Questions & Answers

Question

Answers

Answer
Verified

The given number is \[\dfrac{1}{{14}}\]. We need to find the decimal expansion of the number up to 7 decimal places.

We have to convert a fraction to a decimal is just a division operation. So the fraction \[\dfrac{1}{{14}}\] means \[1 \div 14\].

Then we get,

Here we see that after carrying division \[8\] times we return with a remainder \[10\] which is the dividend at the second step of division continuing after 6 steps. Thus the string \[714825\] respects in the quotient infinitely often. Thus the decimal expansion of \[\dfrac{1}{{14}}\] looks like \[\dfrac{1}{{14}} = 0.0714285714285....\](Here remainder is never \[0\]).

Since the number after in the \[{8^{th}}\] decimal place is \[7\](greater than \[5\]) by round of rule the \[{7^{th}}\] place digit will be \[5 + 1 = 6\].

Thus we can make it up to \[7\] decimal places,\[\dfrac{1}{{14}} = 0.0714286\]

Now, we need to express this as decimal expansion so we get,

\[0.0714286 = 0 \times {10^{ - 1}} + 7 \times {10^{ - 2}} + 1 \times {10^{ - 3}} + 4 \times {10^{ - 4}} + 2 \times {10^{ - 5}} + 8 \times {10^{ - 6}} + 6 \times {10^{ - 6}}\]

\[r = \sum\limits_{i = 0}^\infty {\dfrac{{{a_i}}}{{{{10}^i}}}} \]

where \[{a_0}\] is a nonnegative integer, and \[{a_1},{a_{2,}}........\] are integers satisfying \[0 \leqslant {a_i} \leqslant 9\] , called the digits of the decimal representation.

Rules of rounding a number up to a certain decimal places:

If the number you are rounding is followed by \[5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ or }}9\] round the number up.

If the number you are rounding is followed by \[0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ or }}4\] leave the rounded number as it is.