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Questions and Answers
What does the Fundamental Theorem of Arithmetic state?
What does the Fundamental Theorem of Arithmetic state?
- Every integer can be expressed as a sum of primes.
- Every composite number can be factorized uniquely into primes. (correct)
- Every prime number can be expressed as a sum of composites.
- All integers are either prime or composite.
Define a composite number.
Define a composite number.
A composite number is a positive integer that has at least one positive divisor other than one or itself.
If a prime number divides a square, it must also divide the original number.
If a prime number divides a square, it must also divide the original number.
True (A)
Match the following terms with their definitions:
Match the following terms with their definitions:
The number ____ is an example of a prime number.
The number ____ is an example of a prime number.
Which of the following is an example of a composite number?
Which of the following is an example of a composite number?
The number 1 is considered a composite number.
The number 1 is considered a composite number.
Give an example of a number that is both prime and irrational.
Give an example of a number that is both prime and irrational.
What does the Fundamental Theorem of Arithmetic state?
What does the Fundamental Theorem of Arithmetic state?
The first correct proof of the Fundamental Theorem of Arithmetic was given by Archimedes.
The first correct proof of the Fundamental Theorem of Arithmetic was given by Archimedes.
What unique property does the factorization of a composite number have according to the Fundamental Theorem of Arithmetic?
What unique property does the factorization of a composite number have according to the Fundamental Theorem of Arithmetic?
The prime factorization of 32760 is __________.
The prime factorization of 32760 is __________.
Match the mathematicians to their contributions.
Match the mathematicians to their contributions.
Which of the following numbers is a composite number?
Which of the following numbers is a composite number?
List one composite number and its prime factorization.
List one composite number and its prime factorization.
The numbers 3803 and 3607 are primes.
The numbers 3803 and 3607 are primes.
What does the Fundamental Theorem of Arithmetic state?
What does the Fundamental Theorem of Arithmetic state?
The Euclid’s division algorithm can be used to compute the LCM of two numbers.
The Euclid’s division algorithm can be used to compute the LCM of two numbers.
What is a composite number?
What is a composite number?
Every prime number greater than 2 is __________.
Every prime number greater than 2 is __________.
Match the type of number with its definition:
Match the type of number with its definition:
Which of the following is a prime number?
Which of the following is a prime number?
All prime numbers are also composite numbers.
All prime numbers are also composite numbers.
What is the significance of the number 1 in the context of prime and composite numbers?
What is the significance of the number 1 in the context of prime and composite numbers?
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Study Notes
Introduction to Real Numbers
- Exploration of real numbers continues, following the introduction to irrational numbers in Class IX.
- Focus on important properties of positive integers: Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s Division Algorithm
- A positive integer ( a ) can be divided by another positive integer ( b ) such that the remainder ( r ) is less than ( b ).
- Fundamental for understanding divisibility among integers.
- Utilized primarily to compute the highest common factor (HCF) of two positive integers.
Fundamental Theorem of Arithmetic
- Every composite number can be expressed uniquely as a product of prime numbers, except for the order of factors.
- Provides a basis for proving the irrationality of certain numbers (e.g., ( \sqrt{2}, \sqrt{3}, \sqrt{5} )).
- Important for determining whether the decimal expansion of a rational number is terminating or non-terminating repeating.
Proof of Irrationality
- To show that ( \sqrt{5} - \sqrt{3} ) is irrational, assume it is rational, leading to a contradiction with the properties of integers.
- Similar proof structure used to demonstrate ( \sqrt{3}^2 ) is irrational, using the assumption of rationality and ending in contradiction.
Examples and Exercises
- Exercises encourage proving various forms of irrational numbers:
- ( \sqrt{5} )
- ( 3 + 2\sqrt{5} )
- Other combinations of irrational numbers.
Summary of Key Concepts
- Fundamental Theorem of Arithmetic:
- Unique prime factorization of composite numbers.
- Example: ( 32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13 ).
- Properties of Primes:
- If ( p ) is prime and divides ( a^2 ), then ( p ) divides ( a ).
- Historical Context:
- Theorem first recorded by Euclid and formally proven by Carl Friedrich Gauss, a pivotal figure in mathematics.
Significance of the Theorem
- Provides an essential foundation for number theory and integer arithmetic.
- Highlights the uniqueness of prime factorization and its broad applications in mathematics and science.
Important Figures
- Carl Friedrich Gauss (1777 – 1855): Known as the "Prince of Mathematicians," recognized for his extensive contributions to mathematics and science.
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