Decimal Fraction: Definition, Place Value, Operation

A decimal fraction has a denominator of 10, such as ten, hundred, thousand, and so on, and is often stated with the decimal point. It is easier to carry out Mathematical Operations on fractions when they are written in decimal form. For example, any fraction with a power of ten denominators, such as 53/100, can be expressed in decimal as 0.53. A fraction is a portion of the total. For example, it indicates how many slices of pizza are left or eaten in relation to the entire pizza, such as one-half, three-quarters, and so on. Decimal Fractions can be written with a decimal point and no denominator, making computations like division, subtraction, addition, and multiplication on fractions easier. 1/10, 1/100, and 1/1000, for example, are decimal fractions. We can express fractions in decimal forms, such as 0.1, 0.01, 0.001, and so on if we simplify them.

Key Takeaways: Decimal Fraction, Operation, Types of Decimal Fraction, Place Value, Decimal Fraction Conversion

Also read: Isosceles Triangle Theorems

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What is Decimal Fraction?

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A decimal fraction is one with a denominator as a power of ten, such as 10, 100, 1000, 10000, and so on. The relationship between a part and the whole is called a fraction. As a result, in a decimal fraction, the entire is always split into parts of a power of ten, such as 10, 100, 1000, and so on. 7/10, for example, denotes that we are considering 7 pieces out of a total of 10. The first step in converting a decimal to a fraction is to express the denominator as a power of ten, with the number of zeros equal to the number of decimal places in the provided value.

2.5, for example, can be expressed as 25/10, indicating that it is a decimal fraction. 

It's one of the fraction types that may be used to convert decimal fractions to fractions. With the assistance of examples, look at the figure below to learn what decimal fractions are.

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Numerator and Denominator of Fraction

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Fractions, as we all know, are pieces of a larger whole. It's written as a/b, with a and b being integers. The numerator is the number above the bar, while the denominator is the integer below the bar.

The numerator is the number of equal parts, whereas the denominator is the number of parts in a particular number. Let's look at some of the conditions that apply to fractions:

1. The fraction is 1 if the numerator and denominator have the same value.
2. The fraction is 0 if the numerator is equal to 0.
3. If the denominator is 0, the fraction will go to infinity.

4. When the digit after the decimal keeps repeating, the decimal fraction is called a recurring decimal.

Also read: Differentiation and Integration Formula

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Place Value of Decimal Fraction

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The place value of digits begins with ones, tens, hundreds, thousands, ten-thousands, and so on, progressing from right to left for the complete number. Because we are considering the decimal point in the case of decimal fractions, the place value of digits is considered in the order of:

• Tenths
• Hundredths
• Thousandths
• Ten-thousandths

The place value of 5 in 0.25, for example, is tenths. Learn the place value of digits in a number by looking at the table below.

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Operations in Decimal Fraction

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Multiplication, subtraction, addition, and division are the four basic operations of mathematics. Performing arithmetic operations on decimal fractions is simple. Let's look at the many operations that may be performed on decimal fractions.

Addition of Decimal Fraction

You should know by now that decimal fractions contain denominators of 10, 100, 1000, and so on. There are two ways to add two or more decimal fractions, as seen below:

1. By first converting decimal fractions to decimals, then adding them together.
2. By first converting the specified decimal fractions to like fractions, then adding them together.

We convert decimal fractions to decimals and then add those numbers using the first approach. Let's use 2/10 + 34/100 as an example. 

2/10 is equivalent to 0.2, and 34/100 is equivalent to 0.34. 

0.2 + 0.34 = 0.54 now. 

As a result, 2/10 + 34/100 = 0.54, often known as 54/100. 

Let's use the second way to add the same integers.

We use the LCM of the denominators to transform the provided fractions (2/10 and 34/100) into similar fractions. The least frequent multiple of ten and one hundred is one hundred. 

As a result, multiply 2/10's numerator and denominator by ten.

⇒2/10 is equal to (2 *10)/(10 *10)

⇒20/100 = 2/10

So, 20/100 plus 34/100 equals 54/100. 

As a result, 2/10 + 34/100 equals 54/100.

Subtraction of Decimal Fractions

Subtracting decimal fractions is done in the same way that adding decimal fractions is done. 44/100 - 1/10, for example, may be solved as 0.44 - 0.1, which is 0.34 or 34/100. 

