Question
numbers between rational numbers practice
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GPT-4o mini
To work with numbers between rational numbers, you can utilize the concept of rational numbers and the density property of rational numbers in the real number line. Here’s a brief explanation and some practice problems.
### Key Concepts:
1. **Rational Numbers**: A rational number is any number that can be expressed as the quotient of two integers \( \frac{a}{b} \) where \( b \neq 0 \).
2. **Density of Rational Numbers**: Between any two distinct rational numbers, there exists another rational number. This means that if you have two rational numbers \( r_1 \) and \( r_2 \) (where \( r_1 < r_2 \)), you can always find a number \( r \) such that \( r_1 < r < r_2 \).
3. **Midpoint**: A simple way to find a rational number between two rational numbers is to calculate the midpoint:
\[
r = \frac{r_1 + r_2}{2}
\]
### Practice Problems:
1. Find a rational number between \( \frac{1}{3} \) and \( \frac{1}{2} \).
2. Find two rational numbers between \( 0 \) and \( 1 \).
3. Calculate a rational number between \( -\frac{3}{4} \) and \( -\frac{1}{2} \).
4. Identify a rational number between \( 2 \) and \( 3 \).
5. Determine two rational numbers between \( \frac{4}{5} \) and \( 1 \).
### Solutions:
1. **Problem:** Find a rational number between \( \frac{1}{3} \) and \( \frac{1}{2} \).
**Solution:** Calculate the midpoint:
\[
r = \frac{\frac{1}{3} + \frac{1}{2}}{2} = \frac{\frac{2}{6} + \frac{3}{6}}{2} = \frac{\frac{5}{6}}{2} = \frac{5}{12}
\]
Thus, \( \frac{5}{12} \) is between \( \frac{1}{3} \) and \( \frac{1}{2} \).
2. **Problem:** Find two rational numbers between \( 0 \) and \( 1 \).
**Solution:** \( \frac{1}{4} \) and \( \frac{3}{4} \) are two valid rational numbers between \( 0 \) and \( 1 \).
3. **Problem:** Calculate a rational number between \( -\frac{3}{4} \) and \( -\frac{1}{2} \).
**Solution:**
Calculate the midpoint:
\[
r = \frac{-\frac{3}{4} + (-\frac{1}{2})}{2} = \frac{-\frac{3}{4} - \frac{2}{4}}{2} = \frac{-\frac{5}{4}}{2} = -\frac{5}{8}
\]
Thus, \( -\frac{5}{8} \) is between \( -\frac{3}{4} \) and \( -\frac{1}{2} \).
4. **Problem:** Identify a rational number between \( 2 \) and \( 3 \).
**Solution:** \( \frac{5}{2} \) (which is equal to \( 2.5 \)) is a rational number between \( 2 \) and \( 3 \).
5. **Problem:** Determine two rational numbers between \( \frac{4}{5} \) and \( 1 \).
**Solution:** \( \frac{9}{10} \) and \( \frac{19}{20} \) are two rational numbers between \( \frac{4}{5} \) and \( 1 \).
### Additional Practice:
To further enhance your understanding, you can create your own problems by choosing any two rational numbers and finding multiple rational numbers between them using the strategies mentioned above.
### Key Concepts:
1. **Rational Numbers**: A rational number is any number that can be expressed as the quotient of two integers \( \frac{a}{b} \) where \( b \neq 0 \).
2. **Density of Rational Numbers**: Between any two distinct rational numbers, there exists another rational number. This means that if you have two rational numbers \( r_1 \) and \( r_2 \) (where \( r_1 < r_2 \)), you can always find a number \( r \) such that \( r_1 < r < r_2 \).
3. **Midpoint**: A simple way to find a rational number between two rational numbers is to calculate the midpoint:
\[
r = \frac{r_1 + r_2}{2}
\]
### Practice Problems:
1. Find a rational number between \( \frac{1}{3} \) and \( \frac{1}{2} \).
2. Find two rational numbers between \( 0 \) and \( 1 \).
3. Calculate a rational number between \( -\frac{3}{4} \) and \( -\frac{1}{2} \).
4. Identify a rational number between \( 2 \) and \( 3 \).
5. Determine two rational numbers between \( \frac{4}{5} \) and \( 1 \).
### Solutions:
1. **Problem:** Find a rational number between \( \frac{1}{3} \) and \( \frac{1}{2} \).
**Solution:** Calculate the midpoint:
\[
r = \frac{\frac{1}{3} + \frac{1}{2}}{2} = \frac{\frac{2}{6} + \frac{3}{6}}{2} = \frac{\frac{5}{6}}{2} = \frac{5}{12}
\]
Thus, \( \frac{5}{12} \) is between \( \frac{1}{3} \) and \( \frac{1}{2} \).
2. **Problem:** Find two rational numbers between \( 0 \) and \( 1 \).
**Solution:** \( \frac{1}{4} \) and \( \frac{3}{4} \) are two valid rational numbers between \( 0 \) and \( 1 \).
3. **Problem:** Calculate a rational number between \( -\frac{3}{4} \) and \( -\frac{1}{2} \).
**Solution:**
Calculate the midpoint:
\[
r = \frac{-\frac{3}{4} + (-\frac{1}{2})}{2} = \frac{-\frac{3}{4} - \frac{2}{4}}{2} = \frac{-\frac{5}{4}}{2} = -\frac{5}{8}
\]
Thus, \( -\frac{5}{8} \) is between \( -\frac{3}{4} \) and \( -\frac{1}{2} \).
4. **Problem:** Identify a rational number between \( 2 \) and \( 3 \).
**Solution:** \( \frac{5}{2} \) (which is equal to \( 2.5 \)) is a rational number between \( 2 \) and \( 3 \).
5. **Problem:** Determine two rational numbers between \( \frac{4}{5} \) and \( 1 \).
**Solution:** \( \frac{9}{10} \) and \( \frac{19}{20} \) are two rational numbers between \( \frac{4}{5} \) and \( 1 \).
### Additional Practice:
To further enhance your understanding, you can create your own problems by choosing any two rational numbers and finding multiple rational numbers between them using the strategies mentioned above.