Key takeaway: Any value that equals one integer divided by a nonzero integer is a rational number; this covers whole numbers, fractions, and decimals that stop or repeat.
Grade 66.NS.C.6Numbers and Operations
Definition of Rational Number
A rational number is a value you can build by dividing one integer by a second integer that is not zero. The name comes from “ratio”.
Integers such as −9 qualify, because −9 = −9/1. So do fractions like 5/6, terminating decimals like 2.35 (which is 47/20), and repeating decimals like 0.272727… (which is 3/11). Rational numbers fill the number line densely, with negatives on the left of 0 and positives on the right. Numbers whose decimals never end and never repeat are not in this group.
Formula and notation
The pattern to look for is one integer over another:
\[\frac{p}{q} \quad \text{where } p \text{ and } q \text{ are integers and } q \neq 0\]
A negative sign may sit in front of the fraction, on the numerator or on the denominator: \(-\frac{5}{8} = \frac{-5}{8} = \frac{5}{-8}\).
Key facts
- Every integer is in this group, since it can be written over 1.
- A decimal belongs here when it terminates or repeats a block of digits forever.
- Dividing one by another (with a nonzero divisor) always lands back in the same group.
- Numbers like \(\pi\) cannot be written as a fraction of integers, so they are excluded.
Rational Number example
Show that \(2.35\), \(-9\) and \(0.\overline{27}\) (the block 27 repeats) are all rational by writing each as a fraction of integers.
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\(2.35 = \frac{235}{100} = \frac{47}{20}\).
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\(-9 = \frac{-9}{1}\).
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Let \(n = 0.\overline{27}\). Then \(100n = 27.\overline{27}\), so \(99n = 27\) and \(n = \frac{27}{99} = \frac{3}{11}\).
The fractions are \(\frac{47}{20}\), \(\frac{-9}{1}\) and \(\frac{3}{11}\), so all three are rational.
Common mistake
Watch out
Mia says \(0.\overline{27}\) cannot be rational because its digits never end.
Why it is wrong
Endless digits are fine as long as a block repeats. The decimal repeats “27”, and it equals \(\frac{3}{11}\). Only decimals that go on forever without any repeating pattern are not rational.
Related terms and lessons
- Grade 6 · Integers, Opposites and Absolute ValueLessonWorksheetQuiz
- Grade 7 · Rational Numbers: Decimals, Fractions and Word ProblemsLessonWorksheetQuiz
- Grade 8 · Rational and Irrational NumbersLessonWorksheetQuiz
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