Contents

# Terminating Decimal|Definition & Meaning

## Definition

A **terminating** decimal is a **decimal** number that eventually **ends** after a certain number of decimal digits. 1/2 = **0.5**, 1/4 = **0.25**, and 6/25 =**0.24** are examples of terminating decimal numbers. On the other hand, a **non-terminating** decimal (and, by extension, a **recurring** decimal) and all **irrational** numbers go on forever, as in **1/6** = 0.1666…, and **pi** = 3.1415926535….

**Figure 1** shows different examples of **terminating** decimals.

## Conversion From Fraction to Decimal Form

To convert a fraction into **decimal** form, the **numerator**(dividend) is divided by the **denominator**(divisor). **Figure 2** shows the conversion of the fraction **1/8** into **decimal** form by long division, giving the **quotient** as **0.125**.

## Rational Numbers

A **rational** number can be **written** as **u/v** where **v ≠ 0**. The form **u / v** is also known as a **fraction**. The numerator** u** and denominator **v** are **integers,** but the **denominator** is non-zero.

If a **rational** number is **divided**, the answer will always be in **decimal** form. This decimal form can be a **terminating** or a **repeating** decimal. Some examples of rational numbers include **1/5**, **2/7**, **9/4**, and **1/7**. These **fractions** can be written in **decimal** form as:

1/5 = 0.2

2/7 ≅ 0.29

9/4 = 2.25

1/7 ≅ 0.142857

The **approximation** “**≅**” sign indicates that the **decimal** digits are **rounded** off or approximated.

The set of **rational** numbers, usually denoted by **Q,** is a subset of **real** numbers **R**. The integers **Z**, **natural** numbers **N**, and whole numbers** W** are the **subset** of the **rational** numbers as every **integer**, natural number, and **whole** number can be written in the form of **u/v**.

### Non-terminating and Recurring Decimal

A non-terminating and recurring decimal is a **rational** number whose **decimal** digits go on **forever**, and there is a **repetitive** pattern in the decimal digits. A **bar** is placed on top of the repeating **decimal digits**.

For example, **10/3** is a non-terminating and **recurring** decimal equal to **3.333…** which can also be written as 3.$\bar{3}$.

### Standard Form

For a **rational number**, the standard form is the **simplified** form of **u/v**. It means that the **divisor** and the **dividend** have no common **factors** other than one.

For example, the rational number **100/40** has **10** as the common factor. Dividing the numerator and denominator by 10 gives:

100/40 = 10/4

The fraction **10/4** still has **2** as the **common** factor. Dividing 10/4 by 2 gives:

10/4 = 5/2

Hence, the **standard** form of **100/40** is **5/2,** which is a **terminating** decimal equal to **2.5**.

## Types of Fractions

There are two types of fractions: **proper** and **improper** fractions.

### Proper Fractions

Proper fractions are the type of fractions whose **numerator** is **smaller** than its **denominator**. For example, **6/10**, **3/20**, and **8/15** are proper fractions whose decimal forms are:

6/10 = 0.6

3/20 = 0.15

8/15 = 0.53

All the above proper fractions are **terminating** decimals.

### Improper Fractions

Improper fractions are the fractions whose **numerator** is **greater** than its **denominator**. For example, **50/16**, **14/8**, and **66/12** are improper fractions that can be written in decimal forms as:

50/16 = 3.125

14/8 = 1.75

66/12 = 5.5

All the above **decimals** are terminating. **Figure 3** shows examples of **proper** and **improper** fractions with their terminating decimal forms. But not all proper and improper fractions are **terminating** decimals.

## Arithmetic Operations on Terminating Decimals

The different arithmetic **operations** that can be performed on terminating decimals are **addition**, subtraction, multiplication, and **division**.

### Addition

Two or more **terminating** decimals can be added by **aligning** the **decimal** points vertically. The empty spaces are filled with **zeros**. The numbers are added simply as any other **addition**. The decimal point in the obtained **sum** is placed **vertically,** similar to the decimal **addends**.

### Subtraction

One terminating decimal is **subtracted** from the other by the same **vertical** alignment of the **decimal** points. The subtraction is carried out with zeros assumed in the **empty** place. The decimal is placed in the same **position** as in two **decimals** being subtracted.

**Figure 4** shows the **addition** and **subtraction** of two terminating decimals.

### Multiplication

The multiplication of two **decimal** numbers is carried out by considering that there is no decimal **point** in the numbers. After the **product** is attained, the number of decimal **digits** of both decimal numbers is **counted**.

Starting from the **right** of the product and going towards the **left**, the decimal **point** is placed according to the total number of decimal **places** in the **multiplicand** and the **multiplier**.

