1. Write the following in decimal form and say what kind of decimal expansion each has :
      1. \(\frac{36}{100}\)
        Solution:
        \(\frac{36}{100} = 0.36\Rightarrow\color{blue}{\text{Terminating}}\\\\\)
      2. \(\frac{1}{11}\)
        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

    2. \(4\frac{1}{8}\)
      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
    3. \(\frac{3}{13}\)
      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

    4. \(\frac{2}{11}\)
      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

    5. \(\frac{329}{400}\)
      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

      6.  
    6. 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 \[ \begin{align*} \scriptsize\frac{1}{7} &\scriptsize= 0.\overline{142857} \\ \scriptsize\Rightarrow \\ \scriptsize\frac{2}{7} &\scriptsize= 2\times\left(\frac{1}{7}\right) = 2\times 0.\overline{142857} = 0.\overline{285714} \\ \scriptsize\frac{3}{7} &\scriptsize= 3\times\left(\frac{1}{7}\right) = 3\times 0.\overline{142857} = 0.\overline{428571} \\ \scriptsize\frac{4}{7} &\scriptsize= 4\times\left(\frac{1}{7}\right) = 4\times 0.\overline{142857} = 0.\overline{571428} \\ \scriptsize\frac{5}{7} &\scriptsize= 5\times\left(\frac{1}{7}\right) = 5\times 0.\overline{142857} = 0.\overline{714285} \\ \scriptsize\frac{6}{7} &\scriptsize= 6\times\left(\frac{1}{7}\right) = 6\times 0.\overline{142857} = 0.\overline{857142} \end{align*} \]
    7. Express the following in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0
      1. \(0.\bar{6}\)
        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): \[ \require{cancel} \begin{aligned} \scriptsize10x &\scriptsize= 6.6666\ldots\\\scriptsize x &\scriptsize= 0.6666\ldots \\ \hline \scriptsize9x &\scriptsize= 6 \\ \scriptsize x &\scriptsize= \frac{\cancelto{2}6}{\cancelto{3}{9}} = \frac{2}{3} \end{aligned}\]
      2. \(0.4\bar{7}\)

        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): \[ \begin{align} \scriptsize100x &\scriptsize= 47.7777\ldots \\\scriptsize10x&\scriptsize=04.7777\ldots \\\hline \scriptsize90x &\scriptsize= 43 \\ \scriptsize x &\scriptsize= \frac{43}{90} \end{align} \]
      3. \(0.\overline{001}\\\)
        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): \[\begin{align}\scriptsize1000x&\scriptsize=1.001001\ldots\\\scriptsize x&\scriptsize=0.001001\ldots\\\hline\scriptsize 999x&\scriptsize=1\\ \scriptsize x&\scriptsize=\frac{1}{999}\\\\\scriptsize\text{hence, } 0.\overline{001}&\scriptsize = \frac{1}{999} \end{align} \]
  • Express 0.99999... in the form \(\frac{p}{q}\). Are you surprised by your answer? With your teacher and classmates, discuss why the answer makes sense.
    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) \[\begin{align}10x &=9.99999\ldots\\x &=0.99999\ldots\\\hline 9x &=9\\ x &=\frac{\cancel{9}}{\cancel{9}}=1\\\\\text{hence, } 0.99999\ldots& = 1 \end{align} \]
  • What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \( \frac{1}{17}\) ? Perform the division to check your answer.
    Solution:
    \(\scriptsize \require{enclose} \begin{array}{rlc} &&\phantom{000000000}0.588235294117647\\ && \phantom{0000000}17\enclose{longdiv}{1.000000000000000}\\&& \underline{85\phantom{00}}\\&&150\phantom{0}\\&& \underline{136\phantom{0}}\\&&\phantom{0}140\\&& \phantom{000}\underline{136\phantom{0}}\\&&\phantom{0000}40\phantom{}\\&& \phantom{000000}\underline{34\phantom{00}}\\&&\phantom{0000000}60\\&& \phantom{000000000}\underline{51\phantom{00}}\\&&\phantom{000000000}90\\&& \phantom{00000000000}\underline{85\phantom{00}}\\&&\phantom{00000000000}50\\&& \phantom{0000000000000}\underline{34\phantom{00}}\\&&\phantom{000000000000}160\\&& \phantom{00000000000000}\underline{153\phantom{00}}\\&&\phantom{000000000000000}70\\&& \phantom{00000000000000000}\underline{68\phantom{00}}\\&&\phantom{00000000000000000}20\\&& \phantom{0000000000000000000}\underline{17\phantom{00}}\\&&\phantom{000000000000000000}130\\&& \phantom{00000000000000000000}\underline{119\phantom{00}}\\&&\phantom{00000000000000000000}110\\&& \phantom{00000000000000000000}\underline{102}\\&&\phantom{00000000000000000000000}80\\&& \phantom{0000000000000000000000000}\underline{68}\phantom{00}\\&&\phantom{0000000000000000000000000}120\\&& \phantom{000000000000000000000000000}\underline{119\phantom{00}}\\&&\phantom{0000000000000000000000000000}1\\&& \end{array} \) \[\] \(\frac{1}{17}=0.\overline{588235294117647}\)


