Square Root Calculator
Calculate the square root of any number using multiple methods. Find principal square roots for positive numbers and handle complex roots for negative numbers.
Calculate Square Root
Enter a number to find its square root:
For negative numbers, the result is a complex number
What is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 ร 3 = 9.
Mathematical Notation
where y² = x and y ≥ 0 (principal square root)
Calculation Methods
๐งฎ Built-in Function
Uses JavaScript's Math.sqrt() function for fast, accurate calculations
๐๏ธ Babylonian Method
Ancient algorithm: guess = (guess + x/guess) / 2, repeat until convergence
๐ฌ Newton's Method
Modern iterative method using calculus: xโโ = x - f(x)/f'(x)
Perfect Squares
Some numbers have exact square roots called perfect squares:
| Number | Square Root | Verification |
|---|---|---|
| 1 | 1 | 1 ร 1 = 1 |
| 4 | 2 | 2 ร 2 = 4 |
| 9 | 3 | 3 ร 3 = 9 |
| 16 | 4 | 4 ร 4 = 16 |
| 25 | 5 | 5 ร 5 = 25 |
| 36 | 6 | 6 ร 6 = 36 |
| 49 | 7 | 7 ร 7 = 49 |
| 64 | 8 | 8 ร 8 = 64 |
| 81 | 9 | 9 ร 9 = 81 |
| 100 | 10 | 10 ร 10 = 100 |
Complex Square Roots
Negative numbers don't have real square roots. Instead, they have complex roots:
where i is the imaginary unit (iยฒ = -1)
Examples of Complex Roots:
| Number | Complex Square Root | Verification |
|---|---|---|
| -1 | i | i ร i = iยฒ = -1 |
| -4 | 2i | (2i) ร (2i) = 4iยฒ = 4 ร (-1) = -4 |
| -9 | 3i | (3i) ร (3i) = 9iยฒ = 9 ร (-1) = -9 |
| -25 | 5i | (5i) ร (5i) = 25iยฒ = 25 ร (-1) = -25 |
Applications of Square Roots
๐ Geometry
Calculate diagonals, areas of circles, and distances using the Pythagorean theorem
๐ Statistics
Standard deviation calculations often involve square roots
๐๏ธ Engineering
Stress calculations, signal processing, and structural analysis
๐ฐ Finance
Volatility calculations and risk assessment in investments
Historical Methods
The Babylonian method (also known as Heron's method) was used by ancient civilizations over 3,500 years ago. It provides a simple iterative approach to finding square roots without modern calculators.
๐ก Pro Tip: For perfect squares, the result will be an integer. For other numbers, the result will be a decimal that, when squared, gives the original number.
Properties of Square Roots
- Principal Root: Always returns the non-negative square root
- Identity: โ(xยฒ) = |x| (absolute value)
- Product Rule: โ(a ร b) = โa ร โb
- Quotient Rule: โ(a/b) = โa / โb
- Power Rule: โ(aโฟ) = (โa)โฟ = a^(n/2)
Common Square Root Values
| Expression | Approximate Value | Exact Value |
|---|---|---|
| โ2 | 1.414213562 | โ2 (irrational) |
| โ3 | 1.732050808 | โ3 (irrational) |
| โ5 | 2.236067977 | โ5 (irrational) |
| โฯ | 1.772453851 | โฯ |
| โe | 1.648721271 | โe |