Question
Rational numbers can take what forms?
Irrational numbers can take what forms?
5 Problems - SHOW ALL WORK!
Isaac knows that ¼ is written as 0.25 in decimal form. How can he find a decimal for 7/4?
Explain why ⅓ is called a rational number. How do you know?
Predict the decimal for 5/9 using mental math.
Divide to check your prediction, and describe the results.
How would you identify this as a rational or irrational number?
Jackson is practicing basketball. He makes a basket from the free-throw line 9 out of 27 shots.
Write the decimal from the fraction of Jackson’s shot that resulted in the basket.
How can you tell if this number is rational or irrational?
What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
What do I need help with?
Irrational numbers can take what forms?
5 Problems - SHOW ALL WORK!
Isaac knows that ¼ is written as 0.25 in decimal form. How can he find a decimal for 7/4?
Explain why ⅓ is called a rational number. How do you know?
Predict the decimal for 5/9 using mental math.
Divide to check your prediction, and describe the results.
How would you identify this as a rational or irrational number?
Jackson is practicing basketball. He makes a basket from the free-throw line 9 out of 27 shots.
Write the decimal from the fraction of Jackson’s shot that resulted in the basket.
How can you tell if this number is rational or irrational?
What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
What do I need help with?
Answers
GPT-4o mini
Sure! Let's break down your questions step by step and solve each problem carefully.
### Rational Numbers:
Rational numbers can take the forms of:
1. **Fractions** (e.g., \( \frac{1}{2}, \frac{7}{4} \))
2. **Terminating Decimals** (e.g., \( 0.5, 2.75 \))
3. **Repeating Decimals** (e.g., \( 0.333... \) or \( 0.666... \))
### Irrational Numbers:
Irrational numbers can take the forms of:
1. **Non-repeating Decimals** (e.g., \( \sqrt{2} \approx 1.414213...\) which goes on forever without repeating)
2. **Certain Famous Numbers** (e.g., \( \pi \approx 3.14159...\))
### Problems:
1. **Finding a decimal for \( \frac{7}{4} \)**:
- To convert \( \frac{7}{4} \) into decimal form, you can perform long division of 7 by 4.
- \( 7 \div 4 = 1.75 \)
- Therefore, the decimal form of \( \frac{7}{4} \) is **1.75**.
2. **Why \( \frac{1}{3} \) is a rational number**:
- A rational number is defined as any number that can be expressed as the fraction of two integers (where the denominator is not zero).
- \( \frac{1}{3} \) can be written as \( \frac{p}{q} \) with \( p = 1 \) and \( q = 3 \).
- Therefore, it qualifies as a rational number.
**Predicting the decimal for \( \frac{5}{9} \)**:
- You may intuitively know that \( \frac{5}{9} \) will be approximately \( 0.55\) since \( \frac{5}{9} \approx 0.555...\) based on the repeating nature of \( \frac{1}{3} \).
**To check using division**:
- Perform the long division of 5 by 9.
- \( 5 \div 9 = 0.555...\) (indicating it is a repeating decimal).
- The prediction is accurate, so \( \frac{5}{9} = 0.555...\) which is a repeating decimal.
3. **Identifying \( \frac{5}{9} \)**:
- Since \( \frac{5}{9} \) can be expressed as a fraction of two integers, it is classified as a rational number, despite being a repeating decimal.
4. **Decimal from Jackson's basketball shots**:
- Jackson makes \( \frac{9}{27} \) baskets:
- Simplifying: \( \frac{9}{27} = \frac{1}{3} = 0.333...\)
- This decimal \( 0.333...\) is a repeating decimal which is rational, as it can be expressed as the fraction of two integers \( \frac{1}{3} \).
5. **Terminating vs. Repeating Decimals**:
- **Terminating Decimals** are numbers that can be expressed in a decimal form that ends at a certain point (e.g., 0.25).
- **Repeating Decimals** have one or more digits that repeat indefinitely (e.g., 0.333...).
- **Rational numbers** will either terminate or repeat because they can be expressed as fractions. **Irrational numbers** do not terminate or repeat; their decimal expansions are non-repeating and infinite.
### What do you need help with?
If you need further clarification on any of these topics or more examples, please let me know!
### Rational Numbers:
Rational numbers can take the forms of:
1. **Fractions** (e.g., \( \frac{1}{2}, \frac{7}{4} \))
2. **Terminating Decimals** (e.g., \( 0.5, 2.75 \))
3. **Repeating Decimals** (e.g., \( 0.333... \) or \( 0.666... \))
### Irrational Numbers:
Irrational numbers can take the forms of:
1. **Non-repeating Decimals** (e.g., \( \sqrt{2} \approx 1.414213...\) which goes on forever without repeating)
2. **Certain Famous Numbers** (e.g., \( \pi \approx 3.14159...\))
### Problems:
1. **Finding a decimal for \( \frac{7}{4} \)**:
- To convert \( \frac{7}{4} \) into decimal form, you can perform long division of 7 by 4.
- \( 7 \div 4 = 1.75 \)
- Therefore, the decimal form of \( \frac{7}{4} \) is **1.75**.
2. **Why \( \frac{1}{3} \) is a rational number**:
- A rational number is defined as any number that can be expressed as the fraction of two integers (where the denominator is not zero).
- \( \frac{1}{3} \) can be written as \( \frac{p}{q} \) with \( p = 1 \) and \( q = 3 \).
- Therefore, it qualifies as a rational number.
**Predicting the decimal for \( \frac{5}{9} \)**:
- You may intuitively know that \( \frac{5}{9} \) will be approximately \( 0.55\) since \( \frac{5}{9} \approx 0.555...\) based on the repeating nature of \( \frac{1}{3} \).
**To check using division**:
- Perform the long division of 5 by 9.
- \( 5 \div 9 = 0.555...\) (indicating it is a repeating decimal).
- The prediction is accurate, so \( \frac{5}{9} = 0.555...\) which is a repeating decimal.
3. **Identifying \( \frac{5}{9} \)**:
- Since \( \frac{5}{9} \) can be expressed as a fraction of two integers, it is classified as a rational number, despite being a repeating decimal.
4. **Decimal from Jackson's basketball shots**:
- Jackson makes \( \frac{9}{27} \) baskets:
- Simplifying: \( \frac{9}{27} = \frac{1}{3} = 0.333...\)
- This decimal \( 0.333...\) is a repeating decimal which is rational, as it can be expressed as the fraction of two integers \( \frac{1}{3} \).
5. **Terminating vs. Repeating Decimals**:
- **Terminating Decimals** are numbers that can be expressed in a decimal form that ends at a certain point (e.g., 0.25).
- **Repeating Decimals** have one or more digits that repeat indefinitely (e.g., 0.333...).
- **Rational numbers** will either terminate or repeat because they can be expressed as fractions. **Irrational numbers** do not terminate or repeat; their decimal expansions are non-repeating and infinite.
### What do you need help with?
If you need further clarification on any of these topics or more examples, please let me know!