To which subsets of the real numbers does the number belong? (square root) 46
The correct answer and explanation is:
The number 46\sqrt{46} belongs to the subset of real numbers called irrational numbers.
Explanation:
The set of real numbers is composed of several subsets, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. To understand where 46\sqrt{46} fits, let’s break down each subset:
- Natural Numbers (N\mathbb{N}): These are the counting numbers starting from 1 (e.g., 1, 2, 3, …). Since 46\sqrt{46} is not a whole number, it does not belong to this set.
- Whole Numbers: This set includes all natural numbers and zero (e.g., 0, 1, 2, 3, …). Again, since 46\sqrt{46} is not an integer, it is not part of this set.
- Integers: The integers consist of all positive and negative whole numbers, including zero (e.g., -3, -2, -1, 0, 1, 2, 3, …). Since 46\sqrt{46} is not a whole number, it is not an integer either.
- Rational Numbers (Q\mathbb{Q}): Rational numbers can be written as the quotient of two integers (i.e., in the form ab\frac{a}{b}, where aa and bb are integers and b≠0b \neq 0). 46\sqrt{46} is not rational because it cannot be expressed as the ratio of two integers. The square root of a non-perfect square is irrational.
- Irrational Numbers: These are real numbers that cannot be expressed as a ratio of two integers. The square root of any prime number, or a non-perfect square, is irrational. 46\sqrt{46} is an irrational number because it does not have a simple fractional representation. It has an infinite, non-repeating decimal expansion. Specifically, 46≈6.78233…\sqrt{46} \approx 6.78233…, and this decimal continues without repeating.
Conclusion:
Since 46\sqrt{46} cannot be expressed as a fraction and has a non-repeating, infinite decimal expansion, it belongs to the subset of irrational numbers within the real numbers.