# Recurring decimals

**What are recurring decimals?**

A decimal number contains points that work as a separator between the whole number and the fractional part. There are two types of decimal numbers; they are terminating and non-terminating decimal numbers. The terminating decimal numbers are those which contain a finite number of digits after the decimal point.

The non-terminating decimal numbers are those numbers that contain infinite numbers of digits after the decimal point and do not end with a zero. The non-terminating decimal points are divided into two parts; they are recurring and non-recurring decimal points.

**Recurring decimal**

A recurring decimal number are digits that do not end with zero but are repeated at the same periodic interval. The representation of the recurring decimal numbers is $$a.\bar{b}$$.

For example, if someone goes to buy medicines at a pharmacy where one pack of $$15$$ tablets costs $$\$65$$. If the customer buys $$1$$ tablet, it will cost $$\$4.\bar{3}$$.

**Non-recurring decimal**

A non-recurring decimal number are digits that do not end with zero, and they do not get repeated after any periodic interval. The representation of the non-recurring decimal number is $$a.bcdefghshria{\cdots}$$.

**E1.5A: Use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts.**

The recurring decimal numbers can be compared with other recurring or non-recurring numbers. Different operators are used to comparing the different fractions or recurring decimal numbers; they are $$>$$, $$=$$, $$\leqslant$$, $$\geqslant$$, and so on.

**Worked example**

**Example 1:** Compare the numbers $$\frac{11}{3}$$ and $$\frac{42}{13}$$. Which number is greater between them?

**Step 1: Convert the fraction $$\frac{11}{3}$$ into recurring or non-recurring decimal.**

$$11$$ divided by $$3$$ is $$3$$, so the remainder is $$2$$ carrying $$2$$ to the tenth column.

Now, $$20$$ divided by $$3$$ by $$6$$ times, then the remainder will be $$2$$.

Now, $$20$$ divided by $$3$$ by $$6$$ times, then the remainder will be $$2$$.

Similarly, $$20$$ divided by $$3$$ by $$6$$ times, then the remainder will be $$2$$, and so on.

**Step 2: Collect the quotient from the above step.**

The quotient is $$3.666\cdots$$.

**Step 3: Convert the fraction $$\frac{42}{13}$$ into recurring or non-recurring decimal.**

$$42$$ divided by $$13$$ is $$3$$, so the remainder is $$3$$ carrying $$3$$ to the tenth column.

Now, $$30$$ divided by $$13$$ by $$2$$ times, then the remainder will be $$4$$.

Now, $$40$$ divided by $$13$$ by $$3$$ times, then the remainder will be $$1$$.

Similarly, $$10$$ cannot be divided by $$13$$, then the remainder will be $$10$$, and so on.

**Step 4: Collect the quotient from the above step.**

The quotient is $$3.230\cdots$$.

**Step 5: Compare both numbers $$3.666\cdots$$ and $$3.230\cdots$$.**

On comparing both numbers $$3.666\cdots$$ and $$3.230\cdots$$, the conclusion is that $$3.666\cdots$$ is greater than $$3.230\cdots$$, i.e., $$3.666{\cdots} > 3.230{\cdots}$$.

So, $$\frac{11}{3} > \frac{42}{13}$$.

**E1.5B: Recognise equivalence and convert between these forms.**

The non-recurring numbers are the simple decimal numbers that do not end with zero and do not repeat the same interval. But the recurring numbers do not end with zero. The numbers are continuous till infinity and are periodic.

The recurring numbers can be converted into fractions, whereas non-recurring numbers cannot be converted into fractions.

**Worked example**

**Example 1:** Convert $$\frac{1}{6}$$ into a recurring decimal number.

**Step 1: Divide $$1$$ by $$6$$.**

$$1$$ divided by $$6$$ is $$0$$, so the remainder is $$1$$ carrying $$1$$ to the tenth column.

Now, $$10$$ divided by $$6$$ by $$1$$ times, then the remainder will be $$4$$.

Similarly, $$40$$ divided by $$6$$ times, then the remainder will be $$4$$, and so on.

**Step 2: Collect the quotient from the above step.**

The quotient is $$0.1666{\cdots}$$.

So, the recurring decimal is $$0.1\bar{6}$$.