To categorize the given numbers into irrational numbers, rational numbers, real numbers, and integers, we can follow these steps:
1. Identify and List Each Set:
- Set 1: [tex]\(\{-227, 4, 19, 28\}\)[/tex]
- Set 2: [tex]\(\{\sqrt{36}, \frac{1}{6}, 0.245, -15\}\)[/tex]
- Set 3: [tex]\(\{\pi, \frac{\sqrt{3}}{2}, 6.283185, -\sqrt{6}\}\)[/tex]
- Set 4: [tex]\(\{-\frac{3}{25}, \frac{2 \pi}{3}, \sqrt{16}, \frac{2}{\sqrt{5}}\}\)[/tex]
2. Categorize Each Number:
- Integers: Numbers that have no fractional or decimal part.
- Set 1: [tex]\(-227, 4, 19, 28\)[/tex]
- Set 2: [tex]\(\sqrt{36}=6\)[/tex], [tex]\(-15\)[/tex]
- Set 3: None
- Set 4: [tex]\(\sqrt{16}=4\)[/tex]
Therefore, the integers are: [tex]\(-227, 4, 19, 28, 6, -15\)[/tex].
- Rational Numbers: Numbers that can be expressed as a fraction of two integers. Includes integers.
- Set 1: All integers (already listed)
- Set 2: [tex]\(\frac{1}{6}, 0.245 = \frac{245}{1000}, -15\)[/tex], [tex]\(\sqrt{36} = 6\)[/tex]
- Set 3: None are rational since [tex]\(\pi\)[/tex] and its multiples or roots are irrational
- Set 4: [tex]\(-\frac{3}{25}, \sqrt{16}=4\)[/tex]
Therefore, the rational numbers are: [tex]\(-227, 4, 19, 28, 6, -15, \frac{1}{6}, 0.245, -\frac{3}{25}\)[/tex].
- Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-ending decimal parts.
- Set 1: None
- Set 2: None
- Set 3: [tex]\(\pi, \frac{\sqrt{3}}{2}, 6.283185 \ldots \approx 2\pi, -\sqrt{6}\)[/tex]
- Set 4: [tex]\(\frac{2\pi}{3}, \frac{2}{\sqrt{5}}\)[/tex]
Therefore, the irrational numbers are: [tex]\(\pi, \frac{\sqrt{3}}{2}, 6.283185, -\sqrt{6}, \frac{2\pi}{3}, \frac{2}{\sqrt{5}}\)[/tex].
- Real Numbers: This category includes all rational and irrational numbers.
- All given numbers from each set that are rational or irrational are real numbers.
- Combining rational and irrational numbers provides us the list of real numbers.
Therefore, the real numbers are: [tex]\(-227, 4, 19, 28, 6, -15, \frac{1}{6}, 0.245, -\frac{3}{25}, \pi, \frac{\sqrt{3}}{2}, 6.283185, -\sqrt{6}, \frac{2\pi}{3}, \frac{2}{\sqrt{5}}\)[/tex].
4. Result Organization:
- Irrational Numbers:
- [tex]\(\pi, \frac{\sqrt{3}}{2}, 6.283185, -\sqrt{6}, \frac{2\pi}{3}, \frac{2}{\sqrt{5}}\)[/tex]
- Rational Numbers:
- [tex]\(-227, 4, 19, 28, 6, -15, \frac{1}{6}, 0.245, -\frac{3}{25}\)[/tex]
- Real Numbers:
- [tex]\(-227, 4, 19, 28, 6, -15, \frac{1}{6}, 0.245, -\frac{3}{25}, \pi, \frac{\sqrt{3}}{2}, 6.283185, -\sqrt{6}, \frac{2\pi}{3}, \frac{2}{\sqrt{5}}\)[/tex]
- Integers:
- [tex]\(-227, 4, 19, 28, 6, -15\)[/tex]
Each number is categorized into its respective set according to the provided data, ensuring we correctly place the numbers in irrational numbers, rational numbers, real numbers, and integers.
