What is a square root?

The square root of a number x is the value that, when multiplied by itself, equals x. In symbols:

• √x = y means y² = x.
• The symbol √ is called the radical, and x is the radicand.

Square roots show up everywhere because “squaring” is how we measure area and magnitude. If a square has area 81, its side length is √81 = 9. If a point is 3 units over and 4 units up from the origin, its distance is √(3² + 4²) = √25 = 5.

• √(a·b) = √a · √b for a ≥ 0, b ≥ 0.
• √(a²) = |a| (absolute value matters).
• √(a/b) = √a / √b for a ≥ 0, b > 0.
• √x is only real if x ≥ 0. If x < 0, the real square root does not exist.

That last line is why this calculator gives you two modes: real-only, or “allow complex results.” In complex numbers, √(-x) can be written as i√x where i² = -1.

If you multiply x by 4, you only double the square root: √(4x) = 2√x. This “slow growth” is why square roots are used in science and statistics — they compress big ranges into smaller ones.

How this calculator computes √x

Your browser computes the square root using built-in floating-point math (similar to a scientific calculator). Here’s what happens behind the scenes:

• 1) Parse the input as a number (supports negatives and decimals).
• 2) If x is negative:
  • Real-only mode: show an explanation error.
  • Complex mode: compute √(|x|) and add i.
• 3) Compute √x using Math.sqrt for non-negative x.
• 4) Smart rounding so perfect squares look clean.
• 5) Provide a “check” by squaring the result (approx if rounded).

For “auto” rounding, the calculator uses up to 6 decimals and trims trailing zeros. You can override it if your worksheet requires a specific number of decimals.

Square root examples (with intuition)

Examples make square roots feel obvious. Here are a few that cover the most common cases.

√144 = 12 because 12 × 12 = 144.

√0.04 = 0.2 because 0.2 × 0.2 = 0.04.

√2 ≈ 1.41421356… (never ends). Rounding to 3–6 decimals is usually enough.

√(-9) = 3i because (3i)² = -9.

√50 is between √49 and √64, so it’s between 7 and 8 — and closer to 7. The calculator gives ~7.0711.

Want a viral mini-challenge? Ask people to guess √2 to 3 decimals (1.414) and see who’s closest.

Frequently Asked Questions

• What does √x mean in plain English?

  “The number that multiplied by itself equals x.” If √x = y, then y² = x.

• Why does √(a²) equal |a|?

  Because both +a and -a square to a², but √ returns the principal (non-negative) root.

• Can a negative number have a square root?

  Not in real numbers. In complex numbers, √(-x) = i√x. Use complex mode for answers like 3i.

• How many decimals should I round to?

  Homework: 3–4 decimals. More sensitive work: 6+ decimals (depends on your use-case).

• Does this calculator simplify radicals like √50 = 5√2?

  This page focuses on numerical values. For symbolic simplification, use a roots/simplifying radicals tool.

Make this page shareable

• Post “√2025 = 45” as a quick math flex screenshot.
• Run a “guess √2 to 3 decimals” challenge with friends.
• Use the negative-input mode to introduce imaginary numbers in one screenshot.
• Pair with Pythagorean theorem examples: √(a²+b²) distance problems.

Teacher tip: ask students to predict the result range (between which two integers) before calculating.