Terminating Decimals

A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example, 0.5, 1.25, and 0.333 are all terminating decimals.

How to Identify a Terminating Decimal

There are two ways to identify a terminating decimal:

• The denominator of the fraction that represents the decimal is a power of 10. For example, the fraction 1/2 can be written as 0.5, which is a terminating decimal because 2 is a power of 10.
• The decimal has a repeating pattern of digits after the decimal point. For example, the decimal 0.333 has a repeating pattern of 3s after the decimal point, so it is a terminating decimal.

Examples of Terminating Decimals

Here are some examples of terminating decimals:

• 0.5
• 1.25
• 0.333
• 0.666
• 0.75

Applications of Terminating Decimals

Terminating decimals are used in a variety of applications, including:

• Currency: Terminating decimals are used to represent currency amounts. For example, the United States dollar is represented by the symbol $, and the decimal point is used to separate the dollars from the cents.
• Measurement: Terminating decimals are used to represent measurements. For example, the meter is the standard unit of length in the metric system, and the decimal point is used to represent fractions of a meter.
• Science: Terminating decimals are used to represent scientific data. For example, the temperature of water at sea level is 100 degrees Celsius, and the decimal point is used to represent the fraction of a degree.

Terminating decimals are a useful way to represent numbers that have a finite number of digits after the decimal point. They are used in a variety of applications, including currency, measurement, and science.

Steps to Identify Terminating Decimals

A terminating decimal is a decimal that ends after a finite number of digits. In other words, it is a decimal that does not repeat.

There are three steps to identify terminating decimals:

1. Look at the denominator of the fraction. If the denominator is a power of 10 (10, 100, 1000, etc.), then the decimal will terminate.
2. If the denominator is not a power of 10, then look at the prime factors of the denominator. If all of the prime factors of the denominator are 2 or 5, then the decimal will terminate.
3. If any of the prime factors of the denominator are other than 2 or 5, then the decimal will not terminate.

Here are some examples of terminating decimals:

• 1/2 = 0.5
• 1/4 = 0.25
• 1/8 = 0.125
• 1/10 = 0.1
• 1/20 = 0.05

Here are some examples of non-terminating decimals:

• 1/3 = 0.333…
• 1/7 = 0.142857142857…
• 1/9 = 0.1111…
• 1/11 = 0.090909…
• 1/12 = 0.083333…

Additional Notes

• A decimal that terminates is also called a finite decimal.
• A decimal that does not terminate is also called an infinite decimal or a repeating decimal.
• The period of a repeating decimal is the group of digits that repeats. For example, the period of the decimal 0.333… is 3.

Terminating by Decimal Division

Decimal division is a method for dividing two numbers that may have decimal points. It is similar to long division, but it uses decimals instead of whole numbers.

Steps for Decimal Division

1. Place the dividend (the number being divided) on top and the divisor (the number dividing the dividend) on the bottom, separated by a division symbol (÷).
2. Move the decimal point in the dividend all the way to the right.
3. Move the decimal point in the divisor all the way to the right until it is directly below the decimal point in the dividend.
4. Bring down any zeros that are needed to make the decimal points line up.
5. Divide the leftmost digit of the dividend by the leftmost digit of the divisor.
6. Write the quotient (the answer to the division) above the dividend.
7. Multiply the divisor by the quotient.
8. Subtract the product from the dividend.
9. Bring down the next digit from the dividend.
10. Repeat steps 5-9 until there are no more digits to bring down.

Example of Decimal Division

Let’s divide 12.34 by 3.45.

12.34 ÷ 3.45 = 3.5768…

1. Place the dividend (12.34) on top and the divisor (3.45) on the bottom, separated by a division symbol (÷).

12.34 ÷ 3.45

2. Move the decimal point in the dividend all the way to the right. 123.4

3. Move the decimal point in the divisor all the way to the right until it is directly below the decimal point in the dividend. 3.45

4. Bring down any zeros that are needed to make the decimal points line up. 123.40

5. Divide the leftmost digit of the dividend (1) by the leftmost digit of the divisor (3). 1 ÷ 3 = 0

6. Write the quotient (0) above the dividend. 0 123.40

7. Multiply the divisor (3.45) by the quotient (0). 3.45 × 0 = 0

8. Subtract the product (0) from the dividend (123.40). 123.40 - 0 = 123.40

9. Bring down the next digit from the dividend (3). 123.403

10. Repeat steps 5-9 until there are no more digits to bring down. 123.403 ÷ 3.45 = 35.768…

The final answer is 35.768…

Difference between Terminating and Non-Terminating Decimals

Decimals are numbers that have a decimal point and a fractional part. They can be classified into two types: terminating decimals and non-terminating decimals.

