Asked by Loser123
List the rational numbers: 3, 1/3, 2 1/5, -3, 2, or 3/1
Answers
Answered by
Loser123
Explain how it is, so there isn’t ANY wrong answers?
Answered by
GPT-4o mini
Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Let's analyze the numbers you've provided:
1. **3** - This can be expressed as \( \frac{3}{1} \), so it is rational.
2. **1/3** - This is already in the form of a fraction, so it is rational.
3. **2 1/5** - This is a mixed number and can be converted to an improper fraction: \( 2 + \frac{1}{5} = \frac{10}{5} + \frac{1}{5} = \frac{11}{5} \), so it is rational.
4. **-3** - This can be expressed as \( \frac{-3}{1} \), so it is rational.
5. **2** - This can be expressed as \( \frac{2}{1} \), so it is rational.
6. **3/1** - This is already in the form of a fraction, so it is rational.
All the numbers listed are rational numbers. Thus, the list of rational numbers is:
- 3
- 1/3
- 2 1/5
- -3
- 2
- 3/1
1. **3** - This can be expressed as \( \frac{3}{1} \), so it is rational.
2. **1/3** - This is already in the form of a fraction, so it is rational.
3. **2 1/5** - This is a mixed number and can be converted to an improper fraction: \( 2 + \frac{1}{5} = \frac{10}{5} + \frac{1}{5} = \frac{11}{5} \), so it is rational.
4. **-3** - This can be expressed as \( \frac{-3}{1} \), so it is rational.
5. **2** - This can be expressed as \( \frac{2}{1} \), so it is rational.
6. **3/1** - This is already in the form of a fraction, so it is rational.
All the numbers listed are rational numbers. Thus, the list of rational numbers is:
- 3
- 1/3
- 2 1/5
- -3
- 2
- 3/1
Answered by
GPT-4o mini
Certainly! Let's define what a rational number is to clarify why all the numbers you listed are considered rational, and then I'll break down each individual example:
### Definition of Rational Numbers
A rational number is any number that can be expressed in the form of a fraction \( \frac{a}{b} \), where:
- \( a \) is an integer (positive, negative, or zero).
- \( b \) is a non-zero integer.
This means that **as long as a number can be represented as a ratio of two integers**, it is considered a rational number.
### Breakdown of Each Number
1. **3**
- This can be expressed as \( \frac{3}{1} \) (3 is the numerator, and 1 is the denominator). Since both are integers and the denominator is non-zero, 3 is rational.
2. **1/3**
- This is already in the form of a fraction where 1 is the numerator and 3 is the denominator. Both are integers (and 3 is non-zero), so 1/3 is rational.
3. **2 1/5**
- This is a mixed number. We convert it to an improper fraction:
- \( 2 \) can be expressed as \( \frac{10}{5} \) (since \( 2 = 2 \times \frac{5}{5} \)).
- Adding \( \frac{1}{5} \) gives:
\[
\frac{10}{5} + \frac{1}{5} = \frac{11}{5}
\]
- Both 11 and 5 are integers (and 5 is non-zero), so \( \frac{11}{5} \) is rational.
4. **-3**
- This can be expressed as \( \frac{-3}{1} \). The numerator is -3 (an integer), and the denominator is 1 (also an integer and not zero). Hence, -3 is rational.
5. **2**
- Similar to 3, this can be expressed as \( \frac{2}{1} \). Both 2 and 1 are integers (and 1 is non-zero), so 2 is rational.
6. **3/1**
- This is already in the form of a fraction. The numerator is 3 and the denominator is 1. Both are integers, and the denominator is non-zero, making \( \frac{3}{1} \) rational.
### Conclusion
Since every number you provided can be represented as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \), there are **no wrong answers** in the list. All of them are indeed rational numbers.
### Definition of Rational Numbers
A rational number is any number that can be expressed in the form of a fraction \( \frac{a}{b} \), where:
- \( a \) is an integer (positive, negative, or zero).
- \( b \) is a non-zero integer.
This means that **as long as a number can be represented as a ratio of two integers**, it is considered a rational number.
### Breakdown of Each Number
1. **3**
- This can be expressed as \( \frac{3}{1} \) (3 is the numerator, and 1 is the denominator). Since both are integers and the denominator is non-zero, 3 is rational.
2. **1/3**
- This is already in the form of a fraction where 1 is the numerator and 3 is the denominator. Both are integers (and 3 is non-zero), so 1/3 is rational.
3. **2 1/5**
- This is a mixed number. We convert it to an improper fraction:
- \( 2 \) can be expressed as \( \frac{10}{5} \) (since \( 2 = 2 \times \frac{5}{5} \)).
- Adding \( \frac{1}{5} \) gives:
\[
\frac{10}{5} + \frac{1}{5} = \frac{11}{5}
\]
- Both 11 and 5 are integers (and 5 is non-zero), so \( \frac{11}{5} \) is rational.
4. **-3**
- This can be expressed as \( \frac{-3}{1} \). The numerator is -3 (an integer), and the denominator is 1 (also an integer and not zero). Hence, -3 is rational.
5. **2**
- Similar to 3, this can be expressed as \( \frac{2}{1} \). Both 2 and 1 are integers (and 1 is non-zero), so 2 is rational.
6. **3/1**
- This is already in the form of a fraction. The numerator is 3 and the denominator is 1. Both are integers, and the denominator is non-zero, making \( \frac{3}{1} \) rational.
### Conclusion
Since every number you provided can be represented as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \), there are **no wrong answers** in the list. All of them are indeed rational numbers.