Dividing a real number into its integer part and decimal part makes calculations involving irrational numbers easier to understand.
Definitions of the Integer Part and Decimal Part
The greatest integer not exceeding x is called the integer part, and x − the integer part is called the decimal part. The decimal part is at least 0 and less than 1.
- x = 3.7 → integer part 3, decimal part 0.7
- x = √2 → integer part 1, decimal part √2 − 1
Integer Part of an Irrational Number
The integer part of √n is determined by the perfect squares between which n lies.
√10 → 9 < 10 < 16 → 3 < √10 < 4 → integer part 3
Example ①
Find the integer part a and decimal part b of √7.
Answer: 4 < 7 < 9 → 2 < √7 < 3. a = 2, b = √7 − 2
Find the integer part a and decimal part b of √7.
Answer: 4 < 7 < 9 → 2 < √7 < 3. a = 2, b = √7 − 2
Example ②
Let a be the integer part and b the decimal part of √5. Find b² + 4b.
Answer: a = 2, b = √5 − 2. b² = 9 − 4√5, 4b = 4√5 − 8. b² + 4b = 1
Let a be the integer part and b the decimal part of √5. Find b² + 4b.
Answer: a = 2, b = √5 − 2. b² = 9 − 4√5, 4b = 4√5 − 8. b² + 4b = 1
Key Point
- Integer part = the greatest integer not exceeding x
- Decimal part = x − integer part (0 ≦ decimal part < 1)
- Find the integer part of √n using the perfect squares between which n lies
Practice Problems
- Find the integer part and decimal part of √11.
- Let a be the integer part and b the decimal part of √3. Find b² + 2b.
- Find the integer part and decimal part of 3 + √8.
Answers and Explanations
- Answer: Integer part 3, decimal part √11 − 3
Explanation: 9 < 11 < 16. - Answer: 2
Explanation: a=1, b=√3−1. b²+2b = (4 − 2√3) + (2√3 − 2) = 2. - Answer: Integer part 5, decimal part 2√2 − 2
Explanation: √8 = 2√2 ≈ 2.83, so 3+√8 ≈ 5.83. Decimal part = √8 − 2 = 2√2 − 2.