10 Questions
What is the characteristic of an irrational number?
Non-repeating and non-terminating decimal
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.
non-terminating
What is a real number?
A combination of rational and irrational numbers
Irrational numbers can be expressed as a finite decimal or fraction.
False
What is an example of an irrational number?
π, e, √2, etc.
Euclid's Lemma states that if a prime number divides a product of two numbers, it must divide one of the ______________________ numbers.
two
Match the following with their characteristics:
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
What is the purpose of the Unique Factorization Theorem?
All of the above
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.
This quiz covers the basics of real numbers, including irrational numbers, their properties, and decimal expansions.
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