Let's analyze the given numbers one by one and then determine their order from least to greatest:
1. Number 1: 1.01
- This is a straightforward decimal number.
- Its value is exactly 1.01.
2. Number 2: [tex]\(\frac{21}{19}\)[/tex]
- This is a fraction.
- To find its decimal equivalent, divide 21 by 19.
- [tex]\[\frac{21}{19} \approx 1.105263157894737\][/tex]
3. Number 3: [tex]\(1 . \overline{01}\)[/tex]
- This notation represents a repeating decimal 1.010101...
- To understand its value, observe that it is slightly more than 1.01 but less than any number that exceeds 1.02.
- The best representation for comparison purposes is approximately 1.02.
Now, let's order these numbers from least to greatest based on their decimal values:
- [tex]\(1.01\)[/tex]
- [tex]\(1.02\)[/tex]
- [tex]\(1.105263157894737\)[/tex] ([tex]\(\frac{21}{19}\)[/tex])
Hence, from least to greatest, the correct order of the numbers is:
- [tex]\(1.01, 1 . \overline{01}, \frac{21}{19}\)[/tex]
Looking at the options:
- Option A: [tex]\(1.01, \frac{21}{19}, 1 . \overline{01}\)[/tex] (Incorrect order)
- Option B: [tex]\(\frac{21}{19}, 1.01, 1 . \overline{01}\)[/tex] (Incorrect order)
- Option C: [tex]\(1.01, 1 . \overline{01}, \frac{21}{19}\)[/tex] (Correct order)
- Option D: [tex]\(1 . \overline{01}, 1.01, \frac{21}{19}\)[/tex] (Incorrect order)
Therefore, the correct list that shows the numbers ordered from least to greatest is:
Option C: [tex]\(1.01, 1 . \overline{01}, \frac{21}{19}\)[/tex]
