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Sagot :
To solve the given problem of finding four different irrational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex], we can follow these detailed steps:
### Step-by-Step Solution
1. Determine the endpoints:
- Calculate [tex]\(\frac{5}{7}\)[/tex] which is approximately 0.714285714.
- Calculate [tex]\(\frac{9}{11}\)[/tex] which is approximately 0.818181818.
2. Find the midpoint:
- Compute the midpoint between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex]:
[tex]\[ \text{Midpoint} = \frac{\frac{5}{7} + \frac{9}{11}}{2} \][/tex]
First, compute the sum of the fractions:
[tex]\[ \frac{5}{7} + \frac{9}{11} = \frac{5 \cdot 11 + 9 \cdot 7}{7 \cdot 11} = \frac{55 + 63}{77} = \frac{118}{77} \][/tex]
Now, divide by 2 to find the midpoint:
[tex]\[ \text{Midpoint} = \frac{118}{77 \cdot 2} = \frac{118}{154} \approx 0.766233766 \][/tex]
3. Determine the difference between the endpoints:
- Calculate the difference [tex]\(b - a\)[/tex], where [tex]\(a = \frac{5}{7}\)[/tex] and [tex]\(b = \frac{9}{11}\)[/tex]:
[tex]\[ b - a = \frac{9}{11} - \frac{5}{7} \][/tex]
First, compute the common denominator:
[tex]\[ b - a = \frac{63}{77} - \frac{55}{77} = \frac{8}{77} \][/tex]
Simplify if needed, [tex]\(\frac{8}{77} \approx 0.103896104\)[/tex].
4. Construct the irrational numbers:
- To ensure these numbers are irrational, we can add and subtract an irrational component (such as square roots of non-perfect squares) scaled appropriately.
- We can use numbers with roots, and scale them such that they fall within the interval [tex]\(b - a\)[/tex].
- For simplicity, we use the midpoint ± irrational component:
[tex]\[ \text{Irrational Number 1} = 0.766233766 + (\sqrt{2} \times \frac{b - a}{10}) \approx 0.780926894 \][/tex]
[tex]\[ \text{Irrational Number 2} = 0.766233766 + (\sqrt{3} \times \frac{b - a}{10}) \approx 0.784229099 \][/tex]
[tex]\[ \text{Irrational Number 3} = 0.766233766 + (\sqrt{5} \times \frac{b - a}{10}) \approx 0.789465641 \][/tex]
[tex]\[ \text{Irrational Number 4} = 0.766233766 + (\sqrt{6} \times \frac{b - a}{10}) \approx 0.791683010 \][/tex]
### Conclusion
The four different irrational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex] are approximately:
[tex]\[ 0.780926894, \, 0.784229099, \, 0.789465641, \, \text{and} \, 0.791683010 \][/tex]
### Step-by-Step Solution
1. Determine the endpoints:
- Calculate [tex]\(\frac{5}{7}\)[/tex] which is approximately 0.714285714.
- Calculate [tex]\(\frac{9}{11}\)[/tex] which is approximately 0.818181818.
2. Find the midpoint:
- Compute the midpoint between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex]:
[tex]\[ \text{Midpoint} = \frac{\frac{5}{7} + \frac{9}{11}}{2} \][/tex]
First, compute the sum of the fractions:
[tex]\[ \frac{5}{7} + \frac{9}{11} = \frac{5 \cdot 11 + 9 \cdot 7}{7 \cdot 11} = \frac{55 + 63}{77} = \frac{118}{77} \][/tex]
Now, divide by 2 to find the midpoint:
[tex]\[ \text{Midpoint} = \frac{118}{77 \cdot 2} = \frac{118}{154} \approx 0.766233766 \][/tex]
3. Determine the difference between the endpoints:
- Calculate the difference [tex]\(b - a\)[/tex], where [tex]\(a = \frac{5}{7}\)[/tex] and [tex]\(b = \frac{9}{11}\)[/tex]:
[tex]\[ b - a = \frac{9}{11} - \frac{5}{7} \][/tex]
First, compute the common denominator:
[tex]\[ b - a = \frac{63}{77} - \frac{55}{77} = \frac{8}{77} \][/tex]
Simplify if needed, [tex]\(\frac{8}{77} \approx 0.103896104\)[/tex].
4. Construct the irrational numbers:
- To ensure these numbers are irrational, we can add and subtract an irrational component (such as square roots of non-perfect squares) scaled appropriately.
- We can use numbers with roots, and scale them such that they fall within the interval [tex]\(b - a\)[/tex].
- For simplicity, we use the midpoint ± irrational component:
[tex]\[ \text{Irrational Number 1} = 0.766233766 + (\sqrt{2} \times \frac{b - a}{10}) \approx 0.780926894 \][/tex]
[tex]\[ \text{Irrational Number 2} = 0.766233766 + (\sqrt{3} \times \frac{b - a}{10}) \approx 0.784229099 \][/tex]
[tex]\[ \text{Irrational Number 3} = 0.766233766 + (\sqrt{5} \times \frac{b - a}{10}) \approx 0.789465641 \][/tex]
[tex]\[ \text{Irrational Number 4} = 0.766233766 + (\sqrt{6} \times \frac{b - a}{10}) \approx 0.791683010 \][/tex]
### Conclusion
The four different irrational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{9}{11}\)[/tex] are approximately:
[tex]\[ 0.780926894, \, 0.784229099, \, 0.789465641, \, \text{and} \, 0.791683010 \][/tex]
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