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How do you find the complex number of a square root?

Published in Complex Numbers 2 mins read

To find the complex number of a square root, you typically use a formula that leverages the rectangular form of the complex number. Here's how:

The Formula

The easiest way to determine the square root of a complex number a + ib (where a and b are real numbers and i is the imaginary unit) is using the following formula:

√(a + ib) = ± (√(|z|+a/2) + i sgn(b) √(|z|−a/2))

Where:

  • z is the complex number a + ib
  • |z| is the modulus (or absolute value) of z, calculated as √(a2 + b2)
  • sgn(b) is the sign function of b:
    • If b > 0, then sgn(b) = 1
    • If b < 0, then sgn(b) = -1
    • If b = 0, then sgn(b) = 0 (In this case the imaginary part will be 0, which simplifies the formula)

Step-by-Step Calculation

Here's how to apply the formula:

  1. Identify a and b: Determine the real part (a) and the imaginary part (b) of the complex number.
  2. Calculate the modulus |z|: Find the absolute value (or modulus) of the complex number using the formula |z| = √(a2 + b2).
  3. Calculate √(|z|+a/2): This will be the real part of the square root (before considering the ±).
  4. Determine the sign of b: Find sgn(b) to determine the sign of the imaginary part of the square root.
  5. Calculate √(|z|−a/2): This will be the absolute value of imaginary part of the square root (before considering the ±).
  6. Combine the results: Use the formula to construct the square roots: ± (√(|z|+a/2) + i sgn(b) √(|z|−a/2))

Example

Let's find the square root of 3 + 4i:

  1. a = 3, b = 4
  2. |z| = √(32 + 42) = √(9 + 16) = √25 = 5
  3. √(|z|+a/2) = √(5 + 3/2) = √(10/2 + 3/2) = √(13/2)
  4. sgn(b) = sgn(4) = 1
  5. √(|z|−a/2) = √(5 - 3/2) = √(10/2 - 3/2) = √(7/2)
  6. √ (3 + 4i) = ± (√(13/2) + i * √(7/2))

Therefore, the square roots of 3 + 4i are approximately ± (2.55 + 1.87i).

Explanation

The formula is derived by assuming that the square root of a complex number a + ib is another complex number x + iy. Squaring x + iy and equating the real and imaginary parts leads to a system of equations that can be solved for x and y in terms of a and b. The result is the formula provided above.

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