To which subsets of real numbers does the number √34 belong?
a. Whole numbers, integers, rational numbers
b. Whole numbers, natural numbers, integers
c. Irrational numbers
d. Rational numbers
The Correct Answer and Explanation is:
The correct answer is:
c. Irrational numbers
Explanation:
To understand why ( \sqrt{34} ) is classified as an irrational number, we first need to consider what each subset of real numbers represents. Real numbers can be broken down into categories based on their properties:
- Natural Numbers (Counting Numbers): These are positive whole numbers starting from 1, such as ( 1, 2, 3, ) and so on. They do not include zero, fractions, decimals, or negative numbers.
- Whole Numbers: This set includes all natural numbers and adds zero. It includes numbers like ( 0, 1, 2, 3, ) etc., without fractions or decimals.
- Integers: Integers include all whole numbers as well as their negative counterparts, such as ( -3, -2, -1, 0, 1, 2, 3, ) etc. Integers are still whole values, so they do not include fractions or decimals.
- Rational Numbers: A rational number is any number that can be expressed as the ratio of two integers, where the denominator is not zero. For example, numbers like ( \frac{1}{2}, -3, 0.75, ) and ( 4 ) are all rational because they can be represented as fractions of integers. Rational numbers include terminating decimals and repeating decimals.
- Irrational Numbers: An irrational number cannot be expressed as a simple fraction of two integers. Its decimal form is non-terminating and non-repeating. Examples include numbers like ( \pi, \sqrt{2}, ) and ( e ).
When evaluating ( \sqrt{34} ), notice that it does not simplify to an integer, as 34 is not a perfect square (numbers such as 1, 4, 9, 16, and 25 are perfect squares because they result from squaring whole numbers). Since ( \sqrt{34} ) does not have an exact integer root, it is neither a whole number nor an integer.
Next, we examine whether ( \sqrt{34} ) can be rational. To be rational, ( \sqrt{34} ) would need to be expressible as a fraction (ratio) of two integers. However, it does not meet this criterion, as its decimal representation (approximately 5.831) is non-terminating and non-repeating. Thus, ( \sqrt{34} ) cannot be simplified into a fraction of two integers, making it an irrational number.
In conclusion, because ( \sqrt{34} ) does not belong to the subsets of whole numbers, integers, or rational numbers, it is best classified solely as an irrational number.