Key takeaway: A terminating decimal is a decimal with a finite number of digits after the point, so the division that produces it ends with a remainder of zero.
Grade 77.NS.A.2dNumbers and Operations
Definition of Terminating Decimal
A terminating decimal is a decimal that comes to an end, such as 0.375 or 4.8. After the last digit, every remaining place is zero.
Every one equals a fraction with a power of 10 in the denominator: 0.375 is \(\frac{375}{1000}\). A reduced fraction ends exactly when its denominator is built only from the primes 2 and 5, the two primes hiding inside every power of 10. Then the denominator can be stretched to 10, 100 or 1000 and no remainder is left behind.
Formula and notation
A fraction in lowest terms \(\frac{a}{b}\) gives a terminating decimal when
\[b = 2^m \cdot 5^n\]
for whole numbers \(m\) and \(n\). Here is one case: \(\frac{7}{20} = \frac{7}{2^2 \cdot 5} = 0.35\), which stops after two places.
Key facts
- Denominators such as 2, 4, 5, 8, 10, 16, 20 and 25 always terminate.
- A denominator with a factor of 3, 7 or 11 (after simplifying) never terminates.
- The number of decimal places equals the larger of the exponents \(m\) and \(n\).
- Whole numbers qualify too, since 14 can be written 14.0 with nothing after the zero.
Terminating Decimal example
Without dividing, decide which of these fractions are terminating decimals: \(\frac{21}{40}\), \(\frac{7}{15}\), \(\frac{9}{12}\). Then convert the ones that terminate.
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\(\frac{21}{40}\): \(40 = 2^3 \cdot 5\), so it terminates. \(21 : 40 = 0.525\).
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\(\frac{7}{15}\): \(15 = 3 \cdot 5\), and the 3 stays, so it does not terminate.
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\(\frac{9}{12}\): reduce it to \(\frac{3}{4}\) first. As \(4 = 2^2\), the decimal stops at \(0.75\).
\(\frac{21}{40} = 0.525\) and \(\frac{9}{12} = 0.75\) are terminating; \(\frac{7}{15}\) is not.
Common mistake
Watch out
Kai looks at \(\frac{9}{12}\), sees a 3 inside 12, and declares that the decimal repeats forever.
Why it is wrong
The test works only on a fraction in lowest terms. Dividing top and bottom by 3 turns \(\frac{9}{12}\) into \(\frac{3}{4}\), whose denominator has just 2s, so \(\frac{9}{12} = 0.75\) ends. Always reduce the fraction before checking the prime factors of the denominator.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Numbers and Operations