**Figure 5** shows the multiplication of two **terminating** decimal numbers.

### Division

The division of two **terminating** decimal numbers is performed by converting the **divisor** into a **whole** number.

This is done by moving the **decimal** point in the divisor to the **right, **making it a **whole** number. Also, the dividend’s decimal point is moved the same number of **places** as the divisor. To move the decimal point to the right means **to multiply** the number by **10** or **100**.

Then, the **dividend** is divided by the **divisor** by the same long division method. The decimal point in the **quotient** is placed above the dividend’s **decimal** point.

For example, in 33.856/2.3, **33.856** is the dividend, and **2.3** is the divisor. The **divisor** is multiplied by **10** to remove the **decimal** point as:

2.3 ✕ 10 = 23

Similarly, the **dividend** is also multiplied by **10** to balance the fraction as:

33.856 ✕ 10 = 338.56

The long division of **338.56/23** is shown in **figure 6**.

## Rounding Off Terminating Decimals

A rounded-off decimal is **close** to its original value. The **terminating** decimals are rounded off to a certain degree of **accuracy**. Rounding off terminating decimals helps to **estimate** the result faster and in an easier way. For example, a student’s **percentage** is usually rounded off to estimate a student’s performance in a class.

A terminating decimal can be **rounded** off to the tenths, hundredths, or even thousandths place. It can also be rounded to a **whole** number by rounding it off to the tens or hundreds place. To round **up** or **down** to a certain place, e.g., the tenth or hundredth place, the number **following** the place is checked.

### Rounding Up

If the number following or **next** to a certain place, e.g., the tenth place, is** 5** or **greater** than 5, the decimal is rounded **up**.

For example, in the terminating decimal **35.27**, **2** is in the tenth place, and** 7** is in the hundredth place. To round this decimal to the **tenth** place, the number **next** to the tenth place, i.e., **7**, is checked. As it is **greater** than 5, the decimal is rounded up. So, the number becomes **35.3**.

A decimal can also be rounded to the **hundredth** place. In the number **42.685**, **8** is in the hundredth place. To round it off to the **hundredth** place, the number next to the hundredth place, i.e., **5**, is checked. As it is 5, so the decimal is rounded **up**. The decimal becomes **42.69**.

### Rounding Down

If the number next to a certain place, e.g., the thousandth place, is **less** than **5**, the decimal is rounded **down**.

For example, in the terminating decimal, **64.8652**, the number 5 is in the thousandth place. To round it to the **thousandth** place, the number following the thousandth place, i.e., **2**, is checked. As it is **less** than **5**, the decimal is rounded down. So, the decimal becomes **64.865**.

## Properties of Terminating Decimal

The following are the properties of a terminating decimal:

The result of the **multiplication** of a terminating **decimal** with another terminating decimal is always a **terminating** decimal.

The terminating decimal remains the **same** if both the numerator and denominator are **multiplied** by the same **number** in its fraction form.

If a **zero** is added to a terminating decimal, the result will be the **terminating** decimal.

## Irrational Numbers

The numbers which **cannot** be written in **u/v** form are known as irrational numbers. For example, $\sqrt{2}$, $\sqrt{7}$, and $\sqrt{1.6}$ are all **irrational** numbers.

### Non-terminating and Non-recurring Decimal

**Irrational** numbers are non-terminating and non-recurring. It means that they can be written in **infinite** decimal digits, and **no repetition** in a digit pattern is observed. For example, the **Euler’s** number “**e**” equal to **2.7182818…** is a non-terminating and **non-recurring** irrational decimal.

## Examples of Terminating Decimals

### Example 1

Which of the following numbers are **terminating** decimals?

1.25, $\sqrt{5}$, 2.14

Calculate the **sum** and **product** of the two terminating decimals.

### Solution

The **terminating** decimals are **1.25** and **2.14** as $\sqrt{5}$ is equal to 2.23606797… The sum and the product of 1.25 and 2.14 are calculated in **figure 7**.

Hence, the **sum** of 1.25 and 2.14 is** 3.39,** and their **product** is **2.6750**.

### Example 2

**Round** off the terminating decimal **82.68** to the tenth place, **73.542** to the hundredth place, and **27.8359** to the thousandth place. Are they rounded **up** or **down**?

### Solution

The number **82.68** is rounded to the tenth place, i.e., 6 as **82.7**, as the number following the **tenth** place, i.e., **8**, is greater than 5. It is rounded **up**.

The number **73.542** is rounded to the hundredth place, i.e., 4 as **73.54** as the number following the **hundredth** place, i.e., **2** is less than 5. It is rounded **down**.

The number** 27.8359** is rounded to the thousandth place, i.e., 5 as **27.836** as the number following the **thousandth** place, i.e., **9** is greater than 5. It is rounded **up**.

*All the images are created using GeoGebra.*