Work Out Square Root Calculator

Finding square roots is a fundamental math operation with applications in geometry, algebra, and real-world measurements. Our square root calculator provides an accurate and user-friendly way to compute square roots, along with explanations of the underlying concepts and practical applications.

What is a square root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9. Square roots are denoted by the radical symbol √.

Square roots can be either positive or negative, but by convention, the principal (or positive) square root is typically used unless specified otherwise. For example, √9 = 3, but both 3 and -3 are square roots of 9.

Square roots are used in various mathematical fields, including geometry (calculating lengths and areas), algebra (solving equations), and calculus. They also appear in real-world applications such as measuring distances, calculating areas, and analyzing data.

How to calculate square roots

There are several methods to calculate square roots, ranging from simple mental math to more advanced mathematical techniques. Here are some common approaches:

Estimation method

For numbers between 1 and 100, you can estimate the square root by finding two consecutive perfect squares that the number falls between. For example, to find √45:

Identify perfect squares around 45: 6² = 36 and 7² = 49
45 is between 36 and 49, so √45 is between 6 and 7
Refine the estimate: try 6.7² = 44.89, which is close to 45

Long division method

The long division method is a more precise approach that works well for numbers with many decimal places. Here's a simplified version:

Pair the digits from the decimal point, adding zeros if needed
Find the largest digit whose square is less than or equal to the first pair
Subtract and bring down the next pair
Repeat the process, doubling the divisor and finding the next digit

Using a calculator

For most practical purposes, using a calculator is the most efficient method. Our square root calculator provides an accurate and convenient way to compute square roots with just a few clicks.

Square root formula

The square root of a number x can be expressed mathematically as:

√x = y, where y × y = x

This formula states that the square root of x is a number y such that when y is multiplied by itself, the result is x. The square root function is the inverse of the squaring function.

Note: The square root function is defined for non-negative real numbers. Attempting to find the square root of a negative number in real numbers results in an imaginary number, which involves the imaginary unit i (where i² = -1).

Worked examples

Let's look at some examples to illustrate how to calculate square roots and interpret the results.

Example 1: Perfect square

Find √16.

Identify that 16 is a perfect square (4 × 4 = 16)
Therefore, √16 = 4

Example 2: Non-perfect square

Find √20.

Identify perfect squares around 20: 4² = 16 and 5² = 25
20 is between 16 and 25, so √20 is between 4 and 5
Refine the estimate: try 4.4² = 19.36, which is close to 20
Further refinement: 4.47² ≈ 20.00
Therefore, √20 ≈ 4.472

Example 3: Decimal number

Find √0.81.

Identify that 0.81 is 81/100
√(81/100) = √81 / √100 = 9/10 = 0.9
Therefore, √0.81 = 0.9

FAQ

What is the difference between a square root and a square?: The square of a number is obtained by multiplying the number by itself (e.g., 5² = 25). The square root of a number is a value that, when multiplied by itself, gives the original number (e.g., √25 = 5).
Can I find the square root of a negative number?: In real numbers, no. The square root of a negative number is not defined in the set of real numbers. However, in complex numbers, the square root of a negative number involves the imaginary unit i (e.g., √-1 = i).
How do I calculate the square root of a fraction?: To find the square root of a fraction, take the square root of the numerator and the denominator separately. For example, √(a/b) = √a / √b.
What is the square root of zero?: The square root of zero is zero, because 0 × 0 = 0. This is the only non-negative number that is its own square root.
How can I verify the result from the square root calculator?: You can verify the result by squaring the calculated square root and checking if it matches the original number. For example, if the calculator shows √20 ≈ 4.472, then 4.472 × 4.472 ≈ 20.