Textbook - Real Numbers Questions in English

(B) To determine if a rational number $\frac{p}{q}$ has a terminating decimal expansion,we check the prime factorization of the denominator $q$.
If $q = 2^n \times 5^m$,where $n$ and $m$ are non-negative integers,the decimal expansion is terminating.
Here,$q = 455$.
Prime factorization of $455 = 5 \times 7 \times 13$.
Since the prime factorization contains factors other than $2$ and $5$ (specifically $7$ and $13$),the decimal expansion of $\frac{64}{455}$ is non-terminating repeating.

(N/A) To find the decimal expansion of $\frac{15}{1600}$,we perform long division:
$1$. Divide $15$ by $1600$.
$2$. Since $15 < 1600$,we place a decimal point and add zeros to $15$ to make it $15.000000$.
$3$. $1600$ goes into $15000$ nine times $(1600 \times 9 = 14400)$,leaving a remainder of $600$.
$4$. Bring down $0$ to make it $6000$. $1600$ goes into $6000$ three times $(1600 \times 3 = 4800)$,leaving a remainder of $1200$.
$5$. Bring down $0$ to make it $12000$. $1600$ goes into $12000$ seven times $(1600 \times 7 = 11200)$,leaving a remainder of $800$.
$6$. Bring down $0$ to make it $8000$. $1600$ goes into $8000$ five times $(1600 \times 5 = 8000)$,leaving a remainder of $0$.
Thus,the decimal expansion of $\frac{15}{1600}$ is $0.009375$.

(D) To determine if the rational number $\frac{29}{343}$ has a terminating decimal expansion,we examine the prime factorization of the denominator.
$1$. The denominator is $343 = 7^3$.
$2$. $A$ rational number $\frac{p}{q}$ has a terminating decimal expansion if and only if the prime factorization of $q$ is of the form $2^n \times 5^m$,where $n$ and $m$ are non-negative integers.
$3$. Since the prime factorization of $343$ is $7^3$,which does not contain factors of $2$ or $5$,the decimal expansion is non-terminating and repeating.
$4$. Therefore,$\frac{29}{343}$ does not have a terminating decimal expansion.

(N/A) To find the decimal expansion of the rational number $\frac{23}{2^{3} 5^{2}}$,we first simplify the denominator:
$\frac{23}{2^{3} \times 5^{2}} = \frac{23}{8 \times 25} = \frac{23}{200}$
Now,we perform the division of $23$ by $200$:
$23 \div 200 = 0.115$
Alternatively,we can make the powers of $2$ and $5$ equal:
$\frac{23}{2^{3} \times 5^{2}} = \frac{23 \times 5}{2^{3} \times 5^{2} \times 5} = \frac{115}{2^{3} \times 5^{3}} = \frac{115}{(2 \times 5)^{3}} = \frac{115}{10^{3}} = \frac{115}{1000} = 0.115$

(C) rational number $\frac{p}{q}$ has a terminating decimal expansion if the prime factorization of the denominator $q$ is of the form $2^n 5^m$,where $n$ and $m$ are non-negative integers.
In the given rational number $\frac{129}{2^{2} 5^{7} 7^{5}}$,the denominator contains a factor of $7^5$ in addition to $2$ and $5$.
Since the prime factorization of the denominator includes a prime factor other than $2$ or $5$ (which is $7$),the decimal expansion of this rational number is non-terminating and repeating.
Therefore,it does not have a terminating decimal expansion.

(N/A) To find the decimal expansion of the rational number $\frac{6}{15}$,we first simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor,which is $3$.
$\frac{6}{15} = \frac{2 \times 3}{5 \times 3} = \frac{2}{5}$.
Now,we convert $\frac{2}{5}$ into a decimal by dividing $2$ by $5$:
$2 \div 5 = 0.4$.
Thus,the decimal expansion of $\frac{6}{15}$ is $0.4$.

(N/A) To find the decimal expansion of the rational number $\frac{35}{50}$,we can perform long division or simplify the fraction first.
Method $1$: Long Division
Divide $35$ by $50$:
$35 \div 50 = 0.7$
Method $2$: Converting to a denominator of power of $10$
$\frac{35}{50} = \frac{35 \times 2}{50 \times 2} = \frac{70}{100} = 0.7$
Thus,the decimal expansion of $\frac{35}{50}$ is $0.7$.

(B) First,simplify the fraction $\frac{77}{210}$ by dividing both numerator and denominator by their greatest common divisor,which is $7$.
$\frac{77 \div 7}{210 \div 7} = \frac{11}{30}$.
Next,find the prime factorization of the denominator $30$:
$30 = 2 \times 3 \times 5$.
$A$ rational number has a terminating decimal expansion if and only if the prime factorization of its denominator (in simplest form) is of the form $2^n \times 5^m$,where $n$ and $m$ are non-negative integers.
Since the denominator $30$ contains a factor of $3$ (which is not $2$ or $5$),the decimal expansion of $\frac{11}{30}$ is non-terminating and repeating.
Performing the division $11 \div 30$ gives $0.3666...$ or $0.3\overline{6}$.

(A) The given number is $43.123456789$.
Since this number has a terminating decimal expansion,it is a rational number.
It can be expressed in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \neq 0$.
For a rational number to have a terminating decimal expansion,the prime factorization of the denominator $q$ must be of the form $2^{m} \times 5^{n}$,where $m$ and $n$ are non-negative integers.
Therefore,the prime factors of $q$ are only $2$ or $5$ or both.

(N/A) The given decimal expansion is $0.120120012000120000 \ldots$
$1$. $A$ number is rational if its decimal expansion is either terminating or non-terminating recurring.
$2$. $A$ number is irrational if its decimal expansion is non-terminating and non-recurring.
$3$. In the given number $0.120120012000120000 \ldots$,the pattern of digits does not repeat in a fixed block,and it does not terminate.
$4$. Since the decimal expansion is non-terminating and non-recurring,the given number is an irrational number.
$5$. Because it is irrational,it cannot be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

(N/A) The given number is $43. \overline{123456789}$.
Since the decimal expansion is non-terminating and repeating (recurring),the given number is a rational number.
$A$ rational number has a terminating decimal expansion if and only if the prime factorization of its denominator $q$ is of the form $2^m \times 5^n$,where $m$ and $n$ are non-negative integers.
Since this decimal expansion is non-terminating repeating,the denominator $q$ must have prime factors other than $2$ or $5$.