Repeating Decimals: Decimals with a recurring digit or pattern. (e.g., 0.333... = 1/3).
C. Standard Form
A rational number p/q is in standard form if:
The denominator q is positive.
p and q have no common factors other than 1 (Highest Common Factor = 1).
D. Density Property
Between any two rational numbers, there are infinitely many other rational numbers. To find them, use common denominators or find the mean (average) of the two numbers.
💡 Worked Examples
1. Standard Form: Simplify -15/45. Step: Divide numerator and denominator by their HCF (15). Result: -1/3
2. Decimal to Fraction: Convert 2.5 to standard form. Step: Write as 25/10. Divide by 5. Result: 5/2
3. Finding a number in between: Find a rational number between 1/3 and 1/2. Step: Convert to common denominator 6: 2/6 and 3/6. Now multiply both by 2: 4/12 and 6/12. Result: 5/12 is between them.
✍️ Practice Exercises (20 Questions)
1. Is 0 a rational number?
Yes. It can be written as 0/1, 0/5, etc.
2. What is the denominator of the integer -12 when written as a rational number?
1. (-12/1).
3. Convert 0.75 to a fraction in standard form.
3/4. (75/100 simplified by 25).
4. Is the square root of 4 rational?
Yes. √4 = 2, which is rational (2/1).
5. Is the square root of 2 rational?
No. √2 cannot be written as a simple fraction (it is an irrational number).
6. Convert 1/8 to a decimal.
0.125. This is a terminating decimal.
7. Simplify 24/60 to standard form.
2/5. (Divide by HCF = 12).
8. Is 0.666... rational?
Yes. It is a repeating decimal equal to 2/3.
9. Write 4.2 as a fraction in standard form.
21/5. (42/10 simplified by 2).
10. Why is 5/0 NOT a rational number?
Division by zero is undefined. In the definition p/q, q cannot be 0.
11. Find a rational number between 2 and 3.
2.5 (or 5/2).
12. True or False: All terminating decimals are rational.
True.
13. In p/q, what must p and q be?
Integers.
14. What is standard form for 5/-10?
-1/2. (Standard form requires a positive denominator).
15. Is π (3.1415...) rational?
No. It is a non-terminating, non-repeating decimal.
16. Simplify -40/-80 to standard form.
1/2. (Negative divided by negative is positive).
17. How many rational numbers exist between 1 and 2?
Infinitely many. (Density property).
18. True or False: Every rational number is an integer.
False. 1/2 is rational but not an integer.
19. Standard form of 15/5?
3/1 (or simply 3).
20. What do we call a decimal that goes on forever with no pattern?