To determine whether \(\sqrt{43}\) is rational or irrational, we can analyze the nature of the number 43.
A rational number is defined as a number that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not zero. An irrational number cannot be written in this way.
The square root of a natural number is rational if and only if the number is a perfect square. A perfect square is an integer that is the square of another integer.
To check if 43 is a perfect square, we list the perfect squares of integers close to \(\sqrt{43}\):
- \(6^2 = 36\)
- \(7^2 = 49\)
Since \(43\) lies between \(36\) and \(49\), it is not a perfect square. Therefore, \(\sqrt{43}\) cannot be expressed as a fraction of two integers.
Thus, \(\sqrt{43}\) is an irrational number.
In conclusion, \(\sqrt{43}\) is irrational because 43 is not a perfect square.