How can we identify a fraction as recurring or non-recurring without actually dividing? Here is how:
First we need to reduce the fraction completely. If the denominator of the reduced fraction contains only $2$'s and $5$'s as prime factors, the fraction is non-recurring, but if the denominator contains any prime factor other than $2$ and $5$, it is a recurring fractions. Let us try some small examples:
$\dfrac{2}{5} = 0.4$ (This is a non-recurring fraction. Here denominator is $5$)
$\dfrac{3}{8} = 0.375$ (This is also a non-recurring fraction. Here the denominator $8$ consists of only $2$ as its factor.)
$\dfrac{7}{20} = 0.35$ (Another non-recurring fraction. Here the denominator consists of only $2$ and $5$ as its factor)
$\dfrac{5}{11} = 0.\overline{45}$ (This is a recurring fraction. The denominator is $11$.)
$\dfrac{2}{15} = 0.1\overline{3}$ (This is a recurring fraction. The denominator contains $3$ as a factor.
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So, the steps to identify if a fraction is a recurring decimal or non-recurring decimal, you need to use the following steps:
- Reduce the fraction
- Check if the denominator of the reduced fraction contains any factor other than $2$ or $5$. If it does, then it will give a recurring decimal, otherwise it will be non-recurring.
II. Understanding The Details
Let us try to understand why this works. When you are writing a fraction in its decimal form, you are basically multiplying the denominator and the numerator by some power of $10$, which is $1$ followed by some number of zeros.
If we are able to cancel out the complete denominator except the power of $10$ with the numerator, then we will be left with only the power of $10$ in the denominator and we will be able to write the fraction in its decimal form by just moving the decimal point in the numerator by the number of $0$s in the denominator.
We can try to understand this using an example. Let's convert the fraction $\dfrac{3}{8}$ to it's decimal form. The denominator in this case is $8$ which is equal to $2 \times 2 \times 2$. Therefore, using our rule, we can say that this is a non-recurring decimal form. The denominator contains three $2$s, therefore, we need to have three $10$s in the numerator to cancel out all the $2$s in the denominator.
So, we can write:
$\dfrac{3}{8}$
$= \dfrac{3}{2 \times 2 \times 2}$ (we factored the denominator into prime factors)
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$= \dfrac{3 \times 10 \times 10 \times 10}{2 \times 2 \times 2 \times 10 \times 10 \times 10}$ (since there are three $2$s in the denominator, we multiplied the numerator and denominator by three $10$s so that we can cancel out all the $2$s in the denominator)
$= \dfrac{375}{1000}$ (now we are left with $1$ followed by three $0$s in the denominator)
$= 0.375$ (we shifted the decimal point by as many places, as there are $0$s in the denominator)
Now, let us take another example where the fraction gives us a recurring form of decimal. Let's take the fraction $\dfrac{7}{15}$.
The denominator here, is $15$ which is $3 \times 5$. We can cancel out the $5$ by multiplying the numerator and the denominator by $10$. But there is no way we can eliminate the $3$ from the denominator and make the denominator a number which is $1$ followed by some $0$s.
Therefore, this be a recurring form, which happens to be $0.4666666...$ that is, $0.4\overline{6}$.