Square Root Calculator
Calculate square roots using multiple methods. Essential for mathematics education, geometry, physics, engineering, and scientific calculations.
Calculate Square Root
Find the square root of any non-negative number:
√x = y, where y × y = x and y ≥ 0
Example: √16 = 4, because 4 × 4 = 16
Understanding Square Roots
The square root of a number x is a value that, when multiplied by itself, gives x. It's one of the most fundamental operations in mathematics with applications in geometry, physics, engineering, and statistics.
Square Root Fundamentals
🔢 What is a Square Root?
√x = y means y × y = x
Principal square root is always positive
Symbol: √ (radical symbol)
Every positive number has two square roots
📊 Perfect Squares
Numbers with integer square roots
1² = 1, 2² = 4, 3² = 9, 4² = 16
5² = 25, 6² = 36, 7² = 49, 8² = 64
9² = 81, 10² = 100, 11² = 121
🔬 Irrational Numbers
√2 ≈ 1.41421356237...
√3 ≈ 1.73205080757...
√5 ≈ 2.236067977...
Infinite non-repeating decimals
Calculation Methods
🏛️ Babylonian Method
Ancient algorithm from Mesopotamia
Iterative approximation technique
Formula: xₙ₊₁ = (xₙ + x/xₙ) ÷ 2
Fast convergence for most numbers
🎯 Newton-Raphson Method
Advanced iterative technique
Derivative-based approximation
Quadratic convergence rate
Used in scientific computing
🔍 Binary Search Method
Simple interval halving technique
Guaranteed convergence
Easy to understand and implement
Slower but reliable
Mathematical Properties
➕ Basic Properties
√(x × y) = √x × √y
√(x/y) = √x ÷ √y
√(x²) = |x| (absolute value)
(√x)² = x for x ≥ 0
➗ Fractional Exponents
x^(1/2) = √x
x^(1/3) = ∛x (cube root)
x^(1/n) = nth root of x
General root relationship
⚠️ Complex Numbers
√(-1) = i (imaginary unit)
√(negative number) = complex number
i² = -1
Complex plane representation
Practical Applications
📐 Geometry & Trigonometry
Pythagorean theorem: c = √(a² + b²)
Distance formula calculations
Circle area: A = πr²
Triangle side calculations
⚡ Physics & Engineering
Standard deviation calculations
Root mean square (RMS) values
Wave function amplitudes
Signal processing algorithms
💰 Finance & Statistics
Volatility calculations
Risk assessment metrics
Standard error computations
Correlation coefficient analysis
🧮 Computer Science
Algorithm complexity analysis
Random number generation
Cryptography applications
Computational mathematics
Square Root Examples
| Number | Square Root | Perfect Square? | Application |
|---|---|---|---|
| 1 | 1 | Yes | Unit square side |
| 4 | 2 | Yes | 2×2 square |
| 9 | 3 | Yes | 3×3 square |
| 16 | 4 | Yes | 4×4 square |
| 25 | 5 | Yes | 5×5 square |
| 2 | 1.4142... | No | √2 (irrational) |
| 3 | 1.7320... | No | √3 (irrational) |
| π | 1.7724... | No | Circle calculations |
| 100 | 10 | Yes | 10×10 square |
🔢 Square Root Tip: Square roots of negative numbers don't exist in real numbers, but they do exist in complex numbers. For example, √(-4) = 2i, where i is the imaginary unit (i² = -1).
Historical Development
🏛️ Ancient Methods
Babylonian clay tablets (1800 BCE)
Egyptian approximation methods
Greek geometric constructions
Indian mathematical texts
Chinese calculation techniques
📚 Renaissance Mathematics
Rafal Bombelli and complex numbers
Gerolamo Cardano's cubic formulas
François Viète's algebraic methods
Decimal notation development
Logarithmic tables creation
⚙️ Modern Computing
Isaac Newton-Raphson method
Binary search algorithms
Electronic calculator development
Computer algorithm optimization
Floating-point arithmetic
Common Square Root Values
📐 Perfect Squares 1-20
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10
🔬 Important Constants
√2 ≈ 1.41421356237
√3 ≈ 1.73205080757
√5 ≈ 2.23606797750
√π ≈ 1.77245385091
√e ≈ 1.64872127194
📊 Common Approximations
√10 ≈ 3.16227766017
√15 ≈ 3.87298334621
√20 ≈ 4.47213595499
√50 ≈ 7.07106781187
√1000 ≈ 31.6227766017