Which statement is correct?
A. All intergers are rational numbers.
B. All irrational numbers are whole numbers.
C. A real number must be a rational number.
D. A repeating decimal is an irrational number
I think it's either A or C.
To determine which statement is correct, let's analyze each statement one by one:
A. All integers are rational numbers.
To check if this statement is correct, we need to understand the definitions of integers and rational numbers. Integers are whole numbers (positive, negative, or zero) without any decimal or fractional parts, such as -3, 0, 2. Rational numbers, on the other hand, are numbers that can be expressed as a fraction, where both the numerator and denominator are integers and the denominator is not zero. Examples of rational numbers include 1/2, 3/4, -5/7.
Now, since all integers can be expressed as a fraction (e.g., -3 can be expressed as -3/1), it follows that all integers are indeed rational numbers. Therefore, statement A is correct.
B. All irrational numbers are whole numbers.
To verify this statement, we need to understand the definition of irrational numbers. Irrational numbers are numbers that cannot be expressed as a fraction or a ratio of two integers. They cannot be written as terminating or repeating decimals. Some examples of irrational numbers are √2, π, and e.
Since whole numbers, by definition, are integers without any decimal or fractional parts, it is not true to say that all irrational numbers are whole numbers. In fact, most irrational numbers do have decimal parts. Hence, statement B is incorrect.
C. A real number must be a rational number.
To assess this statement, we need to understand the concepts of real numbers and rational numbers. Real numbers include all the rational numbers (such as integers, fractions, and terminating/repeating decimals) as well as irrational numbers (such as √2 and π).
Since real numbers encompass both rational and irrational numbers, it is true to say that a real number does not have to be a rational number. In fact, irrational numbers are also real numbers. Thus, statement C is incorrect.
D. A repeating decimal is an irrational number.
To evaluate this statement, we need to determine the nature of repeating decimals. A repeating decimal is a decimal representation in which one or more digits repeat infinitely. For example, 0.3333... or 1.272727... are both repeating decimals.
Contrary to what statement D suggests, repeating decimals are actually rational numbers. They can always be expressed as a fraction. For instance, the repeating decimal 0.3333... can be written as 1/3, and 1.272727... can be written as 14/11. Thus, statement D is incorrect.
To summarize:
Statement A is correct. All integers are rational numbers.
Statement B is incorrect. Not all irrational numbers are whole numbers.
Statement C is incorrect. A real number can be rational or irrational.
Statement D is incorrect. A repeating decimal is a rational number.
Therefore, the correct statement is A: All integers are rational numbers.
The correct statement is C. A real number must be a rational number.
Explanation:
A. All integers are rational numbers - This statement is incorrect. Although all integers can be represented as rational numbers, not all rational numbers are integers. Rational numbers include fractions as well.
B. All irrational numbers are whole numbers - This statement is incorrect. Irrational numbers are numbers that cannot be expressed as fractions or as terminating or repeating decimals. Whole numbers are integers starting from zero.
C. A real number must be a rational number - This statement is correct. A real number can be either rational or irrational. Therefore, it includes both rational numbers and irrational numbers.
D. A repeating decimal is an irrational number - This statement is incorrect. A repeating decimal is a rational number because it can be expressed as a fraction. For example, 0.333... can be written as 1/3.