Number Theory Basics

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Questions and Answers

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?

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?

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?

It is unique, but only up to the order of the prime factors. (D) Signup and view all the answers

What is the importance of irrational numbers in mathematics?

They are used to represent quantities that cannot be expressed exactly as a finite decimal or a ratio of integers. (A) Signup and view all the answers

What is the set of irrational numbers?

An infinite and uncountable set. (A) Signup and view all the answers

What is the relationship between the order of the prime factors in the prime factorization of a positive integer?

The order of the prime factors does not matter. (B) Signup and view all the answers

What is the fundamental property of electric charge that explains why like charges repel and opposite charges attract each other?

It is quantized, meaning it comes in discrete packets (quanta) rather than being continuous. Signup and view all the answers

According to Ohm's Law, what is the relationship between voltage, current, and resistance in a conductor?

V = I × R, where V is the potential difference, I is the current, and R is the resistance. Signup and view all the answers

What are the three factors that affect the resistance of a conductor?

The material, length, and cross-sectional area of the conductor. Signup and view all the answers

What is the difference between a series circuit and a parallel circuit?

In a series circuit, devices are connected one after the other, whereas in a parallel circuit, devices are connected between the same two points. Signup and view all the answers

What is the function of a capacitor in an electric circuit?

It stores electric energy. Signup and view all the answers

What is the SI unit of electric charge, and what is the significance of this unit?

The coulomb (C), which is a fundamental unit that quantizes electric charge. Signup and view all the answers

How does the length of a conductor affect its resistance?

The resistance is directly proportional to the length of the conductor. Signup and view all the answers

What is the role of an inductor in an electric circuit?

It stores magnetic energy. Signup and view all the answers

What is the purpose of a resistor in an electric circuit?

It reduces voltage and current. Signup and view all the answers

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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.