All integers are rational numbers. True False
The correct answer and explanation is :
True.
All integers are rational numbers. To understand why this is true, we need to recall the definitions of both types of numbers.
Integers
An integer is any number in the set of whole numbers and their negatives, including zero. This set can be represented as:
[
{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots }
]
These numbers do not have any fractional or decimal parts.
Rational Numbers
A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero. In mathematical terms, a rational number is any number that can be written in the form:
[
\frac{a}{b}
]
where ( a ) and ( b ) are integers, and ( b \neq 0 ).
Why All Integers Are Rational Numbers
Any integer can be expressed as a rational number by writing it as a fraction with the integer as the numerator and 1 as the denominator. For example:
- The integer 5 can be written as ( \frac{5}{1} ).
- The integer -3 can be written as ( \frac{-3}{1} ).
- The integer 0 can be written as ( \frac{0}{1} ).
Since all integers can be expressed in the form ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 ), it follows that all integers are indeed rational numbers.
Conclusion
Thus, the statement “All integers are rational numbers” is true. Integers are a subset of rational numbers because they can be expressed as fractions with a denominator of 1. Therefore, every integer is inherently a rational number.