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SubjectsMathematics
दशमलव निरूपण
In this Class 9 Mathematics topic from the Number Systems chapter, students learn how numbers are expressed in decimal form and how decimal expansions relate to rational and irrational numbers. They examine terminating and non-terminating decimals, identify repeating patterns, and connect decimal representations with fractions. The topic builds accuracy in comparing, interpreting, and converting numerical forms while strengthening understanding of the structure and properties of real numbers.
TOPIC PRACTICE
Up to 6 questions from this page. Select your focus, then start.
In \(\frac{13}{40}\), the denominator is \(40=2^3\times5\), so its decimal expansion terminates. The denominator of \(\frac{7}{18}\) also contains 3, so it is non-terminating recurring. Exam tip: in lowest form, only 2 and 5 may occur in the denominator.
\(\frac{13}{2^2\times5\times7}=\frac{13}{140}\) is in lowest terms because 13 and 140 have no common factor. The denominator \(140=2^2\times5\times7\) also contains the prime factor 7. A rational number has a terminating decimal expansion only if, in lowest terms, its denominator has no prime factors other than 2 and/or 5. Therefore, this decimal expansion is non-terminating recurring. A non-terminating non-recurring decimal represents an irrational number. Exam tip: reduce the fraction first, then inspect the prime factors of its denominator.
\(\frac{21}{84}=\frac14\), and \(4=2^2\); hence its decimal form terminates at \(0.25\). In \(\frac{7}{30}\), the factor 3 remains, so it is non-terminating recurring. Exam tip: simplify the fraction before checking the denominator.
The digit block 285714 repeats continuously, so this is a non-terminating recurring decimal. In fact, \(\frac{2}{7}=0.285714285714\ldots\), so both forms represent the same rational number. Option B is wrong because the decimal does not end, and option D is wrong because every recurring decimal is rational. Exam tip: A decimal that does not end but repeats a fixed block of digits is recurring.
The denominator is \(2^3\times5^5\), and 43 has no common factor with 2 or 5. Multiply the numerator and denominator by \(2^2\): \(\frac{43}{2^3\times5^5}=\frac{172}{10^5}\). Hence, the decimal expansion terminates after 5 decimal places. Option 8 is incorrect because the exponents are not added; the larger exponent determines the required number of places. Exam tip: for a reduced denominator of the form \(2^m5^n\), the decimal terminates in at most \(\max(m,n)\) places.
A rational number has a terminating decimal only when, in lowest form, its denominator has prime factors 2 and/or 5 only. Since \(120=2^3\times3\times5\) includes 3, \(\frac{17}{120}\) is non-terminating recurring. Exam tip: always check the denominator after simplification.
QUIZ COMPLETE
