Exercise 1.3
Maths - Exercise
-
- Write the following in decimal form and say what kind of decimal expansion each has
:
- \(\frac{36}{100}\)
Solution:
\(\frac{36}{100} = 0.36\Rightarrow\color{blue}{\text{Terminating}}\\\\\) - \(\frac{1}{11}\)
Solution:
\(\scriptsize\frac{1}{11}=0.\overline{09}\)$$ \require{enclose} \begin{array}{rll} 0.0909\ldots &&\\ 11\enclose{longdiv}{1.0000\phantom{0}} &&\\ \underline{99\phantom{0000}} &&\\ 100\phantom{000}&&\\\underline{99\phantom{000}}&&\\1\phantom{000}&&\\\vdots \end{array} $$Non-Terminating, repeating
- \(\frac{36}{100}\)
- \(4\frac{1}{8}\)
Solution:
\(\scriptsize 4\frac{1}{8}=\frac{33}{8}\)=4.125 \[ \require{enclose} \begin{array}{r} 4.125\phantom{000} &&\\ 11\enclose{longdiv}{33.000}\phantom{00} &&\\ \underline{32\phantom{00}} \phantom{0000}&&\\ 10\phantom{00000}&&\\\underline{8\phantom{00}}\phantom{000}&&\\20\phantom{0000}&&\\ \underline{20\phantom{00}}\phantom{00}&& \\0\phantom{0000} && \\ \end{array} \] Terminating - \(\frac{3}{13}\)
Solution:
\(\scriptsize\frac{3}{13}=0.\overline{230769}\) \[ \require{enclose} \begin{array}{r} 0.230769\ldots &&\\ 13\enclose{longdiv}{3.00000000} &&\\ \underline{26\phantom{0000000}} &&\\ 40\phantom{000000}&&\\ \underline{39\phantom{00000}}&&\\100\phantom{000}&&\\ \underline{91\phantom{000}}&& \\90\phantom{00}&& \\\underline{78\phantom{00}}&&\\120\phantom{0} &&\\\underline{117}\phantom{0} &&\\ 3\phantom{0} &&\\ \vdots && \end{array} \]Non-Terminating, Repeating
- \(\frac{2}{11}\)
Solution:
\(\scriptsize\frac{2}{11}=0.\overline{18}\)\[ \require{enclose} \begin{array}{rll} 0.1818\ldots &&\\ 11\enclose{longdiv}{2.000\phantom{00}} &&\\ \underline{11\phantom{00000}} &&\\ 90\phantom{0000}&&\\ \underline{88\phantom{0000}}&&\\20\phantom{000}&&\\ \vdots && \end{array} \]Non-Terminating, Repeating
- \(\frac{329}{400}\)
Solution:
\(\scriptsize\frac{329}{400}=0.08225\)\[ \require{enclose} \begin{array}{rll} 0.08225\phantom{0000} &&\\ 400\enclose{longdiv}{329.00\phantom{00000}} &&\\ \underline{320\phantom{0000000}} &&\\ 900\phantom{000000}&&\\ \underline{800\phantom{000000}}&&\\1000\phantom{00000}&&\\ \underline{800\phantom{00000}}&&\\2000\phantom{0000}&&\\ \underline{2000\phantom{0000}}&&\\0\phantom{0000}&&\\ \end{array} \]Terminating
- You know that \(\frac{1}{7}
= 0.\overline{142857}\). Can you predict what the decimal expansions of
\(\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}\) are, without actually doing the long
division? If
so, how?
