Proofs of Irrationality (Syllabus) MCQs Quiz | Class 10

Understanding Proofs of Irrationality

Irrational numbers are a fascinating part of the number system, representing real numbers that cannot be expressed as a simple fraction p/q. Proving a number is irrational often involves an elegant technique called proof by contradiction, a cornerstone of mathematical reasoning.

Key Concepts

• Rational Numbers: Numbers that can be expressed as p/q, where p and q are integers and q is not equal to 0. Their decimal expansions are either terminating (e.g., 0.5) or non-terminating but repeating (e.g., 0.333…).
• Irrational Numbers: Numbers that cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating. Famous examples include sqrt(2), pi, and Euler’s number ‘e’.
• Proof by Contradiction (Reductio ad Absurdum): This is the primary method for proving irrationality.
  1. Assume the opposite: Start by assuming the number in question is rational.
  2. Derive a contradiction: Use logical steps and established mathematical properties (such as the Fundamental Theorem of Arithmetic) to arrive at a statement that is demonstrably false or contradicts the initial assumption.
  3. Conclude: Since the assumption leads to an absurdity or contradiction, the original assumption must be false. Therefore, the number must be irrational.

Standard Proofs of Irrationality (Examples)

Proof for sqrt(2)

This is a classic example of proof by contradiction:

1. Assume sqrt(2) is rational. This means sqrt(2) can be written as a fraction a/b, where ‘a’ and ‘b’ are coprime integers (meaning they have no common factors other than 1, and b is not 0).
2. Square both sides: 2 = a^2 / b^2, which rearranges to a^2 = 2b^2.
3. This equation implies that a^2 is an even number. If a^2 is even, then ‘a’ itself must also be an even number (because the square of an odd number is always odd).
4. Since ‘a’ is even, we can write a = 2c for some integer ‘c’.
5. Substitute a = 2c back into a^2 = 2b^2: (2c)^2 = 2b^2, which simplifies to 4c^2 = 2b^2, and further to 2c^2 = b^2.
6. This new equation means b^2 is an even number. If b^2 is even, then ‘b’ must also be an even number.
7. Contradiction! We initially assumed that ‘a’ and ‘b’ are coprime (have no common factors other than 1). However, we have deduced that both ‘a’ and ‘b’ must be even, meaning they both have a common factor of 2. This directly contradicts our initial assumption.
8. Conclusion: Our initial assumption that sqrt(2) is rational must be false. Therefore, sqrt(2) is irrational.

Proof for numbers like (rational +/- sqrt(irrational))

To prove that 3 + sqrt(2) is irrational:

1. Assume 3 + sqrt(2) is rational. Let 3 + sqrt(2) = r, where ‘r’ is some rational number.
2. Rearrange the equation to isolate the irrational term: sqrt(2) = r – 3.
3. Since ‘r’ is rational and ‘3’ is also rational, their difference (r – 3) must also be a rational number.
4. This implies that sqrt(2) is rational.
5. Contradiction! We already know (from the previous proof) that sqrt(2) is an irrational number. This contradicts our conclusion that sqrt(2) is rational.
6. Conclusion: Our initial assumption that 3 + sqrt(2) is rational must be false. Therefore, 3 + sqrt(2) is irrational.

Proof Structure and Reasoning

• Axiomatic Basis: These proofs fundamentally rely on the properties of integers, rational numbers, and particularly the unique prime factorization of integers (Fundamental Theorem of Arithmetic).
• Logical Deduction: Each step in a proof must logically follow from the previous statements or established mathematical truths. The chain of reasoning must be impeccable.
• Reductio ad Absurdum: The power of contradiction lies in its ability to prove a statement by showing that its negation is impossible or absurd.
• Fundamental Theorem of Arithmetic: This theorem is crucial. It states that every integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers, and this representation is unique (apart from the order of the factors). This is why, if a number squared is a multiple of a prime ‘p’, then the number itself must also be a multiple of ‘p’.

Quick Revision Points

• Irrational numbers cannot be written in the form p/q.
• Proof by contradiction is the primary method to prove irrationality.
• The process involves assuming the number is rational and then deriving a logical contradiction.
• Prime factorization plays a vital role in proofs involving square roots of non-perfect squares.
• The sum, difference, product, or quotient of a non-zero rational number and an irrational number is always irrational.

Extra Practice Questions

1. Prove that sqrt(3) is an irrational number.
2. Show that 1/sqrt(2) is irrational.
3. Prove that 5 + 2 * sqrt(3) is an irrational number.
4. Is the number 0.235235235… (where 235 repeats) rational or irrational? Justify your answer.
5. If ‘a’ is a rational number and ‘b’ is an irrational number, what type of number is ‘a/b’ (assuming b is not zero)? Prove your answer.