Racine Carrée Calculator

How to Use the Racine Carrée Calculator

Square Root Formula

What Is a Square Root?

A square root is a number that, when multiplied by itself, produces the original number. If 12 × 12 = 144, then 12 is the square root of 144 — written √144 = 12. The operation reverses squaring: squaring takes a number and multiplies it by itself, while finding a square root asks which number produced that result in the first place.

It helps to keep three related terms distinct. A square is the result of multiplying a number by itself (12² = 144, so 144 is 12's square). A square root is the number that produces a given square when multiplied by itself. And the principal square root — the one the √ (radical) symbol always refers to — is specifically the non-negative one, even though the equation x² = 144 technically has two solutions, +12 and −12.

How to Calculate a Square Root

For a perfect square, calculating a square root by hand just means recalling which whole number multiplies by itself to reach the target: 144 is 12 × 12, so √144 = 12.

Most numbers aren't perfect squares, so their square roots are irrational — the digits after the decimal point continue forever without repeating, and can only ever be rounded to an approximation. That's exactly what the calculator above does automatically: it finds the precise value, then rounds it to whichever precision you choose.

Perfect Squares and Non-Perfect Squares

A perfect square is a number produced by multiplying an integer by itself — 1, 4, 9, 16, 25, and so on. Every perfect square has a whole-number square root, which is why 144, 225, and 400 all return clean integers above. A non-perfect square, like 2, 3, 5, 10, or 50, has no integer square root — its root is an irrational number that the calculator rounds for display (√2 ≈ 1.4142, √3 ≈ 1.7321, √5 ≈ 2.2361, √10 ≈ 3.1623).

The full list of two-digit and three-digit perfect squares, along with their roots, is in the reference table further down this page.

Square Roots of Decimals

Decimal inputs work the same way as whole numbers. Some decimals happen to have an exact, finite square root — for example, √2.25 = 1.5 and √0.25 = 0.5, because 1.5 × 1.5 = 2.25 and 0.5 × 0.5 = 0.25. Others, just like most whole numbers, produce an irrational result that must be rounded, such as √0.3 ≈ 0.5477. The calculator supports decimal inputs of any size, including very small values like 0.0001 (√0.0001 = 0.01).

Square Root of Zero

The square root of zero is zero, since 0 × 0 = 0. Zero is one of only two numbers — along with 1 — whose square root is equal to the number itself.

Square Root of a Negative Number

No real number, positive or negative, produces a negative result when multiplied by itself — a negative times a negative is always positive. That means negative numbers have no real square root, which is why entering one above returns an explanation rather than a real-number result.

That doesn't make the question unanswerable, though. Mathematics defines the imaginary unit i as the number for which i² = −1, which extends square roots into the complex number system. Under that system, √−25 = 5i, because (5i)² = 25i² = 25 × (−1) = −25. The calculator above shows this equivalent for any negative input.

Exact vs Approximate Square Roots

A square root is exact when it can be written as a finite decimal or whole number that, multiplied by itself, reproduces the original value precisely — like √144 = 12 or √2.25 = 1.5. A square root is approximate when the true value is irrational, meaning its decimal digits continue forever without repeating, so any displayed value is necessarily rounded — like √2 ≈ 1.4142. The calculator labels every result as either exact or approximate, so it's always clear which one you're looking at.

Common Square Roots

A quick-reference table of perfect squares from 1 to 400 and their whole-number square roots.

Square Root Examples

Frequently Asked Questions

Common Mistakes

• ❌ Assuming √n has two answers

  Solving x² = n does have two solutions (±x), but the √ symbol itself always refers to the single, non-negative principal square root. √144 is 12, not ±12.

• ❌ Treating squaring and square rooting as the same operation

  Squaring a number multiplies it by itself (5² = 25); finding a square root reverses that (√25 = 5). Mixing the two up is a common source of calculation errors.

• ❌ Rounding a square root too early in a multi-step calculation

  If a square root feeds into further math — geometry, statistics, physics — round only the final answer. Rounding √50 to 7.07 first and then multiplying introduces avoidable error compared to carrying more decimal places until the last step.

• ❌ Assuming every decimal has an exact square root

  Some decimals, like 2.25, have a clean exact square root (1.5). Most others, like 0.3, produce an infinite non-repeating decimal and can only ever be approximated.

• ❌ Dismissing negative inputs as just "an error"

  There's no real number that squares to a negative result, but that doesn't mean the question is meaningless — it has a defined answer in the complex number system, explained below.