Question
true or false: every rational number is a natural number,
every natural number is also a integer,
every irrational number is also an integer,
0.128128128... is an irrational number?
every natural number is also a integer,
every irrational number is also an integer,
0.128128128... is an irrational number?
Answers
GPT-4o mini
Let's evaluate each statement one by one:
1. **Every rational number is a natural number**: **False**. Rational numbers include integers, fractions, and decimals that can be expressed as a ratio of two integers. Natural numbers are the positive integers (1, 2, 3, ...), and not all rational numbers fall into this category (e.g., \( \frac{1}{2} \) is rational but not a natural number).
2. **Every natural number is also an integer**: **True**. Natural numbers (1, 2, 3, ...) are part of the set of integers (which also include 0 and negative integers).
3. **Every irrational number is also an integer**: **False**. Irrational numbers cannot be expressed as a ratio of two integers and therefore are not integers. For example, \( \sqrt{2} \) and \( \pi \) are irrational numbers, but they are not integers.
4. **0.128128128... is an irrational number**: **False**. The number 0.128128128... is a repeating decimal, which can be expressed as a fraction (specifically, \( \frac{128}{999} \)). Therefore, it is a rational number, not an irrational number.
In summary:
1. False
2. True
3. False
4. False
1. **Every rational number is a natural number**: **False**. Rational numbers include integers, fractions, and decimals that can be expressed as a ratio of two integers. Natural numbers are the positive integers (1, 2, 3, ...), and not all rational numbers fall into this category (e.g., \( \frac{1}{2} \) is rational but not a natural number).
2. **Every natural number is also an integer**: **True**. Natural numbers (1, 2, 3, ...) are part of the set of integers (which also include 0 and negative integers).
3. **Every irrational number is also an integer**: **False**. Irrational numbers cannot be expressed as a ratio of two integers and therefore are not integers. For example, \( \sqrt{2} \) and \( \pi \) are irrational numbers, but they are not integers.
4. **0.128128128... is an irrational number**: **False**. The number 0.128128128... is a repeating decimal, which can be expressed as a fraction (specifically, \( \frac{128}{999} \)). Therefore, it is a rational number, not an irrational number.
In summary:
1. False
2. True
3. False
4. False