To determine the sets of numbers to which [tex]\(\sqrt{14}\)[/tex] belongs, we need to consider the properties of this number. Here’s a step-by-step explanation:
1. Real Numbers:
- Definition: Real numbers include all the numbers on the number line. This encompasses both rational numbers (like [tex]\(2\)[/tex], [tex]\(\frac{1}{2}\)[/tex], and [tex]\(\sqrt{4}\)[/tex]) and irrational numbers (like [tex]\(\pi\)[/tex] and [tex]\(\sqrt{2}\)[/tex]).
- Conclusion: Since [tex]\(\sqrt{14}\)[/tex] can be represented on the number line and has a specific position (approximately [tex]\(3.7417\)[/tex]), it is a real number.
2. Irrational Numbers:
- Definition: Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. Their decimal representations are non-terminating and non-repeating.
- Conclusion: Since [tex]\(14\)[/tex] is not a perfect square, its square root [tex]\(\sqrt{14}\)[/tex] cannot be expressed as a fraction, implying that it is an irrational number.
In summary:
- [tex]\(\sqrt{14}\)[/tex] is a real number because it is a point on the real number line.
- [tex]\(\sqrt{14}\)[/tex] is an irrational number because it cannot be expressed as a ratio of two integers and its decimal expansion is non-terminating and non-repeating.
So, [tex]\(\sqrt{14}\)[/tex] belongs to the sets of real numbers and irrational numbers.
