irrational numbers cannot be expressed as a finite decimal or fraction.
    • They have an infinite number of digits that never repeat in a predictable pattern.

    Decimal Expansions

    • Decimal expansions are ways to represent numbers in base 10.
    • Decimal expansions can be terminating or non-terminating.
    • Terminating decimals have a finite number of digits after the decimal point.
    • Non-terminating decimals have an infinite number of digits after the decimal point.
    • Non-terminating decimals can be recurring (repeating) or non-recurring (non-repeating).

    Fundamental Theorem of Arithmetic

    • The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way.
    • This theorem is also known as the Unique Factorization Theorem.
    • The theorem can be used to find the prime factorization of a number.
    • Prime factorization is used to simplify fractions, find the greatest common divisor (GCD), and solve Diophantine equations.

    Key Points

    • Euclid's Lemma: If a prime number divides a product of two numbers, it must divide one of the two numbers.
    • The Fundamental Theorem of Arithmetic is used to prove that the square root of 2 is irrational.
    • Irrational numbers cannot be expressed as a finite decimal or fraction.

    Real Numbers

    • Real numbers are a combination of rational and irrational numbers and can be represented on the number line.
    • They can be either rational or irrational.

    Irrational Numbers

    • Irrational numbers are non-repeating, non-terminating decimals with infinite digits that never repeat in a predictable pattern.
    • Examples of irrational numbers include π, e, and √2.
    • They cannot be expressed as a finite decimal or fraction.

    Decimal Expansions

    • Decimal expansions are ways to represent numbers in base 10.
    • They can be terminating or non-terminating, with terminating decimals having a finite number of digits after the decimal point.
    • Non-terminating decimals have an infinite number of digits after the decimal point and can be recurring (repeating) or non-recurring (non-repeating).

    Fundamental Theorem of Arithmetic

    • The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way.
    • This theorem is also known as the Unique Factorization Theorem.
    • It can be used to find the prime factorization of a number, which is used to simplify fractions, find the greatest common divisor (GCD), and solve Diophantine equations.

    Key Points

    • Euclid's Lemma states that if a prime number divides a product of two numbers, it must divide one of the two numbers.
    • The Fundamental Theorem of Arithmetic is used to prove that the square root of 2 is irrational.
    • Irrational numbers cannot be expressed as a finite decimal or fraction.

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    Description

    This quiz covers the basics of real numbers, including irrational numbers, their properties, and decimal expansions.

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