Square Root of -124

The square root of a negative number involves the imaginary unit 'i', where i² = -1. Thus, the square root of -124 is expressed as √-124 = √124 * i. In its simplest form, √124 is approximated, and the complete expression becomes √124 * i ≈ 11.1355i, making it an imaginary number.

When dealing with negative numbers under the square root, the result is not a real number but an imaginary number. The common methods for calculating square roots, such as the long division method or approximation, focus on the positive part of the number. For -124, we first find the square root of 124 and then multiply by 'i'.

To find the square root of 124, which is the positive component of -124, we can use methods like prime factorization or approximation:

Step 1: Prime factorization of 124: 2 x 2 x 31.

Step 2: Since 124 is not a perfect square, we approximate: √124 ≈ 11.1355.

The square root of -124 is then √124 * i ≈ 11.1355i.

Imaginary numbers are used to handle the square roots of negative numbers. The formula i = √-1 is applied here:

Step 1: Calculate √124 ≈ 11.1355.

Step 2: Multiply by 'i' for the imaginary part: √-124 = 11.1355i.

To approximate √124, we identify two perfect squares it is between:

Step 1: The closest perfect squares are 121 (11²) and 144 (12²).

Step 2: √124 is between 11 and 12. Using linear interpolation: (124 - 121) / (144 - 121) = 3 / 23 ≈ 0.1304, thus √124 ≈ 11 + 0.1304 = 11.1304.

Complex numbers are formed by a real part and an imaginary part. The square root of -124 is entirely imaginary: √-124 = 11.1355i, with no real component.

• Imaginary number: A number that can be written as a real number multiplied by the imaginary unit 'i', which satisfies i² = -1.
   
• Complex number: A number that has both a real part and an imaginary part, expressed as a + bi where a and b are real numbers.
   
• Square root: The value that, when multiplied by itself, gives the original number. For negative numbers, this involves imaginary numbers.
   
• Approximation: The process of finding a number close to the exact value, often used for non-perfect squares.
   
• Prime factorization: Breaking down a number into its prime factors, useful for understanding its structure and simplifying calculations.