Is There a Smallest Positive Rational Number? Proof by Contradiction

Debunking the Myth of the Smallest Rational Number: A Proof by Contradiction

Can you pinpoint the smallest positive fraction? It seems impossible, right? That's because there isn't one! Let's rigorously prove this using the powerful tool of proof by contradiction.

The Setup:

1. Assume the opposite: Suppose, for the sake of contradiction, that there is a smallest positive rational number. Let's call this number 'x'.

2. Rational Representation: Since 'x' is a rational number, we can express it as a fraction 'p/q', where 'p' and 'q' are positive integers sharing no common divisors other than 1 (they're relatively prime), and 'q' is not equal to 0.

The Contradiction:

1. Construct a Smaller Number: Now, consider the number 'y = x/2'. Since 'x' is positive, 'y' must also be positive. We can write 'y' as '(p/2)/q', which simplifies to 'p/(2q)'.

2. Rational and Smaller: This form of 'y' reveals that it is also a positive rational number. Moreover, 'y' is clearly smaller than 'x' because we defined it as 'x/2'.

3. The Conflict: We've stumbled upon a contradiction! Our initial assumption claimed that 'x' was the smallest positive rational number. However, we just constructed a smaller positive rational number, 'y'. This means our initial assumption must be false.

The Conclusion:

Because our assumption led to a contradiction, we can confidently conclude that there cannot be a smallest positive rational number. Q.E.D.