Mathematics solution NCERT
Class 9 - Chapter 3: The World of Numbers
Q1: Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 7/20, 4/15 and 13/250. Then check your answers by expressing these rational numbers as decimals.
Solution:
A rational number has a terminating decimal if the prime factors of its denominator (in simplest form) are only 2 and/or 5.
| Rational Number | Prime Factors of Denominator | Type of Decimal |
|---|---|---|
| 7/20 | 20 = 2² × 5 | Terminating |
| 4/15 | 15 = 3 × 5 | Repeating |
| 13/250 | 250 = 2 × 5³ | Terminating |
Checking by division:
7/20 = 0.35
4/15 = 0.26666...
13/250 = 0.052
Answer:
- 7/20 → Terminating decimal
- 4/15 → Repeating decimal
- 13/250 → Terminating decimal
Q2: Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 2/13, 3/13, 4/13, etc.? What do you notice?
Solution:
1/13 = 0.076923076923...
The repeating block is:
076923
Now observe:
| Fraction | Decimal Form |
|---|---|
| 1/13 | 0.076923... |
| 2/13 | 0.153846... |
| 3/13 | 0.230769... |
| 4/13 | 0.307692... |
| 5/13 | 0.384615... |
| 6/13 | 0.461538... |
The digits repeat in a cyclic order.
Each decimal contains the same six digits:
076923
but starting from a different position.
Answer: The repeating block is 076923, and the decimals exhibit a cyclic pattern.
Q3: Classify the following numbers as rational or irrational. Find the explicit fractions in case they are rational.
| Number | Rational / Irrational | Reason |
|---|---|---|
| √81 | Rational | √81 = 9 = 9/1 |
| √12 | Irrational | 12 is not a perfect square |
| 0.33333... | Rational | 0.333... = 1/3 |
| 0.123451234512345... | Rational | Repeating block 12345 |
| 1.01001000100001... | Irrational | Non-terminating and non-repeating |
| 23.560185612239874790120 | Rational | Terminating decimal |
Explicit Fractions:
√81 = 9 = 9/1
0.33333... = 1/3
23.560185612239874790120
= 23560185612239874790120 / 1000000000000000000000
(a terminating decimal can always be written as a fraction).
Q4: The number 0.99999... is a rational number. Using algebra, explain why 0.99999... is exactly equal to 1.
Solution:
Let
x = 0.99999...
Multiply both sides by 10:
10x = 9.99999...
Subtract the first equation from the second:
10x - x = 9.99999... - 0.99999...
9x = 9
x = 1
But x was defined as 0.99999...
Therefore,
0.99999... = 1
Answer: 0.99999... and 1 represent the same real number.
Q5: We have seen that the repeating block of 1/7 is a cyclic number. Try to find more numbers (n) whose reciprocals (1/n) produce decimals with repeating blocks that are cyclic.
Solution:
A cyclic number is a repeating block whose successive multiples are cyclic rearrangements of the same digits.
Examples include:
| n | 1/n | Repeating Block |
|---|---|---|
| 7 | 0.142857... | 142857 |
| 17 | 0.0588235294117647... | Cyclic pattern |
| 19 | 0.052631578947368421... | Cyclic pattern |
| 23 | 0.0434782608695652173913... | Cyclic pattern |
These numbers are examples where the repeating decimal exhibits cyclic behavior.
Answer: Examples include 7, 17, 19, 23 and several other special prime numbers whose reciprocals generate cyclic repeating decimals.