Finding the LCM of the denominators and converting them into like fractions is another technique to remove 1/10 from 44/100. 

100 and 10 have an LCM of 100. 

As a result, divide 1/10 by 10 for the numerator and denominator.

⇒1/10 is (1 *10)/(10* 10)

⇒10/100 = 1/10

Now 44/100 - 10/100 = 34/100. 

As a result, 44/100 minus 1/10 is 34/100.

Multiplication of Decimal Fractions

When multiplying decimal fractions, the numerators and denominators are multiplied independently. We just add the number of zeros to multiply powers of ten. 

7/10* 3/100, for example, is 

(7* 3)/(10 *100) = 21/1000. 

Division of Decimal Fractions

Follow the instructions below to divide two decimal fractions:

Step 1: Determine the second fraction's reciprocal.

Step 2: Multiply the first fraction by the second fraction's reciprocal. That is the required response.

This is the same as dividing fractions normally. 

25/10 ÷ 5/100 = 25/10 * 100/5. 

As a result, 5 x 10 Equals 50. 

As a result, 25/10 ÷ 5/100 = 50.

Also read: Differential Equation 

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Type of Decimal Fraction

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The decimal fractions that have been described are those with denominators that are multiples of ten. In mathematics, we studied different forms of decimals, such as:

• Decimals with a finite number of digits following the decimal are known as terminating decimals.
• Infinite or non-terminating digits follow the decimal point in non-terminating decimals.
• After the decimal, there are repeated digits called recurring decimals.
• Non-Recurring Decimals have digits that do not repeat after the decimal.

We may conclude that decimal fractions are more likely to be terminating and non-repeating based on these factors. Because the denominator is a power of ten, the outcome will be a terminating decimal. When it comes to decimal fractions, we know that any decimal fraction may be expressed as a decimal with a limited number of decimal places equal to the denominator's number of zeros in the power of ten. Decimal fractions are classified as terminating decimals.

Also read: Calculus Formula

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Conversion of Decimal Fraction

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We'll learn how to convert a fraction or decimal to a decimal fraction in this section. If a fraction's denominator can be represented as a prime factorization of 2 or 5 or both, the fraction can be transformed to a decimal fraction. 

The denominator of the fraction 3/4, for example, is 4. The number 4 may be factored in as 2* 2. As a result, we may convert this fraction to a decimal fraction. We'll look at the next power of 10, which is 100 because 10 isn't a multiple of 4. The number 100 is the 25th multiple of four. To turn 3/4 into a decimal fraction, we must multiply the numerator and denominator by 25.

As a result, 3/4 = (3 *25)/(4 *25) = 75/100.

Let's have a look at how to convert a decimal to a decimal fraction in a few easy steps.

Step 1: Determine how many decimal places there are in the supplied decimal number.

Step 2: Subtract the decimal place from the number and divide it by a power of ten, with the number of zeros equal to the number of decimal places noted in the previous step.

Let's make a fraction out of 0.42. There are two decimal places in 0.42. We 

We must divide it by 100, which is the second exponent of 10 (102). As a result, 0.42 = 42/100.

To understand how to convert certain fractions and decimals to decimal fractions, look at the decimal fraction chart below.

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Things to Remember

• A decimal fraction is a fraction with a denominator of 10 or a multiple of 10 such as 100, 1,000, 10,000, etc.
• As the name implies, non-recurring decimal refers to numbers that do not recur or repeat, but rather end or come to a conclusion, such as 0.5, 0.25, and 0.125. 
• To convert these fractions, just divide the numerator by the denominator. For instance, if you need to convert 1/4 to decimal, divide 1 by 4 and you'll receive 0.25.
• As the name implies, non-terminating decimal refers to recurring or repeating numbers that do not terminate or come to an end, such as 0.333..., 0.545454...., and so on. To convert these fractions, just divide the numerator by the denominator. For instance, if you need to convert 1/3 to decimal, divide 1 by 3 and you'll receive 0.333...
• Pure recurring decimals are those in which all or part of the digits (after the decimal point) repeat. 0.7777.... or 2.373737....., for example.
• Mixed recurring decimals are those in which one or more digits after the decimal point do not repeat, but the remaining digits or set of digits (after the decimal point) do.

Also read: Difference between Sequence and Series