Sure, let's match each number on the left with the appropriate description on the right. Here is a detailed step-by-step explanation for each number:
1. [tex]\(\mathbf{4.8 \overline{3}}\)[/tex]:
- This notation represents the decimal [tex]\(4.8\)[/tex] followed by an infinite repetition of the digit [tex]\(3\)[/tex], i.e., [tex]\(4.8333\ldots\)[/tex].
- This number can be expressed as a fraction because it is a repeating decimal.
- Therefore, this number is a rational number, but it is not an integer.
- Match: "This is a rational number, but not an integer."
2. [tex]\(\mathbf{\sqrt{2}}\)[/tex]:
- The square root of 2 is a well-known example of an irrational number.
- It cannot be expressed as a fraction because its decimal representation is non-terminating and non-repeating.
- Therefore, this number is an irrational number.
- Match: "This is an irrational number."
3. [tex]\(\mathbf{-7}\)[/tex]:
- The number [tex]\(-7\)[/tex] is a whole number that can be expressed without a fractional or decimal component.
- It is a perfect example of an integer.
- Therefore, this number is an integer.
- Match: "This is an integer."
4. [tex]\(\mathbf{-6.175}\)[/tex]:
- The number [tex]\(-6.175\)[/tex] is a terminating decimal, meaning it has a finite number of digits after the decimal point.
- Terminating decimals can always be expressed as fractions (rational numbers).
- Therefore, this number is a rational number, but it is not an integer.
- Match: "This is a rational number, but not an integer."
5. [tex]\(\mathbf{\frac{1}{3}}\)[/tex]:
- The fraction [tex]\(\frac{1}{3}\)[/tex] represents a repeating decimal, [tex]\(0.\overline{3}\)[/tex].
- Repeating decimals can be expressed exactly as fractions, making them rational numbers.
- Therefore, this number is a rational number, but it is not an integer.
- Match: "This is a rational number, but not an integer."
Here is the final matching of the numbers with their descriptions:
- [tex]\(4.8 \overline{3}\)[/tex]: "This is a rational number, but not an integer."
- [tex]\(\sqrt{2}\)[/tex]: "This is an irrational number."
- [tex]\(-7\)[/tex]: "This is an integer."
- [tex]\(-6.175\)[/tex]: "This is a rational number, but not an integer."
- [tex]\(\frac{1}{3}\)[/tex]: "This is a rational number, but not an integer."
I hope this detailed explanation helps!
