Question
1. Which statement is true about the relationships between the number sets?
A. All irrational numbers are also integers.
B. Every number is a whole number.
C. Only positive numbers make up the set of integers.
D. Whole number include all natural numbers and 0.
A. All irrational numbers are also integers.
B. Every number is a whole number.
C. Only positive numbers make up the set of integers.
D. Whole number include all natural numbers and 0.
Answers
Answered by
GPT-4o mini
The true statement about the relationships between the number sets is:
**D. Whole numbers include all natural numbers and 0.**
To clarify the other options:
A. This is false because irrational numbers are not integers (e.g., √2, π).
B. This is false because not every number is a whole number; for example, negative numbers and fractions are not whole numbers.
C. This is false because integers include both positive and negative whole numbers, as well as zero (e.g., -1, 0, 1).
**D. Whole numbers include all natural numbers and 0.**
To clarify the other options:
A. This is false because irrational numbers are not integers (e.g., √2, π).
B. This is false because not every number is a whole number; for example, negative numbers and fractions are not whole numbers.
C. This is false because integers include both positive and negative whole numbers, as well as zero (e.g., -1, 0, 1).
Answer
explain why
Answered by
GPT-4o mini
Certainly! Let’s break down the reasoning for each statement to understand why D is true and the others are false:
### A. All irrational numbers are also integers.
**Explanation:**
- **Irrational Numbers**: These are numbers that cannot be expressed as a simple fraction of two integers. Examples include √2, π, and e. They have non-repeating, non-terminating decimal expansions.
- **Integers**: These are whole numbers that can be positive, negative, or zero (e.g., -3, 0, 5).
- **Conclusion**: Since irrational numbers cannot be expressed as integers, this statement is false.
### B. Every number is a whole number.
**Explanation:**
- **Whole Numbers**: These are non-negative integers, which include 0 and all positive integers (0, 1, 2, 3, ...).
- **Other Types of Numbers**: There are many types of numbers that are not whole numbers, such as negative integers (-1, -2, ...), fractions (1/2, 3/4), and irrational numbers (√2, π).
- **Conclusion**: Since not every number is included in the set of whole numbers, this statement is false.
### C. Only positive numbers make up the set of integers.
**Explanation:**
- **Integers**: As previously mentioned, integers include positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...), and zero (0).
- **Conclusion**: Since integers also consist of negative numbers and zero, this statement is false.
### D. Whole numbers include all natural numbers and 0.
**Explanation:**
- **Natural Numbers**: These are typically the set of positive integers (1, 2, 3, ...).
- **Whole Numbers**: This set includes all natural numbers plus zero (0, 1, 2, 3, ...).
- **Conclusion**: The definition of whole numbers correctly incorporates all natural numbers along with zero, making this statement true.
In summary, option D accurately reflects the relationship between whole numbers and natural numbers, while options A, B, and C contain inaccuracies about the definitions of number sets.
### A. All irrational numbers are also integers.
**Explanation:**
- **Irrational Numbers**: These are numbers that cannot be expressed as a simple fraction of two integers. Examples include √2, π, and e. They have non-repeating, non-terminating decimal expansions.
- **Integers**: These are whole numbers that can be positive, negative, or zero (e.g., -3, 0, 5).
- **Conclusion**: Since irrational numbers cannot be expressed as integers, this statement is false.
### B. Every number is a whole number.
**Explanation:**
- **Whole Numbers**: These are non-negative integers, which include 0 and all positive integers (0, 1, 2, 3, ...).
- **Other Types of Numbers**: There are many types of numbers that are not whole numbers, such as negative integers (-1, -2, ...), fractions (1/2, 3/4), and irrational numbers (√2, π).
- **Conclusion**: Since not every number is included in the set of whole numbers, this statement is false.
### C. Only positive numbers make up the set of integers.
**Explanation:**
- **Integers**: As previously mentioned, integers include positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...), and zero (0).
- **Conclusion**: Since integers also consist of negative numbers and zero, this statement is false.
### D. Whole numbers include all natural numbers and 0.
**Explanation:**
- **Natural Numbers**: These are typically the set of positive integers (1, 2, 3, ...).
- **Whole Numbers**: This set includes all natural numbers plus zero (0, 1, 2, 3, ...).
- **Conclusion**: The definition of whole numbers correctly incorporates all natural numbers along with zero, making this statement true.
In summary, option D accurately reflects the relationship between whole numbers and natural numbers, while options A, B, and C contain inaccuracies about the definitions of number sets.