Square Root Calculator

√ What Is a Square Root?

The square root of a non-negative number is a value that, when multiplied by itself, produces the original number.

For example, the principal square root of 64 is 8 because:

When people ask for “the square root” of a non-negative real number, they normally mean its principal square root, which is the non-negative root.

⚙️ How to Use the Square Root Calculator

1. Enter a non-negative number in the calculator.
2. Select how many decimal places you want displayed.
3. The square root appears automatically.
4. Check whether the input is identified as a perfect square.
5. Use the squared check to verify the calculated result.
6. Select Copy Result if you want to copy the answer.

You can enter whole numbers or decimals. For example, the calculator can evaluate √25, √2, √0.81 or √123.45.

🔢 Perfect Squares and Exact Square Roots

A perfect square is a number obtained by multiplying an integer by itself. Perfect squares have integer square roots.

≈ Square Roots That Are Not Whole Numbers

Not every number is a perfect square. For many inputs, the square root is an irrational number whose decimal representation continues without terminating or repeating.

A familiar example is √2:

The calculator’s decimal-place selector controls how many digits are displayed after the decimal point. Increasing the displayed precision can be useful when you need a closer decimal approximation.

🔣 Square Roots of Decimals

Square roots are not limited to whole numbers. Positive decimal values can also have exact or approximate square roots.

For a positive number smaller than 1, its principal square root is larger than the original number but still smaller than 1. For example, √0.04 = 0.2.

− What About the Square Root of a Negative Number?

This calculator is designed for real-number square roots. A negative number does not have a real square root because multiplying a real number by itself cannot produce a negative result.

For example, √−9 is not a real number. In the complex-number system it can be written using the imaginary unit i, where i² = −1:

± Why Is √25 Equal to 5, Not ±5?

Both 5 and −5 satisfy the equation x² = 25 because:

• 5² = 25
• (−5)² = 25

However, the radical expression √25 specifically means the principal square root, which is non-negative. Therefore:

This distinction is useful when moving between square-root notation and equations involving squared variables.

🧠 How to Estimate a Square Root

You can estimate a square root by locating the number between two nearby perfect squares.

For example, consider √50:

The calculator gives approximately 7.0711 when displayed to four decimal places. This agrees with the quick estimate because 7.0711 lies between 7 and 8.

This technique is useful for checking whether a calculator result is reasonable before relying on the displayed digits.

📚 Where Square Roots Are Used

Square roots appear throughout mathematics and in calculations built from mathematical formulas. Common examples include:

• finding a side length from the area of a square;
• using the Pythagorean theorem in geometry;
• calculating distances from coordinate differences;
• working with standard deviation and other statistical formulas;
• solving equations containing squared quantities;
• checking mathematical homework and manual calculations.

Example: Side of a Square

If a square has an area of 225 square units, its side length is:

Example: Pythagorean Theorem

For a right triangle with legs of 3 and 4 units:

🔒 Browser-Based Calculation

This square root calculator is designed to perform the calculation using JavaScript in your browser. The number you enter is processed on the page to produce the result; the calculator itself does not require a separate calculation service or account.

For unusually large or very precise numbers, remember that browser JavaScript uses finite numeric precision. The tool is suitable for common square-root calculations, learning and everyday mathematical work.