Whether a square root is a rational number depends on the number under the radical symbol.
Here's the breakdown:
-
Square roots of perfect squares are rational numbers. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). The square root of a perfect square is an integer, and all integers are rational numbers because they can be expressed as a fraction with a denominator of 1 (e.g., 2 = 2/1).
- Example: √4 = 2 (rational)
- Example: √9 = 3 (rational)
- Example: √16 = 4 (rational)
-
Square roots of numbers that are not perfect squares are irrational numbers. These numbers, when expressed as decimals, neither terminate nor repeat.
- Example: √2 ≈ 1.41421356... (irrational)
- Example: √3 ≈ 1.7320508... (irrational)
- Example: √5 ≈ 2.2360679... (irrational)
In Summary:
- Rational: The square root of a number that results in an integer or a fraction.
- Irrational: The square root of a number that results in a non-repeating, non-terminating decimal.
