Square Root Calculator

Find real or complex principal square roots, distinguish both equation solutions, and check perfect squares.

Enter your values

Change any field. The result is recalculated on this device.

Formula, example and assumptions
Principal square root12

Positive, zero or negative. A negative number gives a complex root using i, where i² = −1.

Your result

Local calculation
Principal square root12

Nonnegative inputs have a nonnegative real root. Negative inputs have no real root; their principal complex root uses i, where i² = −1. Solving x² = your number usually gives two roots.

Number
144
See how this is calculated

Working with your values

Find the principal root

Your number
144
Square-root rule
r² = 144; choose r ≥ 0
Principal root (rounded when needed)
12

Result breakdown

Perfect square
Yes

The principal square root is the positive root. Solving x² = your number gives both the positive and negative roots. Decimal values are rounded.

Check the inputs and method

These are your current inputs, not the worked example further down the page.

Number
144

Method

For a ≥ 0, √a is the unique x ≥ 0 satisfying x² = a. For a < 0, the principal complex root is i√|a|, where i² = −1. If a ≠ 0, x² = a has two solutions: √a and −√a; if a = 0, the only solution is zero.

Read the worked example and assumptions

How this calculation works

For zero or a positive number, the principal square root is the nonnegative number whose square gives the input. A negative number has no real square root, so its principal root is written using i, where i² = −1. The calculator separates this single principal root from the solutions to x² = a and checks for a perfect-square integer.

For a ≥ 0, √a is the unique x ≥ 0 satisfying x² = a. For a < 0, the principal complex root is i√|a|, where i² = −1. If a ≠ 0, x² = a has two solutions: √a and −√a; if a = 0, the only solution is zero.

Worked example

This example uses illustrative inputs. Your current result is shown above.

For a = 144, 12 × 12 = 144: the principal square root is 12 and the perfect-square result is Yes. For a = −2, √(−2) = i√2 ≈ 1.414214i. The equation x² = −2 has two solutions, i√2 and −i√2, but the principal root is only i√2.

Assumptions and limits

  • Enter a finite real number, including a negative number or decimal. General complex inputs such as 3 + 4i are not supported.
  • For a negative input, the principal root has a positive imaginary coefficient. This is a complex answer, not a real number.
  • “Perfect square” means the square of an integer: zero qualifies, but negative numbers and noninteger inputs do not.
  • Nonterminating roots are displayed as numerical approximations; the calculator does not simplify arbitrary radicals symbolically.

Questions people ask

Why is the answer 12 rather than ±12?

The square-root symbol selects the nonnegative principal root. Both 12 and −12 solve x² = 144, but √144 denotes only 12.

Can zero have a square root?

Yes. Zero multiplied by zero is zero, and the principal root of zero is zero.

What does i mean in a negative square root?

The imaginary unit i satisfies i² = −1. For example, (i√2)² = i² × 2 = −2. No real number squared can be negative, but this complex number gives a valid root.

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