Chapter 1: Number Systems

Exercise 1.3 - Detailed Solutions

Exercise 1.3 - Step-by-Step Solutions

❓Question 1: Write in decimal form

(i) 36/100

Step 1: Divide 36 ÷ 100 = 0.36

Answer: 0.36 (Terminating) ✓

(ii) 1/11

Step 1: Long division 1 ÷ 11 = 0.090909...

Step 2: Identify pattern The digits 09 repeat forever!

Answer: 0.0909... (Non-terminating recurring) ✓

❓Question 3: Express as p/q

(i) 0.666... (0.6 recurring)

Step 1: Let x = 0.666...

Step 2: Multiply by 10 10x = 6.666...

Step 3: Subtract original 10x - x = 6.666... - 0.666... 9x = 6

Step 4: Solve x = 6/9 = 2/3

Answer: 0.666... = 23\frac{2}{3} ✓

(ii) 0.4747... (0.47 recurring)

Step 1: Let x = 0.4747...

Step 2: Multiply by 100 (two repeating digits) 100x = 47.4747...

Step 3: Subtract 100x - x = 47.4747... - 0.4747... 99x = 47

Step 4: Solve x = 47/99

Answer: 0.4747... = 4799\frac{47}{99} ✓

❓Question 4: Express 0.9999... as p/q

Step 1: Let x = 0.9999...

Step 2: Multiply by 10 10x = 9.9999...

Step 3: Subtract 10x - x = 9.9999... - 0.9999... 9x = 9

Step 4: Solve x = 9/9 = 1

🤯 Surprise: 0.9999... = 1 exactly!

This is not an approximation - they are the SAME number!

❓Question 5: Max digits in 1/17

Key Insight: For fraction 1/n, the repeating block has at most (n-1) digits.

For 1/17: Maximum = 17 - 1 = 16 digits

Actual: 1/17 = 0.0588235294117647... (exactly 16 digits repeat!)

❓Question 9: Classify as Rational or Irrational

NumberTypeReason
√23Irrational23 is not a perfect square
√225Rational√225 = 15 (perfect square)
0.3796RationalTerminating decimal
7.478478...RationalRepeating (478 repeats)
1.101001000100001...IrrationalPattern but no repetition
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Step-by-Step Solution
1
Let x = 0.666...
Assign variable to the recurring decimal
2
10x = 6.666...
Multiply both sides by 10
3
10x - x = 6
Subtract: 6.666... - 0.666... = 6
4
9x = 6
Simplify left side
5
x = 6/9 = 2/3
Divide and simplify!

Interactive Visualization