Square Root, Radical & Babylonian Method Calculator
Evaluate real & imaginary roots, simplify radical factors, and explore iterative numerical convergence algorithms
Input any positive, negative, or decimal value
Principal Square Root (√x)
12.000000
Perfect Square Integer
Both Roots: ±12.000000 • (±12)² = 144
Negative Root (-√x)
-12.000000
Secondary algebraic solution
Simplified Radical Form
12
Exact integer root
Nearest Perfect Squares
121 < 144 < 169
11² < x < 13²
Fractional Exponent
144^(1/2)
Base raised to power 0.500
Positive integer to factor and reduce into a√b
Exact Reduced Radical (a√b)
6√2
Decimal: ≈ 8.485281 • a = 6, b = 2
Extracted Square Factor (a²)
36 (6²)
Largest perfect square factor
Square-Free Remainder (b)
2
Contains no square factors > 1
Complete Prime Factorization
2³ × 3²
Canonical prime decomposition
Continued Fraction Expansion
[8; (2, 16)]
Periodic infinite continued fraction
Calculates √S via recursive arithmetic mean
Starting guess approximation
Iter (n)
Approximation (xₙ)
Reciprocal (S / xₙ)
Error (|xₙ² – S|)
Iterative Update Formula
xₙ₊₁ = ½(xₙ + S/xₙ)
Newton-Raphson order-2 convergence
Convergence Speed
Quadratically Convergent
Doubles correct decimal digits each step