Certainly! Let's break down the statement "all the integers are rational numbers, but not all rational numbers are integers" with clear examples.
### Definitions
1. Integers: These are the set of whole numbers including negative numbers, zero, and positive numbers. For example: -3, -2, -1, 0, 1, 2, 3, etc.
2. Rational Numbers: These are the numbers that can be expressed as a fraction or quotient of two integers, where the denominator is not zero. For example: 1/2, 3/4, 5, -1, -2/3, etc.
### Explanation with Examples
1. All Integers are Rational Numbers
To understand why all integers are rational numbers, consider that any integer [tex]\(i\)[/tex] can be written as a fraction where the numerator is [tex]\(i\)[/tex] and the denominator is 1. This means every integer conforms to the definition of a rational number.
Example 1: Integer
- Consider the integer [tex]\(5\)[/tex]. It can be written as [tex]\( \frac{5}{1} \)[/tex]. Clearly, 5 is a rational number because it is expressed as a quotient of two integers where the denominator is not zero.
### Another example of an integer being rational is -3:
- Consider the integer [tex]\(-3\)[/tex]. It can be written as [tex]\( \frac{-3}{1} \)[/tex]. This is also a rational number by the definition.
2. Not All Rational Numbers are Integers
While every integer can be considered a rational number, the converse is not true. There are rational numbers that result from two integers but aren't whole numbers themselves.
Example 2: Rational but not an Integer
- Consider the rational number [tex]\(\frac{3}{2}\)[/tex]. This is clearly a rational number because it is the quotient of two integers (3 and 2). However, [tex]\(\frac{3}{2}\)[/tex] simplifies to 1.5, which is not a whole number; hence, it is not an integer.
### Another example of a rational number that’s not an integer is:
- Consider the rational number [tex]\(\frac{7}{3}\)[/tex]. This can be expressed as 2.333…, which is not an integer but is still a rational number because it is derived from the integers 7 and 3.
### Summary
- Integers such as -3, 0, 1, and 5 can all be expressed as fractions like [tex]\(\frac{-3}{1}\)[/tex], [tex]\(\frac{0}{1}\)[/tex], [tex]\(\frac{1}{1}\)[/tex], and [tex]\(\frac{5}{1}\)[/tex]. Therefore, all integers are rational numbers.
- Rational Numbers such as [tex]\(\frac{3}{2}\)[/tex] or [tex]\(\frac{7}{3}\)[/tex] are not whole numbers but they fit the definition of rational numbers as they are quotients of two integers.
Hence, through these examples, the statement "all the integers are rational numbers, but not all rational numbers are integers" is effectively illustrated.
