To determine whether \( \pi + 2 \) is rational or irrational, we need to consider the properties of the numbers involved.
A rational number is defined as any number that can be expressed as the quotient of two integers (i.e., in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \neq 0 \)). An irrational number, on the other hand, cannot be expressed in such a way.
We know that \( \pi \) is an irrational number. This means that it cannot be expressed as a fraction of two integers.
Now, if we add a rational number (which is \( 2 \), as it can be expressed as \( \frac{2}{1} \)) to an irrational number (which is \( \pi \)), the resulting sum is still irrational. The sum of a rational number and an irrational number is always irrational.
Therefore, since \( \pi + 2 \) is the sum of the irrational number \( \pi \) and the rational number \( 2 \), we can conclude that:
\[ \pi + 2 \text{ is irrational.} \]