Real Numbers — Decimal Fractions, Terminating and Non-Terminating Decimals (Class 8 Maths)

 

 

🎯 Real Numbers — Decimal Fractions, Terminating & Non-Terminating Decimals (Class 8)

Let’s explore real numbers step by step! 🌸 In this lesson, we’ll learn what decimal fractions are and how to recognize terminating, non-terminating, recurring, and non-recurring decimals — with clear examples and tips.

Before you start: If you’re new to sets and number types, check the earlier topics on Understanding Sets 💡

🔸 Introduction

The ability to work with real numbers forms the foundation of many mathematical concepts. Real numbers include all the numbers we use in everyday life — counting numbers, fractions, decimals, and even irrational numbers like √2 or π. Understanding how these numbers connect helps us solve real-world problems and build confidence in advanced topics like algebra and geometry. 🌍

📊 How Real Numbers Relate to Other Numbers

Real numbers are divided into two main types: Rational and Irrational. Rational numbers include integers, whole numbers, and natural numbers — basically any number that can be written as a fraction. Irrational numbers, on the other hand, cannot be written as fractions; they continue endlessly without repeating patterns (like √5 or π). Together, they form the complete system of Real Numbers.

🌸 Fig: Real Numbers — including Rational (with Natural, Whole, and Integers) and Irrational Numbers.

🔹 Decimal Fractions

Decimal fractions are a special form of rational numbers. A decimal fraction is a fraction whose denominator is 10 or a multiple of 10 (like 100, 1000, or 10000). For example, 3/10 = 0.3 and 47/100 = 0.47. Decimal numbers are written using a decimal point to separate the whole number part from the fractional part. For instance, in 13.76, “13” is the whole part and “76” is the fractional part.

(i) Terminating Decimals

A terminating decimal is one that comes to an end. It has a limited number of digits after the decimal point. For example, 1/2 = 0.5 or 7/8 = 0.875 — both have a clear stopping point. Such decimals are easy to work with in daily calculations. ✅

(ii) Non-Terminating Decimals

A non-terminating decimal continues endlessly; the digits after the decimal point go on forever. For example, 13.423579… and 22/7 = 3.142857142857… never stop. These decimals do not end but can still have patterns — and that brings us to their two categories:

🔁 Recurring (Repeating) Decimals

A recurring decimal repeats the same digits again and again after a certain point. For example, 4.121212… and 0.666…. The repeating part can be shown with a bar: 0.6̅ = 0.666… and 2.43̅ = 2.4333…. These are also called periodic decimals.

➿ Non-Recurring (Non-Repeating) Decimals

A non-recurring decimal neither ends nor repeats. It goes on forever without any pattern. For example, √2 = 1.41421356237… or π = 3.14159265…. These numbers are irrational because they cannot be expressed as a fraction.

Remember: Recurring decimals are also called periodic decimals, while non-recurring decimals are called non-periodic and belong to the set of irrational numbers.

✅ Examples

• 0.784 → Terminating (stops after 3 digits)
• 4.344444… → Recurring decimal (digit 4 repeats)
• 7.849214932… → Non-recurring, non-terminating (irrational)
• 9.47 → Terminating (finite digits)
• 134.8 → Terminating

📝 Practice — Exercise

1. Identify which of the following decimals are terminating or non-terminating:

1. 0.784
2. 4.344444…
3. 7.849214932…
4. 9.47
5. 134.8
6. 8.929241231…

2. Separate the recurring and non-recurring decimals:

1. 7.3421…
2. 8.444444…
3. 16.786786786…
4. 19.76767676…
5. 18.48612432169…
6. 108.6524356218…

💡 Tip: If a decimal number ends, it is called a terminating decimal. If digits repeat in a regular pattern, it is called a recurring decimal. If the digits never end and never repeat, it is called a non-recurring decimal (irrational number). Try these: 1/3 = 0.333… → recurring; 1/4 = 0.25 → terminating; 1/7 = 0.142857… → recurring. Practice by converting fractions into decimals and observing which type they belong to!

🧮 Real Numbers — Quick Practice Quiz

Try the quiz below! Click “Show Answer” to check your understanding. 🌷

1. Which of the following numbers is irrational?
   A) 0.25    B) √3    C) 1.5    D) 2/5
   
   ✅ Answer: √3
2. Which of these is a terminating decimal?
   A) 1/3    B) 7/8    C) 2/7    D) 1/11
   
   ✅ Answer: 7/8
3. The decimal expansion of irrational numbers is:
   A) Finite    B) Terminating    C) Non-terminating and non-repeating    D) None
   
   ✅ Answer: Non-terminating and non-repeating
4. The square of an irrational number is sometimes rational and sometimes irrational. Which example shows this?
   A) (√2)² = 2 (rational)    B) (π)² (usually irrational)    C) Both A and B    D) Neither
   
   ✅ Answer: C) Both A and B — Example: (√2)²=2 (rational), while π² is still irrational in standard results.
   Note: Some irrationals when squared become rational (e.g. √2 → 2). Others remain irrational (e.g. π).
5. Every natural number is also:
   A) Whole number    B) Rational number    C) Real number    D) All of these
   
   ✅ Answer: All of these

🔙 Previous: Understanding Sets 📊 Related: Venn Diagrams

🌟 Great Job!

You have now learned how to identify different types of decimals — terminating, non-terminating, recurring, and non-recurring. Revise the examples and try to explain them in your own words. Keep practicing — every small step makes you stronger in math! 💪✨

© Math with Raabi — Real Numbers (Class 8) • Simple Concepts • Smart Practice • Confident Learning 💫