Terminating Decimals

Terminating decimals are decimals that have a finite number of digits after the decimal point. When you divide the numerator by the denominator of a fraction, if the result is a terminating decimal, it means that the fraction can be expressed as a simple fraction with a finite number of digits in the denominator.

Examples of terminating decimals:

• 0.5
• 0.25
• 0.125
• 0.333 (repeating)

Non-Terminating Decimals

Non-terminating decimals are decimals that have an infinite number of digits after the decimal point. When you divide the numerator by the denominator of a fraction, if the result is a non-terminating decimal, it means that the fraction cannot be expressed as a simple fraction with a finite number of digits in the denominator.

Examples of non-terminating decimals:

• 0.3
• 0.142857…
• 0.666666…
• 0.23456789…

Applications

Terminating and non-terminating decimals have various applications in different fields, including:

• Mathematics: Decimals are used in various mathematical operations, such as addition, subtraction, multiplication, and division.
• Science: Decimals are used to measure and express physical quantities, such as length, mass, and temperature.
• Engineering: Decimals are used in engineering calculations, such as designing and constructing buildings, bridges, and machines.
• Finance: Decimals are used in financial calculations, such as interest rates, currency exchange rates, and stock prices.

Terminating and non-terminating decimals are two important types of decimals that have different properties and applications. Understanding the difference between these two types of decimals is essential for performing various mathematical operations and solving real-world problems.

Solved Examples of Terminating Decimal

A terminating decimal is a decimal that has a finite number of digits after the decimal point. This means that the decimal eventually ends, and there are no repeating digits.

Here are some solved examples of terminating decimals:

Example 1: 0.5

This is a terminating decimal because there are only two digits after the decimal point, and there are no repeating digits.

Example 2: 0.25

This is also a terminating decimal because there are only two digits after the decimal point, and there are no repeating digits.

Example 3: 0.125

This is a terminating decimal because there are only three digits after the decimal point, and there are no repeating digits.

Example 4: 0.0625

This is a terminating decimal because there are only four digits after the decimal point, and there are no repeating digits.

Example 5: 0.03125

This is a terminating decimal because there are only five digits after the decimal point, and there are no repeating digits.

Example 6: 0.015625

This is a terminating decimal because there are only six digits after the decimal point, and there are no repeating digits.

Example 7: 0.0078125

This is a terminating decimal because there are only seven digits after the decimal point, and there are no repeating digits.

Example 8: 0.00390625

This is a terminating decimal because there are only eight digits after the decimal point, and there are no repeating digits.

Example 9: 0.001953125

This is a terminating decimal because there are only nine digits after the decimal point, and there are no repeating digits.

Example 10: 0.0009765625

This is a terminating decimal because there are only ten digits after the decimal point, and there are no repeating digits.

Terminating Decimals FAQs

What is a terminating decimal?

A terminating decimal is a decimal that ends after a finite number of digits. For example, 0.5, 1.25, and 0.0123 are all terminating decimals.

How do you know if a decimal is terminating?

A decimal is terminating if the denominator of the fraction that it represents is a power of 10. For example, the decimal 0.5 is terminating because it can be written as the fraction 1/2, which has a denominator of 2, which is a power of 10.

What is the difference between a terminating decimal and a repeating decimal?

A repeating decimal is a decimal that has a digit or group of digits that repeats indefinitely. For example, 0.3333… is a repeating decimal because the digit 3 repeats indefinitely.

How do you convert a fraction to a terminating decimal?

To convert a fraction to a terminating decimal, you can divide the numerator by the denominator. If the remainder is 0, then the decimal is terminating. If the remainder is not 0, then the decimal is repeating.

What are some examples of terminating decimals?

Some examples of terminating decimals include:

• 0.5
• 1.25
• 0.0123
• 0.0001
• 0.999

What are some examples of repeating decimals?

Some examples of repeating decimals include:

• 0.3333…
• 0.6666…
• 0.142857142857…
• 0.0123456789101112…
• 0.987654321987654321…