Which statements is false?
A. A decimal fraction has a multiple of 10 as a denominator.
B. Every rational number can be associated with a point in the real number line.
C. A terminating decimal cannot be expressed as a repeating decimal.
D. The set of rational numbers includes both the set of whole numbers and the set of integers.
A: true
B: true
C: false
e.g. 0.5 = .499999....
D: true
To determine which statement is false, let's analyze each statement:
A. A decimal fraction has a multiple of 10 as a denominator.
To verify if this statement is true or false, we should first understand what a decimal fraction is. A decimal fraction is a fraction where the denominator is a power of 10, such as 10, 100, 1000, and so on. Therefore, statement A is true, as a decimal fraction indeed has a multiple of 10 as a denominator.
B. Every rational number can be associated with a point on the real number line.
To assess the validity of this statement, we need to understand what rational numbers and the real number line entail. Rational numbers are those that can be written as a fraction, where the numerator and denominator are integers. The real number line is a line that represents all real numbers in a graphical format.
By definition, every rational number can indeed be represented as a point on the real number line. So, statement B is true.
C. A terminating decimal cannot be expressed as a repeating decimal.
To determine the accuracy of this statement, we need to understand the difference between terminating and repeating decimals. A terminating decimal is a decimal that ends after a finite number of digits, such as 0.5 or 0.75. Meanwhile, a repeating decimal is a decimal that has a repeating pattern of digits, such as 0.333... or 0.181818....
Since terminating decimals don't have repeating patterns, statement C is true. Therefore, a terminating decimal cannot be expressed as a repeating decimal.
D. The set of rational numbers includes both the set of whole numbers and the set of integers.
To assess the validity of this statement, we must understand the definitions of the sets mentioned. The set of whole numbers includes all positive integers and zero: {0, 1, 2, 3, ...}. The set of integers includes all positive and negative whole numbers: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
As the set of rational numbers encompasses all numbers that can be expressed as fractions, including whole numbers and integers, statement D is true.
Therefore, the false statement is: C. A terminating decimal cannot be expressed as a repeating decimal.