Question

Is the number
–
1+
–
5 rational or irrational?

Answer 1

To determine whether the number \(-1 + \sqrt{5}\) is rational or irrational, we can analyze the components of the expression.

1. **Rational Numbers**: A rational number can be expressed as the quotient of two integers (i.e., \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b \neq 0\)).

2. **Square Root of a Non-Perfect Square**: The square root of 5, \(\sqrt{5}\), is known to be an irrational number because 5 is not a perfect square.

Now, let's break down the expression:
- We know \(-1\) is a rational number.
- We also know \(\sqrt{5}\) is irrational.

Combining a rational number and an irrational number (in this case, \(-1 + \sqrt{5}\)) will yield an irrational number. Thus, the expression \(-1 + \sqrt{5}\) is irrational.

Therefore, the number \(-1 + \sqrt{5}\) is **irrational**.