M L Aggarwal Solutions for Chapter: Real Numbers, Exercise 4: Exercise 1.4

Without actually performing the long division, state whether $\frac{77}{210}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion. Also, find the number of places after which the decimal expansion terminates.

Write the decimal expansion of $\frac{237}{1500}$ having terminating decimal expansion.

Write the denominator of the rational number $\frac{257}{5000}$ in the form $2^m\times 5^n$, where $m, n$ are non-negative integers. Hence, write its decimal expansion without actual division.

A rational number in its decimal expansion is $327.7081$. What can you say about the prime factorisation of $q$ when this number is expressed in the form $\frac{p}{q}$? Give reason.

The following real number have decimal expansion of $37.09158$. State whether they are rational or not. If they are rational and expressed in the form $\frac{p}{q}$, where $p, q$ are integers, $q\ne 0$ and $p, q$ are co-prime, then what can you say about the prime factors of $q$?

The following real number have decimal expansion of $423.\overline{04567}$. State whether they are rational or not. If they are rational and expressed in the form $\frac{p}{q}$, where $p, q$ are integers, $q\ne 0$ and $p, q$ are co-prime, then what can you say about the prime factors of $q$?

The following real number have decimal expansion of $8.9010010001....$. State whether they are rational or not. If they are rational and expressed in the form $\frac{p}{q}$, where $p, q$ are integers, $q\ne 0$ and $p, q$ are co-prime, then what can you say about the prime factors of $q$?

The following real number have decimal expansion of $2.3476817681...$. State whether they are rational or not. If they are rational and expressed in the form $\frac{p}{q}$, where $p, q$ are integers, $q\ne 0$ and $p, q$ are co-prime, then what can you say about the prime factors of $q$?