Answer :
Sure, let's decide whether each number in the given list is rational or irrational.
### Understanding Rational and Irrational Numbers
- Rational Numbers: These are numbers that can be expressed as the ratio of two integers, [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]. Rational numbers have either a terminating decimal expansion or a repeating decimal expansion.
- Irrational Numbers: These numbers cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating.
### Examining the List of Numbers
We are given the list of numbers:
[tex]\[ \frac{-13}{3}, 0.1234, \sqrt{37}, -77, -\sqrt{100}, -\sqrt{12} \][/tex]
Let's analyze each of these numbers one by one.
1. [tex]\(\frac{-13}{3}\)[/tex]:
- This is a fraction where both the numerator (-13) and the denominator (3) are integers.
- Therefore, [tex]\(\frac{-13}{3}\)[/tex] is a rational number.
2. [tex]\(0.1234\)[/tex]:
- This is a finite decimal.
- Finite decimals are rational because they can be expressed as fractions. For example, [tex]\(0.1234 = \frac{1234}{10000}\)[/tex].
- Therefore, [tex]\(0.1234\)[/tex] is a rational number.
3. [tex]\(\sqrt{37}\)[/tex]:
- The square root of a non-perfect square (37) is irrational.
- Hence, [tex]\(\sqrt{37}\)[/tex] is an irrational number.
4. [tex]\(-77\)[/tex]:
- This is an integer, and any integer can be expressed as a ratio of itself to 1. For example, [tex]\(-77 = \frac{-77}{1}\)[/tex].
- Therefore, [tex]\(-77\)[/tex] is a rational number.
5. [tex]\(-\sqrt{100}\)[/tex]:
- [tex]\(\sqrt{100} = 10\)[/tex], and thus [tex]\(-\sqrt{100} = -10\)[/tex].
- Since -10 is an integer, it is rational.
- Therefore, [tex]\(-\sqrt{100}\)[/tex] is a rational number.
6. [tex]\(-\sqrt{12}\)[/tex]:
- The square root of 12 is irrational because 12 is not a perfect square.
- Hence, [tex]\(-\sqrt{12}\)[/tex] is an irrational number.
### Summary
Let's now classify each given number into rational or irrational.
- Rational:
[tex]\[ \frac{-13}{3}, 0.1234, -77, -10, -3.4641016151377544 \][/tex]
- Irrational:
[tex]\[ \sqrt{37} \][/tex]
Based on this analysis, the number [tex]\( \sqrt{37} \)[/tex] is classified as an irrational number.
Therefore, the final classification is:
Rational:
[tex]\[ \frac{-13}{3}, 0.1234, -77, -10, -3.4641016151377544 \][/tex]
Irrational:
[tex]\[ \sqrt{37} \][/tex]
### Understanding Rational and Irrational Numbers
- Rational Numbers: These are numbers that can be expressed as the ratio of two integers, [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]. Rational numbers have either a terminating decimal expansion or a repeating decimal expansion.
- Irrational Numbers: These numbers cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating.
### Examining the List of Numbers
We are given the list of numbers:
[tex]\[ \frac{-13}{3}, 0.1234, \sqrt{37}, -77, -\sqrt{100}, -\sqrt{12} \][/tex]
Let's analyze each of these numbers one by one.
1. [tex]\(\frac{-13}{3}\)[/tex]:
- This is a fraction where both the numerator (-13) and the denominator (3) are integers.
- Therefore, [tex]\(\frac{-13}{3}\)[/tex] is a rational number.
2. [tex]\(0.1234\)[/tex]:
- This is a finite decimal.
- Finite decimals are rational because they can be expressed as fractions. For example, [tex]\(0.1234 = \frac{1234}{10000}\)[/tex].
- Therefore, [tex]\(0.1234\)[/tex] is a rational number.
3. [tex]\(\sqrt{37}\)[/tex]:
- The square root of a non-perfect square (37) is irrational.
- Hence, [tex]\(\sqrt{37}\)[/tex] is an irrational number.
4. [tex]\(-77\)[/tex]:
- This is an integer, and any integer can be expressed as a ratio of itself to 1. For example, [tex]\(-77 = \frac{-77}{1}\)[/tex].
- Therefore, [tex]\(-77\)[/tex] is a rational number.
5. [tex]\(-\sqrt{100}\)[/tex]:
- [tex]\(\sqrt{100} = 10\)[/tex], and thus [tex]\(-\sqrt{100} = -10\)[/tex].
- Since -10 is an integer, it is rational.
- Therefore, [tex]\(-\sqrt{100}\)[/tex] is a rational number.
6. [tex]\(-\sqrt{12}\)[/tex]:
- The square root of 12 is irrational because 12 is not a perfect square.
- Hence, [tex]\(-\sqrt{12}\)[/tex] is an irrational number.
### Summary
Let's now classify each given number into rational or irrational.
- Rational:
[tex]\[ \frac{-13}{3}, 0.1234, -77, -10, -3.4641016151377544 \][/tex]
- Irrational:
[tex]\[ \sqrt{37} \][/tex]
Based on this analysis, the number [tex]\( \sqrt{37} \)[/tex] is classified as an irrational number.
Therefore, the final classification is:
Rational:
[tex]\[ \frac{-13}{3}, 0.1234, -77, -10, -3.4641016151377544 \][/tex]
Irrational:
[tex]\[ \sqrt{37} \][/tex]
