Math calculator

Square Root Calculator

Calculate the principal square root √x. Fast local calculation with shareable, machine-readable results.

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: ≤2 ULP vs O3 applies to the published tabulated Square Root vectors, including non-negative named and seeded values, principal imaginary of -9 and -2, IEEE-754 min-subnormal, and max-finite. It is not a guarantee over the entire input domain.
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. O3 ULP covers the tabulated non-negative vectors plus principal imaginary -9/-2, min-subnormal, and max-finite.
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 · v1.0.3 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.3 · CVP protocol 1.0.0-proposed · Evidence 2026-09-09.o3-boundary
Verification revision: 2026-09-09.o3-boundary · 4/4 property · digest aa716638904c
Legacy regression: 18/18 tests · Production surface contract 3/3
Reference: O1 model · O3 expected_values · O3 numerical_behavior · 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 not Core CVP. They are covered by npm run test:math-graph-e2e (MATH_GRAPH_E2E_ONLY=square_root).
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 7/7 golden · 7/7 CVP boundary · 7/7 invalid · 4/4 property · 2/2 metamorphic · 3/3 round-trip · 21/21 O3 · 7/7 cross-interface · 5/5 URL→result→graph · 1/1 CVP contract · Manifest
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
NIST DLMF §4.2(iv) — Powers: principal value z^a = exp(a ln z)
CalculatorX mathematical conventions — Principal real root; principal imaginary for x < 0
Sources
Evidence: 11 legacy golden · 7 legacy boundary · legacy regression suite · 7/7 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
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

Enter the radicand x

Any finite real number. Perfect squares are flagged automatically. Share the result URL or call the REST API.

Read the principal root

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

Optional: follow the steps

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

Example calculations

Common configurations with formula and result.

Perfect square

√144

12² = 144

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

√81

Principal root

Solutions of y² = 81: ±9

Estimate √27

3 decimals

Babylonian

≈ 5.196

√2

Irrational

2^(1/2)

≈ 1.414214

Negative

√(−9)

complex

3i (principal)

Square roots table

Common values at a glance.

x	√x
0	0
0.25	0.5
1	1
2	≈ 1.414214
3	≈ 1.732051
4	2
9	3
16	4
25	5
81	9
100	10

i √81 = 9; the equation y² = 81 has two real solutions, ±9.

Square Root calculator specification

Version 1.0.3 · Engine tested

Calculation status

Engine tested 18/18 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.3

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).
IEEE-754 overflow of √x returns RESULT_OVERFLOW, not Infinity.
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 (up to 12 significant digits; scientific notation when |y| ≥ 1e12 or 0 < |y| < 1e-6). Imaginary results display as a trailing i (for example 3i).
Perfect squares that are exact in binary64 (for example √144 = 12) display as integers. Irrational results such as √2 are rounded on the page; the API keeps the full binary64 principal.
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=1.0.3). 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 3/3 · 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: 1.0.3

Background

Interpretation and common distinctions.

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)

Guess a positive b.
Replace b with (b + x/b) / 2.
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)

Math graph pilot under CVP 1.0 Proposed. Profile: core (unitless principal square root, not an engineering model).

Evidence Manifest: /evidence/math.square_root/1.0.3.cvp.json
Independent reference: O1 (NIST DLMF model) + O3 mpmath 1.4.1 python-backend expected values and numerical behavior + O2 live identities
Numerical policy: IEEE-754 binary64; ≤2 ULP vs O3 applies to the published tabulated Square Root vectors, including non-negative named and seeded values, principal imaginary of -9 and -2, IEEE-754 min-subnormal, and max-finite. It is not a guarantee over the entire input domain.
Interfaces: UI (SSR), REST, and MCP must agree. A separate integration check covers URL → result → Live graph. ui-ssr is not a live browser session.
Expert review is optional and is not required for CVP Verified.

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.

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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.

Square Root Calculator

Formulas

How to use

Enter the radicand x

Read the principal root

Optional: follow the steps

Example calculations

Perfect square

√81

Estimate √27

√2

Negative

Square roots table

Square Root calculator specification

Background

What is a square root?

Perfect squares

Estimating by hand (Babylonian)

Graph sketch

CVP (Calculator Verification Protocol)

Related tools

Frequently asked questions