Which statement is correct for ( \sqrt{21} ) and (4.59) on the number line?

Correct answer: ( \sqrt{21}<4.59 )

The correct answer is option A: \(\sqrt{21}<4.59\). Both numbers are positive, so compare their squares. We have \((\sqrt{21})^2=21\), and \(4.59^2=21.0681\). Since \(21<21.0681\), the original positive numbers satisfy \(\sqrt{21}<4.59\). More precisely, \(\sqrt{21}\approx4.582575\), which is just less than 4.59. Option A is therefore correct. Option B reverses the inequality. Option C is wrong because 4.59 is not exactly \(\sqrt{21}\); its square is 21.0681 rather than 21. Option D claims 4.59<4, which is false because 4.59 is greater than 4. On a number line, the square-root point is slightly to the left of 4.59. Memory cue: when comparing positive approximations, squaring both sides is safe and often avoids rounding mistakes.