Write whether the following statements are true or false
a) Every rational number is an integer
b) Every integer is a rational number
c) Every integer is a whole number
d) Every irrational number is a real number
e) All real numbers are rational numbers
The correct answer and explanation is:
a) Every rational number is an integer
False.
A rational number is any number that can be written as a fraction, where both the numerator and denominator are integers, and the denominator is not zero. For example, 1/2 is a rational number but is not an integer because it is not a whole number. Therefore, not every rational number is an integer. Integers are a subset of rational numbers, but there are rational numbers that are not integers, such as 1/2 or 3/4.
b) Every integer is a rational number
True.
Integers are a type of rational number. An integer can be expressed as a fraction with a denominator of 1. For example, the integer 3 can be written as 3/1, which makes it a rational number. Therefore, all integers are rational numbers.
c) Every integer is a whole number
False.
Whole numbers are a set of numbers that includes all the positive integers and zero. However, integers also include negative numbers, and whole numbers do not. For example, -3 is an integer but not a whole number. Thus, not every integer is a whole number.
d) Every irrational number is a real number
True.
Irrational numbers are numbers that cannot be expressed as a simple fraction. They include numbers like √2, π, and e. These numbers do not terminate or repeat when written as decimals. However, they are still part of the real number system. The real number system consists of both rational and irrational numbers, so every irrational number is indeed a real number.
e) All real numbers are rational numbers
False.
Real numbers include both rational and irrational numbers. Rational numbers can be expressed as fractions (e.g., 1/2, 3/4), while irrational numbers cannot be written as fractions (e.g., √2, π). Since irrational numbers exist within the real number system, not all real numbers are rational.
In summary, rational numbers include both integers and fractions, whereas irrational numbers cannot be written as fractions. Real numbers are the broadest category that includes both rational and irrational numbers.