Lecture 3: Real Numbers and Their Decimal Expansions

 .

📘 Class 9 – CBSE Mathematics

Chapter 1: Number System

Lecture 3: Real Numbers and Their Decimal Expansions

🔹 Decimal Expansions of Rational Numbers

When we divide a rational number $\frac{p}{q}$ (with $p, q \in \mathbb{Z}, q \ne 0$), its decimal expansion can be of two types:

🔸 Case I: Terminating Decimal Expansion

• If, during division, the remainder becomes zero after a certain number of steps, the decimal expansion stops or ends.

• Such decimals are called terminating decimals.

✅ Examples:

• $\frac{7}{8} = 0.875$

• $\frac{1}{2} = 0.5$

• $\frac{639}{250} = 2.556$

Observation: The division process ends after a finite number of steps.

🔸 Case II: Non-Terminating Recurring Decimal Expansion

• If the remainder never becomes zero and starts repeating after a certain point, then the decimal goes on forever but with a repeating pattern.

• These are called non-terminating recurring (repeating) decimals.

✅ Examples:

• $\frac{10}{3} = 3.333... = 3.\overline{3}$

• $\frac{1}{7} = 0.142857142857... = 0.\overline{142857}$

• $3.5727272... = 3.57\overline{27}$

Note: The bar $\overline{\phantom{0}}$ indicates the digits that repeat.

🔹 Important Observations from Examples

1. The remainders either:

   • Become zero (Terminating)

   • Or start repeating (Non-Terminating Recurring)

2. In repeating cases, the number of distinct remainders before repetition is less than the divisor.

   • Example: $\frac{10}{3}$ → Divisor is 3 → Only 1 digit (3) repeats

   • Example: $\frac{1}{7}$ → Divisor is 7 → 6 digits (142857) repeat

3. The decimal expansions of rational numbers are either terminating or non-terminating recurring.

🔹 Conclusion

If a number:

• Terminates (e.g., 3.25), or

• Repeats (e.g., 1.272727...)

Then it is a rational number.

So,

• Decimal expansions help us identify rational numbers.

• All rational numbers will either terminate or have repeating decimals.

• If a number has a non-terminating non-repeating decimal (like π or $\sqrt{2}$), it is irrational.