To which subsets of the real numbers does the number square rootof 42 belong? rational numbers irrational numbers whole numbers, integers, rational numbers whole numbers, natural numbers, integers
The Correct Answer and Explanation is:
The number 42\sqrt{42}42 belongs to the subset of irrational numbers.
Here’s why:
- Rational Numbers: A number is considered rational if it can be expressed as the ratio of two integers, i.e., in the form pq\frac{p}{q}qp, where ppp and qqq are integers, and q≠0q \neq 0q=0. However, 42\sqrt{42}42 cannot be written as a fraction of two integers. Since 42 is not a perfect square (it doesn’t have an integer square root), 42\sqrt{42}42 is an irrational number, not a rational one.
- Irrational Numbers: These are numbers that cannot be expressed as the ratio of two integers. The square root of any non-perfect square (like 42) is always irrational because the decimal expansion goes on forever without repeating. For example, 42≈6.4807\sqrt{42} \approx 6.480742≈6.4807, and this goes on without repeating in a predictable pattern. Therefore, 42\sqrt{42}42 is irrational.
- Whole Numbers: Whole numbers are the set of non-negative integers (0, 1, 2, 3, …). Since 42\sqrt{42}42 is not an integer, it is not a whole number.
- Integers: Integers are numbers that can be positive, negative, or zero, and they do not have any fractional or decimal parts. Again, 42\sqrt{42}42 is not an integer because it is not a whole number and is a non-repeating decimal.
- Natural Numbers: Natural numbers are the set of positive integers (1, 2, 3, …). Since 42\sqrt{42}42 is not a positive integer, it is not a natural number.
In conclusion, 42\sqrt{42}42 belongs only to the set of irrational numbers.
