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Proving Irrationality of Numbers

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Questions and Answers

What can be concluded when there exist co-prime positive integers a and b such that an algebraic expression equals a/b?

The algebraic expression is rational.

Why is the assumption that 2√3 - 1 is a rational number proven incorrect?

Because it leads to a contradiction that √3 is rational, which is false.

What is the correct conclusion when √5 + √3 is assumed to be a rational number?

The assumption is incorrect, and √5 + √3 is irrational.

Why is 2 - 3√5 proven to be an irrational number?

Because the assumption that it is rational leads to a contradiction that √5 is rational, which is false. Signup and view all the answers

What is the significance of a and b being co-prime positive integers in the proofs?

It ensures that the fractions formed are in their simplest form, and the numerators and denominators are integers. Signup and view all the answers

How is the expression (a/b) - √3 rationalized in the proof that √5 + √3 is irrational?

By squaring both sides of the equation. Signup and view all the answers

What is the common technique used in the proofs to establish the irrationality of algebraic expressions?

Assuming the expression is rational and arriving at a contradiction. Signup and view all the answers

Why does the expression (a² - 2b²)/2ab represent a rational number?

Because a and b are integers, making the numerator and denominator integers. Signup and view all the answers

What can be inferred about √2 + √3 based on the proof that √5 + √3 is irrational?

It is also irrational. Signup and view all the answers

What is the purpose of the proof by contradiction in these exercises?

To establish the irrationality of algebraic expressions. Signup and view all the answers