Square Root Calculator
Principal square root of any non-negative number, with check and Newton steps.
What it does
Square root asks which non-negative number, times itself, gives the input. The radical sign means the principal root only: root of 25 is 5, never negative 5, since negative 5 is a separate solution to x squared is 25 rather than the root symbol value.
The tool above takes zero or any positive number and returns the root to 6 places with the square check beside it. The default 25 gives 5 with check 25, while 2 gives 1.414214 with check 2.
Hand methods center on Newton iteration, which calculus texts teach as x next = (x + S / x) / 2. Starting at 2 for S is 2, the guesses run 1.5, 1.416666667, 1.414215686, then 1.414213562, matching the OpenStax worked sequence digit for digit.
Roots anchor geometry and statistics: hypotenuses, standard deviations, and quadratic solutions all reduce to them. Perfect squares resolve exactly while the rest round, so the check column proves the rounding landed.
How to use
- Type zero or a positive number.
- Read the root first, the square check beside it.
- Multiply root by itself to confirm the check.
- For hand work, start Newton at half the input.
- Apply (x + S / x) / 2 until digits stop moving.
- Treat negatives as complex territory, not real roots.
Formula
- S
- Input number, zero or positive.
- y
- Principal root, always non-negative.
- x next
- Refined Newton guess from the prior guess.
Zero maps to zero. Between 0 and 1 the root exceeds the input, since small fractions square smaller: root of 0.25 is 0.5.
Worked example
Example: root of 25
Default input: 25.
- Ask: which y >= 0 gives y x y = 25.
- Answer: 5, since 5 x 5 = 25.
- Check: 5 squared is 25.
- Newton view: start 12.5, then (12.5 + 2) / 2 steps down to 5.
- Tool shows 5 with check 25.
Reference table
| Input | Root | Check |
|---|---|---|
| 4 | 2 | 2 x 2 = 4 |
| 9 | 3 | 3 x 3 = 9 |
| 25 | 5 | 5 x 5 = 25 |
| 144 | 12 | 12 x 12 = 144 |
| 196 | 14 | 14 x 14 = 196 |
| 2 | 1.414214 | Newton x4 is 1.414213562 |
| 5 | 2.236068 | Rounds to 2.24 |
| 17 | 4.123106 | Rounds to 4.12 |
Limits
- Real roots only; negatives need complex numbers.
- Six places shown; further digits truncated.
- Inputs cap at 10^15 for display sanity.
- Newton counts shown for teaching, engine uses direct math.
Common mistakes
- Reporting negative 5 as the root of 25.
- Stopping Newton one step early and keeping error.
- Confusing root with half: root of 25 is 5, not 12.5.
- Squaring the rounded root and expecting exact S.
- Typing commas or units into the number box.
Try it yourself
Quick checks. Answers sit tucked under each question.
Root of 144?
12, since 12 x 12 = 144.
Root of 2 to 6 places?
1.414214, Newton x4 is 1.414213562.
Root of 0.25?
0.5, since 0.5 x 0.5 = 0.25.
Questions
What is the square root of 25?
5, with check 25.
Why not negative 5?
The radical sign means the principal non-negative root; negative 5 solves x squared is 25 but is not the symbol value.
How does Newton method work here?
Repeat x next = (x + S / x) / 2. For 2 from x0 2: 1.5, 1.416666667, 1.414215686, 1.414213562.
Root of a fraction?
Between 0 and 1 the root exceeds the input: root of 0.25 is 0.5.
Root of zero?
Zero: 0 x 0 = 0.
Root of a negative?
No real root. Complex math gives an imaginary answer instead.
Sources
- OpenStax Elementary Algebra: Simplify and Use Square Roots
- OpenStax Calculus Volume 1: Newtons Method
- NIST Digital Library of Mathematical Functions: Elementary Functions