Master the art of calculating square roots, understand perfect squares, and learn powerful methods for approximating irrational numbers.
Numbers that are the square of integers
Definition: A perfect square is a number that is the square of an integer.
Examples:
• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• 6² = 36
• 7² = 49
• 8² = 64
• 9² = 81
• 10² = 100
Key Properties:
1
2
3
4
5
1
4
9
16
25
1
2
3
4
5
Tip: Memorizing perfect squares 1-20 helps with quick square root estimation!
Multiple approaches to finding square roots
Example: Find √50
Step 1: Find perfect squares around 50
7² = 49 and 8² = 64, so √50 is between 7 and 8
Step 2: Estimate the decimal part
50 - 49 = 1, 64 - 49 = 15, so 1/15 ≈ 0.07
Estimate: √50 ≈ 7.07
(Actual: √50 ≈ 7.071)
Example: Find √144
Step 1: Group digits in pairs
√1 44
Step 2: Find largest square ≤ 1
1² = 1, so first digit is 1
Step 3: Bring down next pair
Work with 44, find digit that makes 2_ × _ ≤ 44
Result: √144 = 12
More precise methods for approximating square roots
Formula: xₙ₊₁ = (xₙ + a/xₙ) / 2
Where a is the number and xₙ is the current approximation
Example: Find √10
• Start with x₀ = 3 (close guess)
• x₁ = (3 + 10/3) / 2 = (3 + 3.33) / 2 = 3.17
• x₂ = (3.17 + 10/3.17) / 2 = 3.162
• x₃ = (3.162 + 10/3.162) / 2 = 3.1623
Result: √10 ≈ 3.1623
(Actual: √10 ≈ 3.1623)
Historical Note:
Used by ancient Babylonians over 4000 years ago. Same as Newton's method!
Example: Find √2
• Start with x₀ = 1.5
• x₁ = (1.5 + 2/1.5) / 2 = 1.4167
• x₂ = (1.4167 + 2/1.4167) / 2 = 1.4142
• x₃ = (1.4142 + 2/1.4142) / 2 = 1.4142
Result: √2 ≈ 1.4142
Converges very quickly!
How square roots are used in science and engineering
Kinetic Energy:
KE = ½mv², so v = √(2KE/m)
Pendulum Period:
T = 2π√(L/g), where L is length, g is gravity
Wave Speed:
v = √(T/μ), where T is tension, μ is mass per unit length
Signal Processing:
RMS (Root Mean Square) values for electrical signals
Statistics:
Standard deviation = √(variance)
Computer Graphics:
Distance calculations in 3D space using √(x² + y² + z²)
Apply your square root knowledge
Given: Which of these are perfect squares?
36, 50, 81, 120, 144, 200
Solution:
• 36 = 6² → Perfect square
• 50 = 7.07² → Not a perfect square
• 81 = 9² → Perfect square
• 120 = 10.95² → Not a perfect square
• 144 = 12² → Perfect square
• 200 = 14.14² → Not a perfect square
Given: Approximate √75 using the estimation method.
Solution:
• 8² = 64 and 9² = 81
• √75 is between 8 and 9
• 75 - 64 = 11, 81 - 64 = 17
• 11/17 ≈ 0.65
• √75 ≈ 8.65
Answer: √75 ≈ 8.66 (actual: 8.660)
Congratulations! You've mastered the fundamentals of geometry and number theory