Square Root 123 Calculator

What is the square root of 123?

The square root of 123 (√123) is the positive number that, when multiplied by itself, equals 123. Since 123 is not a perfect square, its square root is an irrational number approximately equal to 11.0905.

Why is √123 important?

Square roots appear in many mathematical and real-world applications, including:

• Geometry (finding side lengths of squares)
• Physics (calculating distances and velocities)
• Engineering (design calculations)
• Finance (risk assessment and probability)

How to calculate √123

There are several methods to find the square root of 123:

1. Using a calculator

The simplest method is to use our online square root calculator above. Just enter 123 and click "Calculate".

2. Using the Babylonian method

This iterative method provides an approximation:

1. Start with an initial guess (e.g., 11)
2. Improve the guess using: (guess + 123/guess)/2
3. Repeat until the result stabilizes

3. Using logarithms

For manual calculation:

1. Find log(123)
2. Divide by 2
3. Find antilog of the result

Square root formula

The square root of a number x is defined as:

For 123, we find y where y² = 123. Since 123 isn't a perfect square, we use approximation methods.

Properties of square roots

• √(a²) = |a|
• √(ab) = √a × √b
• √(a/b) = √a / √b
• √(a + b) ≠ √a + √b

Worked example

Let's calculate √123 using the Babylonian method:

1. Initial guess: 11
2. First iteration: (11 + 123/11)/2 = (11 + 11.1818)/2 ≈ 11.0909
3. Second iteration: (11.0909 + 123/11.0909)/2 ≈ (11.0909 + 11.0905)/2 ≈ 11.0907
4. Third iteration: (11.0907 + 123/11.0907)/2 ≈ (11.0907 + 11.0905)/2 ≈ 11.0906

The result stabilizes at approximately 11.0905.