- What Is the Decimal Representation of Rational Numbers?
- Writing Rational Numbers in the Decimal Form
- Decimal Form of a Rational Number Chart
- Solved Examples on Decimal Representation of Rational Numbers
- Practice Problems on Decimal Representation of Rational Numbers
- Frequently Asked Questions on Decimal Representation of Rational Numbers

## What Is the Decimal Representation of Rational Numbers?

**Decimal representation of rational numbers is a method of finding decimal expansion of the given rational number or converting a rational number into an equivalent decimal number using long division.**

Decimal form of a rational number example: $\frac{5}{4} = 5\div4= 1.25$

Rational numbers are the numbers of the form $\frac{p}{q}$, where p and q are integers and $q\neq0$. Decimals are the numbers that have a whole number part and a fractional part separated by a decimal point. Every rational number is either a terminating or repeating decimal.

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## Writing Rational Numbers in the Decimal Form

To convert the rational number $\frac{p}{q}$ into a decimal, we divide the number p by the number q using the long division process. Writing rational numbers in decimal form involves two cases. The decimal form of a rational number we get can be of 2 types:

**When Remainder**$= 0$**, the decimal expansion is terminating.**

Terminating decimal is the decimal number in which there is an end-digit. There are a finite number of digits after the decimal point.

For example, if we divide 5 by 1, we get 0.2. The number terminates and doesn’t continue after 2. So, it’s a terminating number.

**When Remainder**$\neq0$**, the decimal expansion is non-terminating and repeating.**

In a non-terminating and repeating decimal, a single digit or a block of digits repeat themselves infinitely after the decimal point.

For example, we get $0.\overline{09}=0.09090909$ … on dividing 1 by 11. Here, the group of digits, 09, keep on repeating.

So, how are rational numbers written as decimals? Let’s discuss both the cases.

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## Terminating Decimal Representation of Rational Numbers

When the decimal expansion of $\frac{p}{q}$, $q\neq 0$ comes to an end after a finite number of digits, the decimal expansion is called terminating. In this case, when we divide p by q using the long division method, we get remainder 0.

**Example 1: **Find the decimal representation of the rational number $\frac{4}{5}$ .

Thus, $\frac{4}{5} = 0.8$ … terminating decimal expansion

**Example 2:** Find the decimal representation of the rational number $\frac{3}{4}$.

Here, $\frac{3}{4} = 0.75$ … terminating decimal expansion

## Non-terminating and Repeating Decimal Representation of Rational Numbers

In some cases, when we divide p by q to find the decimal expansion of $\frac{p}{q}$, $q\neq 0$, the remainder never becomes 0. Also, the remainder repeats after some steps, which results in a non-terminating and recurring decimal expansion.

**Example 1: **$\frac{10}{3} = 3.333$…

**Example 2: **$\frac{7}{11} = 0.636363$ …

## Decimal Form of a Rational Number Chart

Rational Number | Decimal form |
---|---|

$\frac{1}{2}$ | 0.5 |

$\frac{99}{2}$ | 49.5 |

$\frac{1345}{100}$ | 13.45 |

$\frac{1}{3}$ | 0.333… |

$\frac{5}{9}$ | 0.5555… |

$\frac{6}{5}$ | 1.2 |

$\frac{8}{100}$ | 0.08 |

## Facts about Decimal Representation of Rational Numbers

- A rational number has either a terminating decimal expansion or a non-terminating and recurring (repeating) decimal expansion.

- The converse of the above statement is also true. If the decimal expansion of a number is terminating or non-terminating and recurring (repeating), then the number is a rational number.

- The decimal expansion of an irrational number is non-terminating and non-repeating.

## Conclusion

In this article, we learned about the two types of decimal expansions of rational numbers. Let’s solve a few examples and practice problems for revision.

## Solved Examples on Decimal Representation of Rational Numbers

**Express**$\frac{3}{8}$**in the form of a decimal.**

**Solution: **

Divide 3 by 8 using the long division method.

The rational number $\frac{3}{8}$** **has a terminating decimal expansion. ** **

$\frac{3}{8} = 0.375$

**What is the decimal expansion of the rational number**$\frac{20}{11}$**?**

**Solution: **

Divide 20 by 11.

The rational number $\frac{20}{11}$** **has a non-terminating and recurring decimal expansion.

$\frac{20}{11} = 0.181818$…

**Write the rational numbers in the decimal form?**

**i) **$\frac{2}{10}$** ii) **$\frac{174}{100}$** ii) **$\frac{56}{1000}

**Solution:**

i) $\frac{2}{10} = 0.2$

ii) $\frac{174}{100} = 1.74$

iii) $\frac{56}{1000} = 0.056$

## Practice Problems on Decimal Representation of Rational Numbers

## Decimal Representation of Rational Numbers: Definition, Types, Facts

### What will be the decimal equivalent of the number $\frac{22}{4}$?

Dividing 22 by 4, we get

$\frac{22}{4} = 5.5$

### The decimal expansion of a rational number is either terminating or

A rational number has either a terminating or a non-terminating and recurring (repeating) decimal expansion.

### Is 1.5 a rational number?

If the decimal expansion of a number is terminating or non-terminating and recurring (repeating), then the number is a rational number. $1.5 = \frac{15}{10} = \frac{3}{2}$

## Frequently Asked Questions on Decimal Representation of Rational Numbers

**How do you know if a decimal is rational?**

If the decimal expansion of a number is “terminating” or “non-terminating and recurring (repeating),” then the number is a rational number.

**What cannot be the decimal representation of a rational number?**

A rational number cannot have a non-terminating and non-repeating decimal expansion.

**What is the difference between rational numbers and fractions?**

Rational numbers are the numbers of the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\neq0$. Fractions represent part of a whole. A fraction is written in the form of $\frac{a}{b}$, where $b\neq0$ and a & b are natural numbers. Here, “a” is the numerator that represents the number of parts taken and “b” is the denominator that represents the total number of parts of the whole.