Answer :
To classify the given numbers into their most specific categories, we'll carefully examine each number. We’ll place them in the most specific category possible according to the hierarchy of numbers in mathematics: natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
### Classification of Each Given Number:
#### Rational Numbers:
1. [tex]\( -\frac{2}{3} \)[/tex]
- This number is a fraction and does not fall into any of the categories of natural numbers, whole numbers, or integers. Hence, it is classified as a rational number.
2. [tex]\( 2.79 \)[/tex]
- This number is a decimal which can be represented as a fraction ( [tex]\( \frac{279}{100} \)[/tex] ), so it is classified as a rational number.
3. [tex]\( 12.6 - 7.1 \)[/tex]
- First, we perform the operation: [tex]\( 12.6 - 7.1 = 5.5 \)[/tex].
- The result [tex]\( 5.5 \)[/tex] is a decimal that can be represented as a fraction [tex]\( \frac{11}{2} \)[/tex], so it is classified as a rational number.
Hence:
[tex]\[ \text{Rational Numbers: } -\frac{2}{3}, 2.79, 5.5 \][/tex]
#### Integers:
There are no specific mention of integers in this problem statement that are separate from the provided rational and irrational numbers.
#### Whole Numbers:
There are no specific mention of whole numbers that are different from integers in this problem statement.
#### Natural Numbers:
The problem statement explicitly lists [tex]\( \sqrt{\pi} \)[/tex] under natural numbers, but there is a misunderstanding here:
- [tex]\( \sqrt{\pi} \)[/tex] is not a natural number because it’s not a whole number and it’s not a number used for counting (usually, natural numbers are positive integers starting from 1).
#### Irrational Numbers:
1. [tex]\( \frac{1}{3} \)[/tex]
- The given value [tex]\( \frac{1}{3} \)[/tex] is actually [tex]\( 0.3333... \)[/tex] (a repeating decimal), which should be classified as a rational number. However, for the purpose of this classification, it is placed under irrational incorrectly. To correct this based on the initial problem, [tex]\( \frac{1}{3} \)[/tex] would be considered rational.
2. [tex]\( \frac{1}{1} - \frac{1}{1} \)[/tex]
- Simplifying this we get [tex]\( 1 - 1 = 0 \)[/tex]. This result is neither a fraction nor a decimal that cannot be expressed as a ratio. So, it is [tex]\( 0 \)[/tex].
- Correction based on classification, it would be better if not categorized under irrational since 0 is actually a whole number and an integer.
3. [tex]\( \sqrt{\pi} \)[/tex]
- The square root of [tex]\(\pi\)[/tex] is irrational.
Enumerating correctly:
[tex]\[ \text{Irrational Numbers: } \sqrt{\pi} \][/tex]
### Summary:
Rational:
[tex]\[ -\frac{2}{3}, 2.79, 5.5 \][/tex]
Natural Numbers:
[tex]\[ \text{(None listed in given question correctly)} \][/tex]
Irrational:
[tex]\[ \sqrt{\pi} \][/tex]
Note: There were inconsistencies originally listed. To clarify, the correct classes are identified properly to fit rational and irrational categories correctly for teaching.
### Classification of Each Given Number:
#### Rational Numbers:
1. [tex]\( -\frac{2}{3} \)[/tex]
- This number is a fraction and does not fall into any of the categories of natural numbers, whole numbers, or integers. Hence, it is classified as a rational number.
2. [tex]\( 2.79 \)[/tex]
- This number is a decimal which can be represented as a fraction ( [tex]\( \frac{279}{100} \)[/tex] ), so it is classified as a rational number.
3. [tex]\( 12.6 - 7.1 \)[/tex]
- First, we perform the operation: [tex]\( 12.6 - 7.1 = 5.5 \)[/tex].
- The result [tex]\( 5.5 \)[/tex] is a decimal that can be represented as a fraction [tex]\( \frac{11}{2} \)[/tex], so it is classified as a rational number.
Hence:
[tex]\[ \text{Rational Numbers: } -\frac{2}{3}, 2.79, 5.5 \][/tex]
#### Integers:
There are no specific mention of integers in this problem statement that are separate from the provided rational and irrational numbers.
#### Whole Numbers:
There are no specific mention of whole numbers that are different from integers in this problem statement.
#### Natural Numbers:
The problem statement explicitly lists [tex]\( \sqrt{\pi} \)[/tex] under natural numbers, but there is a misunderstanding here:
- [tex]\( \sqrt{\pi} \)[/tex] is not a natural number because it’s not a whole number and it’s not a number used for counting (usually, natural numbers are positive integers starting from 1).
#### Irrational Numbers:
1. [tex]\( \frac{1}{3} \)[/tex]
- The given value [tex]\( \frac{1}{3} \)[/tex] is actually [tex]\( 0.3333... \)[/tex] (a repeating decimal), which should be classified as a rational number. However, for the purpose of this classification, it is placed under irrational incorrectly. To correct this based on the initial problem, [tex]\( \frac{1}{3} \)[/tex] would be considered rational.
2. [tex]\( \frac{1}{1} - \frac{1}{1} \)[/tex]
- Simplifying this we get [tex]\( 1 - 1 = 0 \)[/tex]. This result is neither a fraction nor a decimal that cannot be expressed as a ratio. So, it is [tex]\( 0 \)[/tex].
- Correction based on classification, it would be better if not categorized under irrational since 0 is actually a whole number and an integer.
3. [tex]\( \sqrt{\pi} \)[/tex]
- The square root of [tex]\(\pi\)[/tex] is irrational.
Enumerating correctly:
[tex]\[ \text{Irrational Numbers: } \sqrt{\pi} \][/tex]
### Summary:
Rational:
[tex]\[ -\frac{2}{3}, 2.79, 5.5 \][/tex]
Natural Numbers:
[tex]\[ \text{(None listed in given question correctly)} \][/tex]
Irrational:
[tex]\[ \sqrt{\pi} \][/tex]
Note: There were inconsistencies originally listed. To clarify, the correct classes are identified properly to fit rational and irrational categories correctly for teaching.
