Square Root Calculator
Calculate square roots with precision and detailed analysis
Square Root Calculation
Square Root Results
Enter a number to calculate its square root
Understanding Square Roots
What is a Square Root?
A square root of a number x is a number y such that y² = x. In other words, a square root is a value that, when multiplied by itself, gives the original number. Every positive real number has two square roots: one positive and one negative.
Mathematical Notation:
√x = y where y² = x
Example: √25 = 5 because 5² = 25
Types of Square Roots
Square roots can be classified into different categories based on their properties and the nature of the original number.
- •Perfect Squares: Numbers that have integer square roots (1, 4, 9, 16, 25, ...)
- •Irrational Roots: Square roots of non-perfect squares (√2, √3, √5, ...)
- •Complex Roots: Square roots of negative numbers (√-1 = i)
- •Principal Root: The positive square root of a positive number
Common Perfect Squares
| Number | Square Root | Verification | Category |
|---|---|---|---|
| 1 | 1 | 1² = 1 | Unit |
| 4 | 2 | 2² = 4 | Small |
| 9 | 3 | 3² = 9 | Small |
| 16 | 4 | 4² = 16 | Small |
| 25 | 5 | 5² = 25 | Small |
| 36 | 6 | 6² = 36 | Small |
| 49 | 7 | 7² = 49 | Small |
| 64 | 8 | 8² = 64 | Small |
| 81 | 9 | 9² = 81 | Small |
| 100 | 10 | 10² = 100 | Medium |
| 144 | 12 | 12² = 144 | Medium |
Square Root Calculation Methods
Prime Factorization Method
Perfect for perfect squares
- Factor the number into prime factors
- Group factors in pairs
- Take one factor from each pair
- Multiply the selected factors
Example: √36 = √(2² × 3²) = 2 × 3 = 6
Long Division Method
Manual calculation method
- Group digits in pairs from right
- Find largest square ≤ first group
- Subtract and bring down next pair
- Double the result and continue
Used for manual calculations without calculators
Newton's Method
Iterative approximation
- Start with initial guess
- Apply formula: x₁ = (x₀ + n/x₀)/2
- Repeat until convergence
- High precision achievable
Used in computer algorithms
Babylonian Method
Ancient approximation
- Make initial guess
- Divide number by guess
- Average guess and result
- Iterate for accuracy
One of the oldest known methods
Real-World Applications
Geometry & Engineering
Calculate distances and dimensions
- • Pythagorean theorem
- • Diagonal calculations
- • Circle radius from area
- • Triangle side lengths
Physics & Science
Natural phenomena calculations
- • Velocity from kinetic energy
- • Wave calculations
- • Electrical circuits
- • Quantum mechanics
Finance & Statistics
Risk and volatility measures
- • Standard deviation
- • Variance calculations
- • Risk assessment
- • Statistical analysis
Computer Graphics
3D rendering and transformations
- • Distance calculations
- • Normalization
- • Vector operations
- • Scaling transformations
Architecture
Building and design calculations
- • Room diagonal measurements
- • Structural calculations
- • Space planning
- • Material requirements
Medicine & Biology
Medical and biological calculations
- • Dosage calculations
- • Statistical analysis
- • Research data
- • Medical imaging
Famous Irrational Square Roots
Mathematical Constants
Historical Significance
√2 - First discovered irrational
Discovered by ancient Greeks, proved that not all numbers are rational
√3 - Geometry fundamental
Height of equilateral triangle with side length 2
√5 - Golden ratio connection
φ = (1 + √5)/2 ≈ 1.618, the golden ratio