Proof by Contradiction Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Prove by contradiction that there is no smallest positive rational number.

Solution

  1. 1
    Assume for contradiction that there is a smallest positive rational number, call it rr.
  2. 2
    Then r2\frac{r}{2} is also a positive rational number, and r2<r\frac{r}{2} < r, which contradicts the assumption that rr was the smallest.

Answer

ThereĀ isĀ noĀ smallestĀ positiveĀ rationalĀ number.\text{There is no smallest positive rational number.}
Proof by contradiction assumes the statement is false and then shows that assumption leads to an impossibility. Here the contradiction comes from constructing an even smaller positive rational number.

About Proof by Contradiction

Proof by contradiction (reductio ad absurdum) assumes the negation of what you want to prove, then derives a logical contradiction, thereby establishing that the original statement must be true.

Learn more about Proof by Contradiction ā†’

More Proof by Contradiction Examples

Example 1 medium

Prove by contradiction that [formula] is irrational.

Example 2 medium

Prove by contradiction that there are infinitely many prime numbers.

Example 3 hard

Prove by contradiction: There is no largest integer.

ā† All Proof by Contradiction Examples Proof by Contradiction Concept