Key definition
Rational number: A number is rational if it can be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
Step-by-step classification of \( \frac{7}{3} \)
1) Check the definition directly
The number \( \frac{7}{3} \) already has the form \( \frac{p}{q} \) with integers \( p=7 \) and \( q=3 \), and \( 3 \neq 0 \). Therefore, \( \frac{7}{3} \) satisfies the definition of a rational number.
2) Confirm it is in simplest form (optional but common)
A fraction is in simplest form when the numerator and denominator share no common factor greater than 1. Compute the greatest common divisor:
\[ \gcd(7,3)=1 \]
Since the greatest common divisor is 1, \( \frac{7}{3} \) is already reduced, reinforcing that it is a valid rational representation.
3) Show equivalent forms (mixed number and decimal)
Converting an improper fraction to a mixed number:
\[ \frac{7}{3} = 2 + \frac{1}{3} \]
Converting to a decimal uses division:
\[ 7 \div 3 = 2.333\ldots = 2.\overline{3} \]
The repeating decimal \( 2.\overline{3} \) is another hallmark of rational numbers: every rational number has a terminating or repeating decimal expansion.
Quick summary table
| Item | Value / conclusion |
|---|---|
| Form \( \frac{p}{q} \) with integers | \( p=7 \), \( q=3 \), and \( q \neq 0 \) ✅ |
| Simplest form check | \( \gcd(7,3)=1 \) so \( \frac{7}{3} \) is reduced ✅ |
| Mixed number form | \( \frac{7}{3}=2+\frac{1}{3} \) |
| Decimal form | \( \frac{7}{3}=2.\overline{3} \) (repeating) ✅ |
| Final classification | \( \frac{7}{3} \) is a rational number ✅ |
Visualization: number line location of \( \frac{7}{3} \)
Conclusion
Yes—\( \frac{7}{3} \) is a rational number because it is exactly the ratio of two integers, 7 and 3, with a nonzero denominator.