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Questions and Answers
What is the fundamental theorem of arithmetic?
What is the fundamental theorem of arithmetic?
- Every positive integer can be expressed as a sum of prime numbers in a unique way.
- Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed. (correct)
- Every positive integer can be expressed as a difference of prime numbers in a unique way.
- Every positive integer can be expressed as a ratio of prime numbers in a unique way.
What is the main characteristic of an irrational number's decimal expansion?
What is the main characteristic of an irrational number's decimal expansion?
- It is a finite decimal.
- It is a repeating decimal.
- It is a non-terminating and non-repeating decimal. (correct)
- It is a terminating decimal.
Which of the following is an example of an irrational number?
Which of the following is an example of an irrational number?
- The ratio of 3 to 4
- The sum of 1 and 2
- The square root of 2 (correct)
- The square root of 4
What can be said about the uniqueness of the prime factorization of a positive integer?
What can be said about the uniqueness of the prime factorization of a positive integer?
What is the importance of irrational numbers in mathematics?
What is the importance of irrational numbers in mathematics?
What is the set of irrational numbers?
What is the set of irrational numbers?
What is the relationship between the order of the prime factors in the prime factorization of a positive integer?
What is the relationship between the order of the prime factors in the prime factorization of a positive integer?
What is the fundamental property of electric charge that explains why like charges repel and opposite charges attract each other?
What is the fundamental property of electric charge that explains why like charges repel and opposite charges attract each other?
According to Ohm's Law, what is the relationship between voltage, current, and resistance in a conductor?
According to Ohm's Law, what is the relationship between voltage, current, and resistance in a conductor?
What are the three factors that affect the resistance of a conductor?
What are the three factors that affect the resistance of a conductor?
What is the difference between a series circuit and a parallel circuit?
What is the difference between a series circuit and a parallel circuit?
What is the function of a capacitor in an electric circuit?
What is the function of a capacitor in an electric circuit?
What is the SI unit of electric charge, and what is the significance of this unit?
What is the SI unit of electric charge, and what is the significance of this unit?
How does the length of a conductor affect its resistance?
How does the length of a conductor affect its resistance?
What is the role of an inductor in an electric circuit?
What is the role of an inductor in an electric circuit?
What is the purpose of a resistor in an electric circuit?
What is the purpose of a resistor in an electric circuit?
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Study Notes
Fundamental Theorem of Arithmetic
- Statement: Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
- Key points:
- Every positive integer has a unique prime factorization.
- The prime factorization is unique up to the order of the prime factors.
- This theorem provides a way to express every positive integer as a product of prime numbers.
Irrational Numbers
- Definition: A real number that cannot be expressed as a finite decimal or a ratio of integers (i.e., it cannot be expressed as a simple fraction).
- Key properties:
- Irrational numbers have non-terminating and non-repeating decimal expansions.
- Irrational numbers cannot be expressed as a finite decimal.
- The set of irrational numbers is infinite and uncountable.
- Examples:
- π (pi) is an irrational number.
- e (Euler's number) is an irrational number.
- The square root of 2 is an irrational number.
- Importance: Irrational numbers play a crucial role in many mathematical concepts, such as algebra, geometry, and calculus. They are used to represent quantities that cannot be expressed exactly as a finite decimal or a ratio of integers.
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