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Math calculator

Square Root Calculator

Find principal real or imaginary square roots with worked steps and a live graph. Share inputs, export complete results, or use the REST and MCP interfaces.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Core
Declared partition coverage
PASS · 6/6 declared partitions (positive-real, zero, negative-imaginary, tiny-subnormal, large-finite, invalid-domain) · Matrix
Numerical scope
80 fixed and seeded complete results including both signs, zero, subnormal, max finite and rounding neighbors; not exhaustive
Known limitations
  • Declared partitions are positive-real, zero, negative-imaginary, tiny-subnormal, large-finite, invalid-domain — not a whole-domain claim. x < 0 is the published principal imaginary √|x| i, not invalid-domain. Complete-result comparison includes both branches, min-subnormal and max-finite; strict decimal parsing and raw underflow rejection.
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The principal square root √x = x^(1/2). For x ≥ 0 this is the unique nonnegative real root. For x < 0 this page returns the principal imaginary value √|x| · i.
Scope
Principal branch: nonnegative real root when x ≥ 0.
Verification
Engine tested · Source checked · v1.0.4 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.4 · CVP protocol 1.0.0-proposed · Evidence 2026-10-26.square-root-complete-contract-v1
Verification revision
2026-10-26.square-root-complete-contract-v1 · 4/4 property · digest 767ce88755a9
Legacy regression
18/18 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP / URL→result→graph — Success 4/4. Error-path engine·REST·MCP 3/3 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Integration: URL → SSR result → Live graph current point (5/5). Hydration/slider/history are URL-canonical contracts, not a live browser session. Live-browser hydration, slider, Copy JSON, and Overview/Focus are covered in the current full review; Core ui-ssr is not a browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
80/80 golden · 7/7 CVP boundary · 11/11 invalid · 4/4 property · 2/2 metamorphic · 3/3 round-trip · 7/7 cross-interface · 5/5 URL→result→graph · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
11 legacy golden · 7 legacy boundary · legacy regression suite · 80/80 oracle-backed golden · 11/11 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.4 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Definition√x = y ⟺ y² = x, y ≥ 0 (principal)
Two real solutionsy = ±√x (when x > 0)
Exponent√x = x^(1/2)
Babylonian stepb ← (b + x/b) / 2
iA perfect square is a nonnegative integer whose principal square root is an integer (0, 1, 4, 9, 16, …). For cube and nth roots, use the Root calculator.

How to use

1

Enter the radicand x

A finite real number or strict decimal/scientific text. Safe-integer perfect squares are flagged. Share the result URL or call the REST API.

2

Read the principal root

√x is nonnegative when real. Also see ±√x when x > 0. Negatives return the principal imaginary √|x| · i.

3

Optional: follow the steps

Babylonian / Newton iterations show how √x can be estimated by hand.

Example calculations

Common configurations with formula and result.

ϟ

Perfect square

√144

12² = 144
12. The equation y² = 144 has two real solutions: ±12
ϟ

√81

Principal root

9
Solutions of y² = 81: ±9
ϟ

Estimate √27

3 decimals

Babylonian
≈ 5.196
ϟ

√2

Irrational

2^(1/2)
≈ 1.414214
ϟ

Negative

√(−9)

complex
3i (principal)

Square roots table

Common values at a glance.

x√x
00
0.250.5
11
2≈ 1.414214
3≈ 1.732051
42
93
164
255
819
10010
i √81 = 9; the equation y² = 81 has two real solutions, ±9.

