Which of the following statements is false?
A rational number cannot be an irrational number.
A natural number cannot be an integer.
An irrational number is always a real number.
An integer will always be a rational number.
You must know the definitions of each of the number sets, and know
which set contains the other.
e.g. An irrational number is always a real number.
Since the set of real numbers contains all the others, this statement is true
Let me know what you think of the others.
The false statement is: "A natural number cannot be an integer."
To determine which of the statements is false, let's analyze each of them:
1. "A rational number cannot be an irrational number."
To determine if this statement is true or false, it is important to understand the definitions of rational and irrational numbers. A rational number is a number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. An irrational number, on the other hand, is a number that cannot be expressed as a fraction and has an infinite number of non-repeating digits after the decimal point. Since rational and irrational numbers are mutually exclusive, this statement is true. Therefore, this statement is NOT false.
2. "A natural number cannot be an integer."
A natural number is a positive whole number starting from 1. An integer, on the other hand, includes both positive and negative whole numbers, including zero. Since all natural numbers are integers, this statement is false. Therefore, this statement IS false.
3. "An irrational number is always a real number."
This statement is true. All irrational numbers are real numbers because they exist on the real number line and cannot be expressed as fractions or terminating decimals.
4. "An integer will always be a rational number."
This statement is true. All integers can be expressed as the quotient of two integers because they do not have any fractional or decimal components. Therefore, this statement is NOT false.
Therefore, the second statement, "A natural number cannot be an integer," is the statement that is false.