Questions and Answers
What is the statement that every composite number can be written as the product of powers of primes called?
Fundamental Theorem of Arithmetic
What is the significance of the Fundamental Theorem of Arithmetic in the study of integers?
It is of basic crucial importance
What is the condition under which the factorization of a composite number as a product of primes is unique?
Apart from the order in which the prime factors occur
Who is credited with giving the first correct proof of the Fundamental Theorem of Arithmetic?
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Can all natural numbers be factorized as a product of powers of primes?
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Study Notes
Real Numbers
- The chapter begins with a review of real numbers, building upon the discussion of irrational numbers in Class IX.
- Two important properties of positive integers are introduced: Euclid's Division Algorithm and the Fundamental Theorem of Arithmetic.
Euclid's Division Algorithm
- The algorithm states that any positive integer a can be divided by another positive integer b, leaving a remainder r that is smaller than b.
- This result is similar to the long division process and has many applications related to the divisibility properties of integers.
Fundamental Theorem of Arithmetic
- The theorem states that every composite number can be expressed as a product of primes in a unique way.
- This result is easy to state but has significant and deep applications in mathematics.
- The theorem is used to prove the irrationality of certain numbers and to explore the decimal expansions of rational numbers.
Applications of the Fundamental Theorem of Arithmetic
- The theorem is used to prove the irrationality of numbers such as √2, √3, and √5.
- It is also used to determine when the decimal expansion of a rational number is terminating or non-terminating repeating.
Irrationality of Certain Numbers
- The irrationality of a number is proven by assuming it is rational and then reaching a contradiction.
- For example, the irrationality of 5 - √3 is proven by assuming it is rational, then rearranging the equation to show that √3 is rational, which is a contradiction.
Exercises
- Prove that √5 is irrational.
- Prove that 3 + √2 is irrational.
- Prove that certain numbers are irrational, such as 1/√2, 7/√5, and 6 + √2.
Summary of the Chapter
- The chapter covers the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of primes in a unique way.
- The theorem is used to prove the irrationality of certain numbers and to explore the decimal expansions of rational numbers.
- The chapter also introduces Euclid's Division Algorithm, which has applications related to the divisibility properties of integers.
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