Square Root calculator specification

Version 1.0.4 · Engine tested

Calculation status

Review policy · Evidence

Definition
The square root of x is a number y such that y² = x. Any positive x has two real square roots, +√x and −√x; the radical symbol √x denotes the principal (nonnegative) root. √0 = 0. For x < 0 there is no real square root; the principal complex value is √|x| · i.
What it calculates
The principal square root √x = x^(1/2). For x ≥ 0 this is the unique nonnegative real root. For x < 0 this page returns the principal imaginary value √|x| · i.
Inputs
  • Radicand x (finite real)
Outputs
  • For x ≥ 0: principal √x in field principal (also real, imag=0)
  • For x < 0: principal imaginary magnitude in field imag (real=0, principal=null)
Formula
√x = x^(1/2) for x ≥ 0; √(−a) = √a · i for a > 0
Assumptions
  • Principal branch: nonnegative real root when x ≥ 0.
  • Unlike Log / Ln, this engine does not reject x < 0. It returns the principal imaginary √|x| i, not a real error.
  • The other real root for x > 0 is −√x; the other imaginary root for x < 0 is −√|x| i. The published value is the principal one.
Units
  • This calculator treats x as a numeric value and does not parse units. If x represents a squared physical quantity, apply the corresponding square-root unit to the result.
Boundary conditions
  • Missing x returns MISSING_REQUIRED_INPUT.
  • Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.
  • √0 = 0.
  • x < 0 → imag = √(−x), principal = null (not NaN).
  • Strict decimal/scientific strings are limited to 512 characters; hexadecimal, commas, booleans, null, arrays and objects reject. Unknown fields reject UNKNOWN_INPUT; x/xInput aliases must agree.
  • Nonzero decimal text or raw JSON numbers that round to zero reject NUMERIC_UNDERFLOW. Overflowing inputs reject INVALID_NUMBER. Every finite normalized radicand has a finite representable square-root magnitude.
Numerical precision
  • Computation uses IEEE-754 binary64 (JavaScript Number): y = Math.sqrt(x) for x ≥ 0, and imag = Math.sqrt(−x) for x < 0.
  • REST and SSR return the engine object { real, imag, principal }. The on-page result may round for display (12 significant digits, using compact decimal/scientific notation). Imaginary results display as a trailing i (for example 3i).
  • Perfect-square labels require an exactly representable safe-integer radicand and safe-integer root; fractional or tiny radicands are never snapped to an integer. The API keeps the full three-field binary64 result.
  • Input decimal rounding to binary64 is distinct from square-root rounding; the independent Decimal2000 check covers 80 fixed and seeded complete results within 2 ULP, not every possible input. The relative condition number is 1/2 away from zero; the derivative is unbounded at zero.
  • Nonzero decimal text that becomes zero is rejected, rather than reported as an exact zero. Shared URLs and REST/MCP send inputs to the server and may enter logs; ordinary typing computes locally and browser history may retain results.
  • Babylonian working uses a normalized radicand in [1,4) and six iterations, followed by rescaling. This is a hand-method illustration; the published numerical result uses Math.sqrt.
  • Interactive evaluation runs in the browser. Shared URLs and REST use the same engine server-side. Babylonian steps and the ± pair are UI explanations; the published engine returns the principal value (or principal imaginary).
  • Copy link / Copy JSON / SSR machine JSON include calculation_version as attribution of the published engine that rendered the result (for example ?x=144&calculation_version=1.0.4). Query `v` is ignored. The live page always runs the current published engine; pin a reproducible version on REST or MCP with CalculatorX-Spec-Version (mismatch → VERSION_MISMATCH).
  • Nth roots other than square root belong on the Root calculator.
  • The Result Card includes an engine-linked Live graph of y = √x using the same Overview / Focus template as Log (canonical window 0 ≤ x ≤ 9; real domain x ≥ 0).
Example
√144 = 12; √2 ≈ 1.41421356237; √(−9) = 3i
Validation cases

10 published on this page · 18/18 tests · Production surface contract 6/6 · View evidence

  • x=144 → principal 12
  • x=81 → principal 9
  • x=2 → principal ≈1.41421356237
  • x=27 → principal ≈5.19615242271
  • x=0 → principal 0
  • x=0.25 → principal 0.5
  • x=-9 → imag 3 (3i)
  • x=-1 → imag 1 (i)
  • empty x → error MISSING_REQUIRED_INPUT
  • x=abc → error INVALID_NUMBER
Sources
  • NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
    Supports: Square root as the principal value of x^(1/2)
  • NIST DLMF §4.2(iv) — Powers: principal value z^a = exp(a ln z)
    Supports: Principal square root is the principal branch of z^(1/2); nonnegative for real x > 0. The branch cut is the negative real axis; this engine reports the principal imaginary on that cut as √|x| i.
  • CalculatorX mathematical conventions — Principal real root; principal imaginary for x < 0
    Supports: Matches the on-page contract. Cube and nth roots are the Root calculator, not this engine.
Calculation version
1.0.4

Background

Interpretation and common distinctions.

