Which statement about numbers is true?
A: All integers are natural numbers.
B: All rational numbers are integers.
C: All natural numbers are whole numbers.
D: All rational numbers are natural numbers.
natural ⊂ whole ⊂ integer ⊂ rational ⊂ real
better review the topic some more
To determine which statement about numbers is true, let's examine each option one by one:
A: All integers are natural numbers.
To verify if this statement is correct, we need to understand what integers and natural numbers are. Integers are a set of whole numbers, including positive numbers, negative numbers, and zero. Natural numbers, on the other hand, are a set of positive whole numbers (excluding zero). Since integers include both positive and negative numbers, the statement A is incorrect. Not all integers are natural numbers.
B: All rational numbers are integers.
To validate this statement, we need to recognize the definitions of rational numbers and integers. Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Integers, as mentioned earlier, are a set of whole numbers. Since every integer can be expressed as a fraction (by putting it over 1), it is true that all integers are rational numbers. Therefore, statement B is correct.
C: All natural numbers are whole numbers.
To check if this statement is accurate, we need to understand the definitions of natural numbers and whole numbers. Natural numbers are positive integers (excluding zero), whereas whole numbers consist of natural numbers along with zero. Since every natural number is also a whole number (as it does not include negative numbers), statement C is correct.
D: All rational numbers are natural numbers.
Following the definitions we discussed earlier, it is evident that this statement is false. Rational numbers can be expressed as fractions, including fractions that are less than zero. While natural numbers consist of positive integers, rational numbers extend beyond that range and include negative numbers, fractions, and decimals.
In summary, the correct statement among the given options is:
C: All natural numbers are whole numbers.