Which statement is true about the relationship between the number sets?
A: Some irrational numbers are also integers.
B: All rational numbers are also integers.
C: Whole numbers include all natural numbers and 0.
D: Not all natural numbers are real numbers.
C: Whole numbers include all natural numbers and 0.
C: Whole numbers include all natural numbers and 0.
To determine which statement is true about the relationship between number sets, let's examine each option:
A: Some irrational numbers are also integers.
To check if this statement is true, let's first understand these number sets:
- Irrational numbers are numbers that cannot be expressed as a fraction, such as √2 or π.
- Integers are whole numbers (positive, negative, or zero) and their opposites, such as -2, 0, and 5.
Now, since irrational numbers cannot be expressed as fractions and integers can be, it is not possible for an irrational number to also be an integer. Therefore, statement A is false.
B: All rational numbers are also integers.
To evaluate this statement, let's recall the definitions:
- Rational numbers are numbers that can be expressed as fractions, such as 1/2 or 3/4.
- Integers include whole numbers and their opposites.
Since every integer can be written as a fraction (e.g., 5 = 5/1), it is correct to say that all integers are rational numbers. Therefore, statement B is true.
C: Whole numbers include all natural numbers and 0.
To verify this statement, let's understand these number sets:
- Natural numbers, also known as counting numbers, are positive numbers starting from 1, such as 1, 2, 3, and so on.
- Whole numbers include all natural numbers and zero, such as 0, 1, 2, 3, and so on.
As all natural numbers are included in the set of whole numbers, and zero is also included, statement C is true.
D: Not all natural numbers are real numbers.
To assess this statement, it's essential to understand the concept of real numbers:
- Real numbers include all rational numbers (fractions) and irrational numbers.
Since natural numbers are a subset of rational numbers, and rational numbers are a subset of real numbers, it is accurate to say that all natural numbers are real numbers. Therefore, statement D is false.
In summary, the true statement about the relationship between the number sets is:
- Option B: All rational numbers are also integers.