Question

Write $0.0005$ as a fraction.

Answer 1

Hint: To convert $0.0005$ as fraction let’s first convert it into scientific form and then further simplify it. Scientific notation of a number is the representation of numbers in such a way that very large numbers can be read easily without any discrepancy. Also in scientific notation the numbers are represented in the form $p\times {10}^n$ where $p$ should be between 1 and 10 i.e. $$1⩽p\prec 10$$ and $n$ is an integer. So using the above information we can solve the given question.

Complete step-by-step solution:
Given
$0.0005............................\left(i\right)$
So to represent $0.0005$ as a fraction let’s first represent it in scientific form and then further simplify it.
So by the basic definition of scientific notation the numbers are represented in the form $p\times {10}^n$where $$1⩽p\prec 10$$ and $n$ is an integer.
Such that we have to find$p\text{and}n$ for the number $0.0005$
Now in order to find $p$ we have to move the position of decimal such that there is only a single digit before it.
So we get:
$\Rightarrow 0.0005=00005.0................\left(i\right)$
So we can write$p=5$
Now in order to find $n$ we have to determine the position of decimal which has been moved:
So here since it has been moved to the right side by $4$ positions, we can write:
$n={10}^{-4}........................\left(ii\right)$
The sign would be negative since it has been shifted to the right side.
So from (i) and (ii) we can write:
$$\begin{gathered} \Rightarrow p\times {10}^n=5\times {10}^{-4} \\ \Rightarrow 0.0005=5\times {10}^{-4}..........\left(iii\right) \end{gathered}$$
Now let’s further simplify (iii):
We know that: $x^{-n}={\displaystyle \frac{1}{x^n}}$
$$\begin{gathered} \Rightarrow x^{-n}={\displaystyle \frac{1}{x^n}} \\ \Rightarrow {10}^{-4}={\displaystyle \frac{1}{{10}^4}}..........................\left(iv\right) \end{gathered}$$
Now let’s substitute (iv) in (iii) we get:
$$\begin{gathered} \Rightarrow 5\times {10}^{-4}=5\times {\displaystyle \frac{1}{{10}^4}} \\ \Rightarrow 5\times {10}^{-4}={\displaystyle \frac{5}{10000}} \\ \Rightarrow 5\times {10}^{-4}={\displaystyle \frac{5}{5\times 2000}} \\ \Rightarrow 5\times {10}^{-4}={\displaystyle \frac{1}{2000}}........................\left(v\right) \end{gathered}$$

Therefore $0.0005$ as a fraction can be represented as: ${\displaystyle \frac{1}{2000}}$

Note: To convert the general format numbers to scientific notation form the standard rules have to be followed. While representing a number in fraction it’s always best to represent in the simplified form. Also the above prescribed method is one of the most efficient one to represent a decimal in fraction.