No, square roots are not always integers.
The question of whether square roots are integers depends on the number whose square root we are considering. According to the provided reference, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. However, the square roots of positive integers that are not perfect squares are irrational numbers.
Understanding Perfect Squares
A perfect square is an integer that can be obtained by squaring another integer. For instance:
- 0 is a perfect square because 0 * 0 = 0
- 1 is a perfect square because 1 * 1 = 1
- 4 is a perfect square because 2 * 2 = 4
- 9 is a perfect square because 3 * 3 = 9
- 16 is a perfect square because 4 * 4 = 16
Square Roots of Perfect Squares
The square root of a perfect square is an integer. For example:
- √0 = 0
- √1 = 1
- √4 = 2
- √9 = 3
- √16 = 4
Square Roots of Non-Perfect Squares
If we take the square root of a positive integer that is not a perfect square (like 2, 3, 5, 6, 7, 8, 10, etc.), the result is an irrational number. Irrational numbers are numbers that cannot be expressed as a simple fraction and have a non-repeating, non-terminating decimal representation.
For instance:
- √2 ≈ 1.41421356...
- √3 ≈ 1.73205081...
- √5 ≈ 2.23606797...
These decimals go on forever without repeating any pattern.
Summary
| Number | Square Root | Integer? |
|---|---|---|
| 0 | 0 | Yes |
| 1 | 1 | Yes |
| 2 | ≈ 1.414 | No |
| 3 | ≈ 1.732 | No |
| 4 | 2 | Yes |
| 5 | ≈ 2.236 | No |
| 9 | 3 | Yes |
| 16 | 4 | Yes |
In conclusion, only the square roots of perfect squares are integers. The square roots of other positive integers are irrational numbers.