  • Look at several examples of rational numbers in the form \(\frac{p}{q}\) (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
    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  
  • Write three numbers whose decimal expansions are non-terminating non-recurring.Solution:
    1. \(\pi\approx 3.1415926535\ldots\)
    2. \(e\approx 2.7182818284\ldots\)
    3. \(\sqrt[2]{2}\approx 1.4142135623\ldots\)
  • Find three different irrational numbers between the rational numbers \(\frac{5}{7} \text{ and } \frac{9}{11}\)
    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.

  • Classify the following numbers as rational or irrational :
    1. \(\sqrt{23}\)
      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
    2. \[ \begin{array}{r|l} \scriptsize 5 & \scriptsize 225 \\ \hline \scriptsize 5 & \scriptsize 45 \\ \hline \scriptsize 3 & \scriptsize 9 \\ \hline \scriptsize 3 & \scriptsize 3 \end{array} \]
      \( \begin{align*} \scriptsize\sqrt{225} &\scriptsize= \sqrt{5\times 5\times 3\times 3} \\ &\scriptsize= 5\times 3 \\ &\scriptsize= 15 \\ \Rightarrow\;\; \color{blue}{\text{Rational Number}} \end{align*} \)
    3. 0.3796 \(\Rightarrow\color{blue}{\text{Terminating Decimal,}\\\text{hence Rational Number}}\)
    4. 7.478478 \(\Rightarrow 7.478478=7.\overline{478} \Rightarrow\color{blue}{\text{Repeating,}\\\text{ Non- Terminating decimal,}\\\text{Hence Rational Number}}\)
    5. 1.101001000100001... \(\Rightarrow\) Non-Repeating and Non-Terminating Decimal, hence Irrational Number

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      Frequently Asked Questions

      A number system is a way of expressing numbers using symbols and rules. It includes natural numbers, whole numbers, integers, rational, and irrational numbers.

      Real numbers include both rational and irrational numbers that can be represented on the number line.

      Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and \(q \neq 0.\)

      Irrational numbers cannot be written as a simple fraction and have non-terminating, non-repeating decimals, like v2 or p.

      Rational numbers can be expressed as p/q, while irrational numbers cannot. Rational decimals terminate or repeat; irrational decimals do not.

      Natural numbers are counting numbers starting from 1, 2, 3, and so on.

      Whole numbers include all natural numbers and 0, i.e., 0, 1, 2, 3, 4, ...

      Integers include all whole numbers and their negatives, such as … -3, -2, -1, 0, 1, 2, 3 …

      The decimal expansion of rational numbers is either terminating or non-terminating repeating.

      The decimal expansion of irrational numbers is non-terminating and non-repeating.

      Yes, every real number, whether rational or irrational, can be represented on the number line.

      All rational numbers are real, but not all real numbers are rational. Real numbers include both rational and irrational types.

      Construct a right-angled triangle with both legs of 1 unit each; the hypotenuse represents v2 when plotted on the number line.

      A non-terminating decimal continues infinitely without ending, like 0.333... or 0.142857142857...

      A repeating decimal has digits that repeat in a pattern, for example, 0.666… or 0.142857142857…

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