What does this square root calculator compute?

The square root of x is a number y such that y² = x. Any positive x has two real square roots, +√x and −√x; the radical symbol √x denotes the principal (nonnegative) root. √0 = 0. For x < 0 there is no real square root; the principal complex value is √|x| · i.

The published formula or method is √x = x^(1/2) for x ≥ 0; √(−a) = √a · i for a > 0. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses √144 with 12² = 144. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.square_root v1.0.4 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

What is a square root?

The square root of a number x is a number y such that

y² = x.

The radical symbol √(x) means the principal square root: the unique nonnegative root when x ≥ 0.

For x > 0 there are two real solutions to y² = x:

y = ±√(x).

Example: √(64) = 8, and both 8 and −8 square to 64.

Perfect squares

A perfect square is a nonnegative integer whose principal square root is an integer (0,1,4,9,16,…).

n n²
0–10 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Estimating by hand (Babylonian)

  1. Guess a positive b.
  2. Replace b with (b + x/b) / 2.
  3. Repeat until the desired decimals stabilize.

Graph sketch

Plotting this calculator’s radicand x against the principal real root for x ≥ 0:

y = √(x)

  • At x = 0, y = 0
  • The graph lies in the first quadrant; √(x) is never negative
  • As x → +∞, y grows without bound, more slowly than any positive power x^p with p > 1/2

Negative x is omitted from this sketch (principal value is imaginary). For cube and nth roots, use the Root calculator.

CVP (Calculator Verification Protocol)

CVP 1.0 remains proposed. This version checks the complete typed real, imag and principal result against an independent CPython Decimal implementation at 2000-digit precision on exact normalized binary64 inputs. The numerical policy allows at most 2 ULP on the declared 80 fixed and seeded vectors, including both branches, zero, subnormal and maximum finite radicands. It is not an exhaustive-domain guarantee or third-party certification.

The public Evidence summary identifies the version, verification revision and digest. Full private cases and generators are withheld under the publication policy. Real browser, graph, clipboard, exports and keyboard checks are separate from the CVP SSR interface checks.

Other calculators in this family: Antilog Calculator, Exponent Calculator, Exponential Growth Calculator, Log Calculator, Natural Log (ln) Calculator, Root Calculator . Explore all Powers, Roots & Logarithms.

→ Antilog Calculator Calculate x = b^y with base 10, e, 2 or any positive base except 1. Explore examples, precision limits and comparison tables, then share or export results. › → Exponent Calculator Calculate real powers, solve for a base or exponent, and keep both real roots when needed. Check domain limits, view graphs, share results, and export data. › → Exponential Growth Calculator Calculate exponential growth or decay with periodic or continuous models. Solve value, time, initial amount, or rate, and compare all four inputs online. › → Log Calculator Solve logarithms for the result, argument or base. Explore real graphs, view calculation steps, and share or export complete results for REST and MCP use. › → Natural Log (ln) Calculator Calculate ln(x) for positive numbers with steps, a live graph and an exponential reverse check. Share results or export full precision for REST and MCP use. › → Root Calculator Calculate square, cube, and nth real roots with a live graph. Check negative radicands and index limits, then share inputs or export full precision results. ›

Frequently asked questions

Key distinctions behind the calculation.

What is a square root?

A number y such that y² = x. Written √x for the principal (nonnegative) root when x ≥ 0.

Why are there two square roots?

If y² = x and x > 0, then (−y)² = x as well. The symbol √x means only the principal positive root; write ±√x for both.

What is a perfect square?

A nonnegative integer whose square root is an integer — for example 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Can √x be negative?

The principal value √x is never negative. The other real root of x > 0 is −√x.

What about square roots of negative numbers?

Not real. This calculator reports the principal imaginary value √|x| · i (e.g. √(−9) = 3i).

How can I estimate √x by hand?

Babylonian method: guess b, replace b with (b + x/b)/2, and repeat until stable.

How is √x related to exponents?

√x = x^(1/2). Squaring undoes the root: (√x)² = x for x ≥ 0.