## Understand Rational Numbers

### What is a Rational Number?

A rational number is a number that is of the form, where p and q are integers and q ≠ 0. Rational numbers are denoted by Q.

How to identify rational numbers?

To identify if a number is rational or not, check the below conditions.

- It is represented in the form of p/q, where q≠0.
- The ratio p/q can be further simplified and represented in decimal form.
- All whole numbers are rational numbers.

### Decimal Representation of Rational Numbers:

Rational numbers can be expressed in the form of decimal fractions.

A rational number can have two types of decimal representations:

- Terminating
- Non-terminating

#### Terminating Decimals:

Terminating decimals are those numbers that come to an end after a few repetitions after the decimal point.

**Example:** 0.5, 2.456, 123.456, etc. are all examples of terminating decimals.

#### Non-terminating Decimals (repeating):

Non-terminating decimals are those that keep on continuing after the decimal point. They do not come to an end or if they do it is after a long interval.

**Example:** 1/7= 0.1428571…. which is a non-terminating repeating decimal.

**Example 1:**

Convert the fraction, 5/8 to a decimal.

So,5/8 = 0.625. This is a terminating decimal.

**Example 2:**

Convert the fraction 7/12 to a decimal.

7/12= 0.583. This is a repeating decimal.

The bar over the number, in this case, 3, indicates the number or block of numbers that repeat unendingly.

**Example 3:**

The length and breadth of a rectangle are 7.1 inches and 2.5 inches respectively. Determine whether the area of the rectangle is a terminating decimal or not.

**Solution:** Given, that the length of the rectangle is 7.1 inches and the breadth of the rectangle = is 2.5 inches.

Area of Rectangle = Length × Breadth = 7.1 inches × 2.5 inches =17.75 inches.

As the number of digits is finite after the decimal point, the area of a rectangle is a terminating decimal expansion.

**Example 4:**

Write 5/3 in decimal form.

Using the long division method, we will observe the steps in calculating 5/3

.

Therefore, 1.666… is a non-terminating repeating decimal and can be expressed as 1.6.

## Exercise:

Classify the following decimal fractions as terminating and non-terminating recurring decimals.

- 0.777…
- 0.777
- 4.7182
- 4.7182
- 9.1651651…….
- 9.165
- 0.52888…….
- 0.528
- 72.13
- 10.605

### Concept Map

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