write the decimal expansion of root 2 , root 3 , root 5
The correct answer and explanation is:
The decimal expansions of the square roots of 2, 3, and 5 are as follows:
- √2 ≈ 1.414213562
- √3 ≈ 1.732050808
- √5 ≈ 2.236067977
These values are irrational numbers, meaning they cannot be expressed as a finite decimal or a simple fraction. Their decimal expansions go on indefinitely without repeating.
Explanation:
The square roots of numbers like 2, 3, and 5 are irrational because there are no exact fractions that represent their values. When attempting to calculate their square roots, the result is a non-repeating and non-terminating decimal. For example:
- √2: The square root of 2 is approximately 1.414213562. This value was famously proven to be irrational by the ancient Greeks using a method of contradiction. The decimal never ends or repeats.
- √3: Similarly, the square root of 3 is approximately 1.732050808, and like √2, it also has a non-repeating, non-terminating decimal. The irrationality of √3 was established in much the same way.
- √5: The square root of 5 is approximately 2.236067977. It also cannot be expressed as a fraction of two integers and has an infinite decimal expansion.
Mathematically, an irrational number cannot be accurately written down in decimal form beyond a certain number of digits. However, these approximations are commonly used when precise calculations are not required. To get more digits for more accuracy, one would need to use a calculator or a numerical method to compute the square roots to the desired precision.
These irrational square roots are essential in many areas of mathematics and physics, and they often appear in geometric problems, such as those involving the Pythagorean theorem.