123 LearnersLast updated on July 23rd, 2025
Decimal Representation of Rational Numbers
A rational number is a number that can be written as p/q. , where p and q are integers, and q ≠ 0. We can denote a collection of rational numbers by Q, and rational numbers can be represented as decimals. The converted decimal number and the rational number have the same mathematical value. In this topic, we will explore the decimal representation of rational numbers in detail.
What is the Decimal Representation of Rational Numbers
A decimal point separates the fractional part and the whole part of decimal numbers. To represent a rational number in a decimal form, we must divide the numerator by the denominator and the quotient represents the decimal form. The two types of decimal representations of a rational number are terminating and non-terminating decimals. If the decimal places end after a finite number, then it is a terminating decimal. Decimal places in non-terminating decimals repeat.
Representing Rational Numbers in the Decimal Form
We divide the integer p by the integer q, to convert a rational number written in the form pq to a decimal form. A rational number can have either a terminating decimal or a non-terminating decimal. If the remainder is zero, the rational number has a terminating decimal. The decimal numbers of a terminating decimal have a fixed number of digits after the decimal point and end without repeating. For example, the decimal representation of a rational number, 38 is:
3 ÷ 8 = 0.375
Here, the decimal number after the decimal point (.375) ends after three decimal places. Hence, it is a terminating decimal.
If the remainder is not zero and continues infinitely, it is called a non-terminating decimal. The digits of a non-terminating decimal can repeat themselves after the decimal point. For instance, divide 1 by 3.
1 ÷ 3 = 0.333…
The number 3 repeats infinitely and it is a non-terminating decimal.
How to Represent Terminating Rational Numbers in Decimals
We can represent a terminating rational number in a decimal form that ends after a finite number of digits. It can be written in the form:
p(2n × 5m). If the denominator of a rational number has only the prime factors of 2 or 5, it results in a terminating decimal. For example, the given rational number is 5/16.
5 ÷ 16 = 0.3125
The digits after the decimal point end after four numbers. The denominator 16 contains the prime factors of 2 (24).
Decimal Form of a Rational Number Chart
A rational number chart represents the decimal form of different rational numbers.
Real-life Applications of Decimal Representation of Rational Numbers
Rational numbers in the form of pq are represented by decimals for accurate calculations, better financial transactions, and comparisons. The real-world applications of the decimal representation of rational numbers are countless.
- In finance and economics, the decimal representation of rational numbers is used for various purposes. To calculate the accurate value of items, bank balances, interest rates, and discounts, the prices are written in decimals.
- In medicine and research, professionals can represent the rational numbers related to height, weight, amount, and measurement in decimals to easily interpret the quantities. For example, doctors prescribe a tonic for daily consumption of 2.5 milliliters.
- For time management, rational numbers are usually used for representation. For example, a sports coach is recording the timing of an athlete. It may appear in decimals, like the 100m race is completed in 4.15 seconds.
- In the field of technology and innovation, rational numbers that are difficult to understand are represented in decimal form. The storage capacity of electronic devices can be accurately represented in decimal form. For example, the processor speed of a computer can be represented as 3.05 GHz. The decimal representation of rational numbers is often used to denote the speed of data transfer and file sizes.
Common Mistakes and How to Avoid Them on Decimal Representation of Rational Numbers
The process of converting a rational number to its decimal form is sometimes tricky and confuses students. They make some common errors that lead them to incorrect conclusions. Here are some common mistakes and helpful solutions to avoid these errors.
Mistake 1
Misunderstanding Terminating and Non-terminating Decimals
By checking the decimal places, students should check whether the decimal number is a terminating or non-terminating decimal. A terminating decimal has a fixed number of decimal places and is finite. However, non-terminating decimals go on infinitely and repeat again and again. For example,
3 ÷ 8 = 0.375 is a terminating decimal.
1 ÷ 3 = 0.333… is a non-terminating decimal.
Mistake 2
Incorrect Placement of Decimal Point
Before finalizing the answer, double-check the position of the decimal point in the result. If students incorrectly place the decimal point, it leads to wrong values. For instance, 8 ÷ 5 = 1.6.
If a student mistakenly writes 1.06 instead of 1.6, the result becomes incorrect.
Mistake 3
Rounding off a Non-terminating Decimal Too Early
After the decimal point, a non-terminating decimal’s numbers may repeat again and again. Therefore, students should not round them too early. For instance, 8 ÷ 3 = 2.666…
If a student rounds the result as 2.66 instead of 2.666… the result becomes incorrect. A bar notation is used to indicate a non-terminating decimal, such as 2.6
Mistake 4
Misunderstanding the Decimal Place Value
The place values of each digit should be learned by students to avoid mistakes in interpreting numbers. If they place the wrong decimal place values it will affect the entire answer and decimal representation. Tenths, hundredths, and so on are the values of the digits after the decimal point. For example, 0.45 means 45 hundredths, and 0.036 means 36 thousandths.
Mistake 5
Forgetting to Simplify the fraction
The given fraction must be simplified to its lowest terms before converting it to a decimal form. Sometimes, students fail to simplify the fractions, which leads to incorrect conclusions. For instance, if the given fraction is 40/100, it can be simplified to 2/5.
Solved Examples of Decimal Representation of Rational Numbers
Problem 1
Convert 2/3 into its decimal representation.
0.6
Explanation
First, we can divide 2 by 3:
2 ÷ 3 = 0.666…
The digit 6 repeats infinitely, so it is a non-terminating decimal.
We can write it as:
0.6
Thus, the decimal representation of 2/3 is 0.6
Problem 2
Convert 3/4 into a decimal form.
0.75
Explanation
To convert 3/4 into a decimal, we must divide 3 by 4:
3 ÷ 4 = 0.75
Here, the division ends after two decimal places, so it is a terminating decimal.
Thus, the decimal representation of 3/4 is 0.75
Problem 3
Convert 7/8 into a decimal form.
0.875
Explanation
First, we can divide 7 by 8:
7 ÷ 8 = 0.875
The decimal ends after three places, so it is a terminating decimal.
Thus, the decimal representation of 7/8 is 0.875
Problem 4
Verify that the decimal representation of 16 is a non-terminating repeating decimal.
Yes, it is a non-terminating repeating decimal.
Explanation
To check the decimal representation of 16, we perform a long division.
First, divide 1 by 6.
1 ÷ 6 = 0.1666…which is a non-terminating repeating decimal.
1/6 = 0.16
Thus, the decimal representation of 1/6 is a non-terminating repeating decimal.
Problem 5
Check whether the decimal representation of 11/5 is a terminating decimal.
11/5 is a terminating decimal.
Explanation
First, we must perform the division:
11 ÷ 5 = 2.2
The quotient is 2.2, which ends after one decimal place. Here, the denominator is 5, a prime factor of 5. The denominator of a terminating decimal has only 2 or 5 as prime factors.
Therefore, 11/5 is a terminating decimal.
FAQs of Decimal Representation of Rational Numbers
1.What do you mean by decimal representation of rational numbers?
2.How is a rational number converted to a decimal?
3.Can irrational numbers have terminating or non-terminating decimals?
4.How can you determine whether a decimal is rational or not?
5.How can children in United Kingdom use numbers in everyday life to understand Decimal Representation of Rational Numbers?
6.What are some fun ways kids in United Kingdom can practice Decimal Representation of Rational Numbers with numbers?
7.What role do numbers and Decimal Representation of Rational Numbers play in helping children in United Kingdom develop problem-solving skills?
8.How can families in United Kingdom create number-rich environments to improve Decimal Representation of Rational Numbers skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
