Question

Express \[0.99999.....\] in the form \[\dfrac{p}{q}\]. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Answer 1

Hint: We will first let the given number equal to \[x\] and mark it as equation 1. Then we will multiply both sides of the equation with 10 as we can see that only 9 is repeating, so the bar is over 9 only hence, we will multiply 10 on both sides of the equation and name it as equation 2. After this we will subtract equation 2 from equation 1 and thus evaluate the value of \[x\] from it and thus get the result.

Complete step-by-step answer:
We will first consider the given number that is \[0.99999.....\].
Now, we will let the number equal to \[x\].
Thus, we get,
\[ \Rightarrow x = 0.99999..... - - - - \left( 1 \right)\]
Next, we will multiply equation (1) with 10 since there is one digit in the right-hand side which is repeating itself implies that the bar is over 9 only that is the one digit.
Hence, we have,
\[
   \Rightarrow 10x = 10\left( {0.99999.....} \right) \\
   \Rightarrow 10x = 9.9999..... - - - - \left( 2 \right) \\
 \]
Next, we will subtract equation (1) from equation (2),
\[
   \Rightarrow 10x - x = 9.9999..... - 0.9999.... \\
   \Rightarrow 9x = 9 \\
   \Rightarrow x = \dfrac{9}{9} \\
   \Rightarrow x = \dfrac{1}{1} \\
 \]
Thus, we can conclude that the \[\dfrac{p}{q}\] form of \[0.99999.....\] is \[1\].

Note: Remember that the bar means the digit will repeat itself again and again. Do multiply the equation with 10 accordingly to the bar over the digits. Simplify the equation carefully and do not make any calculation mistakes in that. When multiplied the expression with 10 does not get confused with decimal point if multiplied by 10 then the decimal will shift to one place to the right.