Question

Put the steps in order to the proof that the sum t=r+s is irrational if r is a rational and s is rational in the correct sequence.
Step 1. Subtract, writing s as a fraction.
Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.
Step 3. Realize a contradiction.
Step 4. Find a common denominator.

Answer 1

To order the steps for proving that the sum \( t = r + s \) is irrational if \( r \) is rational and \( s \) is rational, you would arrange the steps as follows:

1. **Step 2.** For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.
2. **Step 1.** Subtract, writing \( s \) as a fraction.
3. **Step 4.** Find a common denominator.
4. **Step 3.** Realize a contradiction.

This order follows the logical flow of assuming \( t \) is rational, expressing the relevant quantities as fractions, manipulating them to find a common basis for comparison, and ultimately arriving at a contradiction that shows \( t \) cannot be rational.