Which statement is true about the relationships between the number sets?(1 point)
Responses
Some irrational numbers are also integers.
Not all natural numbers are real numbers.
All integers are also rational numbers.
Whole numbers include all positive integers and negative integers.
All integers are also rational numbers.
incorrect
The true statement about the relationships between the number sets is: "Not all natural numbers are real numbers."
To determine which statement is true about the relationships between the number sets, let's analyze each option:
1. "Some irrational numbers are also integers":
To understand this statement, we need to know the definitions of irrational numbers and integers.
- Irrational numbers: These are numbers that cannot be expressed as a ratio of two integers. They have decimal representations that go on forever without repeating patterns, such as π (pi) or √2 (square root of 2).
- Integers: These are whole numbers (including zero) and their negative counterparts, such as -3, -2, -1, 0, 1, 2, 3.
Based on these definitions, it is not possible for irrational numbers, which are non-repeating decimals, to also be integers, which are whole numbers or their negatives. Therefore, the statement "Some irrational numbers are also integers" is false.
2. "Not all natural numbers are real numbers":
To determine the validity of this statement, we need to understand the definitions of natural numbers and real numbers.
- Natural numbers: These are the counting numbers, starting from 1 and going up infinitely: 1, 2, 3, 4, ...
- Real numbers: These include all rational and irrational numbers, covering all possible points on the number line.
Since natural numbers are a subset of real numbers (which include all possible points on the number line), it is true that not all natural numbers are real numbers. For example, natural numbers cannot include irrational numbers like π or √2. Thus, the statement "Not all natural numbers are real numbers" is true.
3. "All integers are also rational numbers":
Let's review the definitions of integers and rational numbers.
- Integers: These are whole numbers (including zero) and their negative counterparts: -3, -2, -1, 0, 1, 2, 3.
- Rational numbers: These are numbers that can be expressed as a ratio of two integers, such as -3/2 or 4/5.
Since every integer can be represented as a ratio of itself divided by 1, it is true that all integers are also rational numbers. Thus, the statement "All integers are also rational numbers" is true.
4. "Whole numbers include all positive integers and negative integers":
The definition of whole numbers encompasses the set of positive integers, zero, and the set of negative integers. Therefore, the statement "Whole numbers include all positive integers and negative integers" is true.
In summary, the true statements about the relationships between the number sets are:
- "Not all natural numbers are real numbers."
- "All integers are also rational numbers."
- "Whole numbers include all positive integers and negative integers."