Question
Which are considered irrational?
2.77777…
-7/3
2.7182818459…
3
2.77777…
-7/3
2.7182818459…
3
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which of the numbers listed are considered irrational, we need to know the definitions of rational and irrational numbers:
- **Rational numbers** are numbers that can be expressed as the quotient of two integers (i.e., a fraction a/b where a and b are integers, and b ≠ 0). Rational numbers can either terminate (like 0.5) or repeat (like 0.333...).
- **Irrational numbers** are numbers that cannot be expressed as a simple fraction, meaning they cannot be written as a ratio of two integers. They have non-repeating, non-terminating decimal expansions.
Now, let's analyze each of the numbers you've listed:
1. **2.77777…**: This number is a repeating decimal (2.7 with the 7 repeating), which can be expressed as a fraction. Therefore, it is a **rational number**.
2. **-7/3**: This is a fraction where both the numerator and denominator are integers. It can be written as -2.333..., which is also a repeating decimal. Hence, this is a **rational number**.
3. **2.7182818459…**: This is a non-repeating, non-terminating decimal. It is an approximation of the mathematical constant \( e \), which is known to be irrational. Therefore, this is an **irrational number**.
4. **3**: This is an integer and can also be expressed as the fraction 3/1. Therefore, it is a **rational number**.
In summary:
- **Irrational number:** 2.7182818459…
- **Rational numbers:** 2.77777…, -7/3, 3
- **Rational numbers** are numbers that can be expressed as the quotient of two integers (i.e., a fraction a/b where a and b are integers, and b ≠ 0). Rational numbers can either terminate (like 0.5) or repeat (like 0.333...).
- **Irrational numbers** are numbers that cannot be expressed as a simple fraction, meaning they cannot be written as a ratio of two integers. They have non-repeating, non-terminating decimal expansions.
Now, let's analyze each of the numbers you've listed:
1. **2.77777…**: This number is a repeating decimal (2.7 with the 7 repeating), which can be expressed as a fraction. Therefore, it is a **rational number**.
2. **-7/3**: This is a fraction where both the numerator and denominator are integers. It can be written as -2.333..., which is also a repeating decimal. Hence, this is a **rational number**.
3. **2.7182818459…**: This is a non-repeating, non-terminating decimal. It is an approximation of the mathematical constant \( e \), which is known to be irrational. Therefore, this is an **irrational number**.
4. **3**: This is an integer and can also be expressed as the fraction 3/1. Therefore, it is a **rational number**.
In summary:
- **Irrational number:** 2.7182818459…
- **Rational numbers:** 2.77777…, -7/3, 3
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