The exact square roots of 113 are positive and negative the square root of 113, represented precisely as $\sqrt{113}$ and $-\sqrt{113}$.
Understanding the Roots of 113
When we talk about the "roots of 113," we are referring to its square roots. A square root of a number is a value that, when multiplied by itself, yields the original number. Every positive number possesses two real square roots: one positive and one negative.
Since 113 is a prime number, it is not a perfect square (meaning it cannot be expressed as the product of an integer multiplied by itself). This characteristic significantly impacts its square roots, preventing them from being simple whole numbers or fractions.
The Irrational Nature of $\sqrt{113}$
The square root of 113 is classified as an irrational number. This means that its decimal representation is non-terminating and non-repeating, extending infinitely without any discernible pattern. Therefore, it cannot be precisely expressed as a simple fraction.
Although an exact decimal value is impossible to state, we can provide an approximation for practical purposes. The square root of 113 is approximately 10.63. It is important to note that this value is a rounded estimate, not the precise, exact answer. For true exactness in mathematical contexts, the radical form ($\sqrt{113}$) is always used.
Key Characteristics of the Square Roots of 113
Here are the main properties of the square roots of 113:
- Exact Forms:
- Positive square root: $\sqrt{113}$
- Negative square root: $-\sqrt{113}$
- Numerical Classification: Both are real numbers.
- Rationality: They are definitively irrational numbers.
- Approximate Value: Approximately $\pm 10.63$.
Finding the Approximate Value
To estimate the square root of 113, you can consider perfect squares that are close to 113:
- $10^2 = 100$
- $11^2 = 121$
Since 113 falls numerically between 100 and 121, its square root must logically be between 10 and 11. Given that 113 is closer to 121 than to 100, its square root is expected to be closer to 11, which aligns with the approximate value of 10.63.
