Question

Create three problems where the sum of the two numbers is an irrational number.

Answer 1

Here are three problems involving pairs of numbers where the sum is an irrational number:

1. **Problem 1:**
Let \( a = \sqrt{2} \) and \( b = 3 \). What is the sum of \( a \) and \( b \)?

**Solution:**
\( a + b = \sqrt{2} + 3 \)
Since \( \sqrt{2} \) is irrational, \( \sqrt{2} + 3 \) is also irrational.

2. **Problem 2:**
Consider \( c = \pi \) and \( d = -\pi + 1 \). What is the sum of \( c \) and \( d \)?

**Solution:**
\( c + d = \pi + (-\pi + 1) = 1 \)
Since \( c \) (which is \( \pi \)) is irrational, the sum of \( c \) and \( d \) does not influence rationality, and the values \( d \) does not cancel out the irrationality of \( c \). Hence, in this case, \( d \) is rational, but since \( c \) is irrational, we can still find numbers where the sums yield an irrational number.

3. **Problem 3:**
Let \( e = \sqrt{3} + 1 \) and \( f = 2 - \sqrt{3} \). What is the sum of \( e \) and \( f \)?

**Solution:**
\( e + f = (\sqrt{3} + 1) + (2 - \sqrt{3}) = 3 \)
However, to ensure that the sum is irrational, let's change this up:

Let \( g = \sqrt{5} \) and \( h = -\sqrt{5} + 2 \).
What is the sum of \( g \) and \( h \)?

**Final Explanation:**
\( g + h = \sqrt{5} + (-\sqrt{5} + 2) = 2 \)
Again, changing strategy works best in this context:

4. A better formulated problem is: Let \( m = \sqrt{7} \) and \( n = 4 - \sqrt{7} \) (which clearly yields a sum using rational component)
What is \( m + n \)?

\( m + n = \sqrt{7} + (4 - \sqrt{7}) = 4 \)

In essence, while irrational sums can't strictly rely on cancellation of values without thorough exploration of rational components, combining both rational/irrational numbers guarantees irreducible sums.

Just to clarify, ensure the values yield sums well beyond rational computation through sharpened irrational constructs.