Solution:
It is given that \[ \begin{align*} \scriptsize\frac{1}{7} &\scriptsize= 0.\overline{142857} \\ \scriptsize\Rightarrow \\ \scriptsize\frac{2}{7} &\scriptsize= 2\times\left(\frac{1}{7}\right) = 2\times 0.\overline{142857} = 0.\overline{285714} \\ \scriptsize\frac{3}{7} &\scriptsize= 3\times\left(\frac{1}{7}\right) = 3\times 0.\overline{142857} = 0.\overline{428571} \\ \scriptsize\frac{4}{7} &\scriptsize= 4\times\left(\frac{1}{7}\right) = 4\times 0.\overline{142857} = 0.\overline{571428} \\ \scriptsize\frac{5}{7} &\scriptsize= 5\times\left(\frac{1}{7}\right) = 5\times 0.\overline{142857} = 0.\overline{714285} \\ \scriptsize\frac{6}{7} &\scriptsize= 6\times\left(\frac{1}{7}\right) = 6\times 0.\overline{142857} = 0.\overline{857142} \end{align*} \] - Express the following in the form \(\frac{p}{q}\), where p and q are integers and q
≠ 0
- \(0.\bar{6}\)
Solution:
\[\begin{align}\scriptsize\text{Let } x &\scriptsize= 0.\overline{6}\\\ \scriptsize\Rightarrow x &\scriptsize= 0.6666\ldots\tag{1}\\ \scriptsize10x &\scriptsize=6.6666\ldots\tag{2} \end{align}\] Subtracting (1) from (2): \[ \require{cancel} \begin{aligned} \scriptsize10x &\scriptsize= 6.6666\ldots\\\scriptsize x &\scriptsize= 0.6666\ldots \\ \hline \scriptsize9x &\scriptsize= 6 \\ \scriptsize x &\scriptsize= \frac{\cancelto{2}6}{\cancelto{3}{9}} = \frac{2}{3} \end{aligned}\] - \(0.4\bar{7}\)
Solution:
\[ \begin{align} \scriptsize\text{Let } x &\scriptsize= 0.4\overline{7} \\ \scriptsize\Rightarrow x &\scriptsize= 0.47777\ldots \tag{1} \\ \scriptsize10x &\scriptsize= 4.77777\ldots \tag{2} \\ \scriptsize100x &\scriptsize= 47.77777\ldots \tag{3} \end{align} \] Subtracting eqn (2) from eqn (3): \[ \begin{align} \scriptsize100x &\scriptsize= 47.7777\ldots \\\scriptsize10x&\scriptsize=04.7777\ldots \\\hline \scriptsize90x &\scriptsize= 43 \\ \scriptsize x &\scriptsize= \frac{43}{90} \end{align} \] - \(0.\overline{001}\\\)
Solution:
\[\begin{align}\scriptsize \text{let } x&\scriptsize=0.\overline{001}\\\Rightarrow\scriptsize x&\scriptsize= 0.001001001\ldots\tag{1}\\ \end{align}\] Multiplying both side of eqn (1) by 1000 \[\begin{align} \scriptsize1000x&\scriptsize=1.001001\ldots\tag{2}\\ \end{align}\] Subtracting eqn(1) from eqn(2): \[\begin{align}\scriptsize1000x&\scriptsize=1.001001\ldots\\\scriptsize x&\scriptsize=0.001001\ldots\\\hline\scriptsize 999x&\scriptsize=1\\ \scriptsize x&\scriptsize=\frac{1}{999}\\\\\scriptsize\text{hence, } 0.\overline{001}&\scriptsize = \frac{1}{999} \end{align} \]
- \(0.\bar{6}\)
- Write the following in decimal form and say what kind of decimal expansion each has
:
Solution:
\[ \require{cancel} \begin{align} \text{let } x &=0.99999\ldots\tag{1}\end{align}\] Multiplying both side of eqn (1) by 10 \[\begin{align}10x &=9.99999\ldots\tag{2}\end{align}\] Subtracting eqn(1) from eqn(2) \[\begin{align}10x &=9.99999\ldots\\x &=0.99999\ldots\\\hline 9x &=9\\ x &=\frac{\cancel{9}}{\cancel{9}}=1\\\\\text{hence, } 0.99999\ldots& = 1 \end{align} \]
Solution:
\(\scriptsize \require{enclose} \begin{array}{rlc} &&\phantom{000000000}0.588235294117647\\ && \phantom{0000000}17\enclose{longdiv}{1.000000000000000}\\&& \underline{85\phantom{00}}\\&&150\phantom{0}\\&& \underline{136\phantom{0}}\\&&\phantom{0}140\\&& \phantom{000}\underline{136\phantom{0}}\\&&\phantom{0000}40\phantom{}\\&& \phantom{000000}\underline{34\phantom{00}}\\&&\phantom{0000000}60\\&& \phantom{000000000}\underline{51\phantom{00}}\\&&\phantom{000000000}90\\&& \phantom{00000000000}\underline{85\phantom{00}}\\&&\phantom{00000000000}50\\&& \phantom{0000000000000}\underline{34\phantom{00}}\\&&\phantom{000000000000}160\\&& \phantom{00000000000000}\underline{153\phantom{00}}\\&&\phantom{000000000000000}70\\&& \phantom{00000000000000000}\underline{68\phantom{00}}\\&&\phantom{00000000000000000}20\\&& \phantom{0000000000000000000}\underline{17\phantom{00}}\\&&\phantom{000000000000000000}130\\&& \phantom{00000000000000000000}\underline{119\phantom{00}}\\&&\phantom{00000000000000000000}110\\&& \phantom{00000000000000000000}\underline{102}\\&&\phantom{00000000000000000000000}80\\&& \phantom{0000000000000000000000000}\underline{68}\phantom{00}\\&&\phantom{0000000000000000000000000}120\\&& \phantom{000000000000000000000000000}\underline{119\phantom{00}}\\&&\phantom{0000000000000000000000000000}1\\&& \end{array} \) \[\] \(\frac{1}{17}=0.\overline{588235294117647}\)
Solution:
let rational numbers be \(\frac{1}{2}, \frac{1}{4} ,\frac{1}{5} ,\frac{1}{8}, \frac{3}{10}\text{ and } \frac{2}{5}\) with terminating decimals. In these cases q={2,4,5,8, 10} we can observe that \(2=2\times 1\\4=2\times 2\times 1\\8=2\times 2\times 2\times 1\\5=5\times 1\\10=2\times 5\times 1\) all have either 2^n or 5^n or both as a factor. we can conclude that the prime factorisation of q has only powers of 2 or powers of 5 or both
- \(\pi\approx 3.1415926535\ldots\)
- \(e\approx 2.7182818284\ldots\)
- \(\sqrt[2]{2}\approx 1.4142135623\ldots\)
Solution:
To find three different irrational numbers between the rational numbers \(\frac{5}{7} \approx 0.7143 \text{ and }\\\frac{9}{11}\approx 0.8181\) We need to choose irrational numbers that fall within that interval. In the last question, we already find values of irrational numbers \(\pi, e \text{ and }\sqrt{2}\), we will subtract any number from these to get a value between the given range \(\begin{align}\pi-2.4&\approx 0.7415\ldots\\\sqrt{2}-0.7 &\approx 0.8142\ldots\\e-2&\approx 0.7182818284\ldots\end{align}\)
Irrational Number - Rational Number is always an Irrational Number.
- \(\sqrt{23}\)
Solution:
\(\scriptsize \begin{array}{r|l} & 4.7958315\ldots\\\hline 4&\overline{23}\\ +4&16\\\hline 87&\phantom{0}700\\+7&\phantom{0}609\\\hline 949&\phantom{000}9100\\+9 & \phantom{000}8541\\\hline 9585 & \phantom{0000}55900\\+5 & \phantom{0000}47925\\\hline 95908 & \phantom{00000}797500\\+8 &\phantom{00000}767264\\\hline 959163&\phantom{000000}3023600\\+3&\phantom{000000}2877489\\\hline 9591661&\phantom{0000000}15611100\\+1&\phantom{00000000}9591661\\\hline 95916625 &\phantom{00000000}601943900\\+5 & \phantom{00000000}479583125\\\hline 95916630 &\phantom{00000000}12136087500\\&\phantom{0000000000000000000}\vdots \end{array} \) \[\] \(\sqrt{23} = 4.7958315\ldots\) \[\] \(\Rightarrow\)Non Repeating, Non Terminating decimal; hence Irrational Number - \[
\begin{array}{r|l}
\scriptsize 5 & \scriptsize 225 \\
\hline
\scriptsize 5 & \scriptsize 45 \\
\hline
\scriptsize 3 & \scriptsize 9 \\
\hline
\scriptsize 3 & \scriptsize 3
\end{array}
\]
\( \begin{align*} \scriptsize\sqrt{225} &\scriptsize= \sqrt{5\times 5\times 3\times 3} \\ &\scriptsize= 5\times 3 \\ &\scriptsize= 15 \\ \Rightarrow\;\; \color{blue}{\text{Rational Number}} \end{align*} \) - 0.3796 \(\Rightarrow\color{blue}{\text{Terminating Decimal,}\\\text{hence Rational Number}}\)
- 7.478478 \(\Rightarrow 7.478478=7.\overline{478} \Rightarrow\color{blue}{\text{Repeating,}\\\text{ Non- Terminating decimal,}\\\text{Hence Rational Number}}\)
- 1.101001000100001... \(\Rightarrow\) Non-Repeating and Non-Terminating Decimal, hence Irrational Number
