## How to use the IMSQRT function

The IMSQRT function calculates the square root of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

### Table of Contents

## 1. IMSQRT Function Syntax

IMSQRT(*inumber*)

## 2. IMSQRT Function Arguments

inumber |
Required. A complex number in x+yi or x+yj text format. |

## 3. IMSQRT function example

The image above demonstrates a formula in cell B28 that calculates the square root of a complex number specified in cell B25.

Formula in cell D3:

I have also included the second solution to square root of 3+4i in cell B31. The chart shows complex number 3+4i in light blue, the solutions to square root of 3+4i which are 2+i and -2-i in green and orange respectively.

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMSQRT(*inumber*)

becomes

IMSQRT(B25)

#### Step 2 - Evaluate the IMSQRT function

IMSQRT(B3)

becomes

IMSQRT("3+4i")

and returns

"2+i".

**Note!** The IMSQRT function returns only one of two possible solutions.

The solutions to square root of 3+4i are 2+i and -2-i. Change the sign of both the real and imaginary parts of the complex number to get the second square root.

You can verify these solutions using the IMPOWER function. IMPOWER("2+i",2) equals 3+4i, however, IMPOWER("-2-i",2) also equals 3+4i.

## 4. How to calculate the second square root of a complex number?

The formula in cell B31 calculates the second square root of a given complex number specified in cell B25.

Formula in cell B31:

It takes the value in cell B25, which is a complex number, and calculates its square root using the IMSQRT function. This returns one of the two possible square roots of the complex number.

It then multiplies the result of the IMSQRT function by -1 using the IMPRODUCT function. This changes the sign of both the real and imaginary parts of the complex number, and effectively returns the other square root.

### Explaining formula

#### Step 1 - Calculate the first square root

The IMSQRT function calculates the square root of a complex number in x + yi or x + yj text format.

Function syntax: IMSQRT(inumber)

IMSQRT(B25)

becomes

IMSQRT("3+4i")

and returns "2+i".

#### Step 2 - Second square root

The IMPRODUCT function calculates the product of complex numbers in x + yi or x + yj text format.

Function syntax: IMPRODUCT(inumber1, [inumber2], ...)

IMPRODUCT(IMSQRT(B25),-1)

becomes

IMPRODUCT("2+i",-1)

and returns "-2-i".

## 5. How is the square root of a complex number in rectangular form calculated in detail?

The following math formula shows how to calculate the square root of a complex number.

Z = x + yi

y â‰ 0 (zero)

iy/|y| gives the sign of the imaginary part of the square root. |y| is the absolute value of y. For example, |4| = 4 and |-4| = 4.

iy/|y| = i if y is positive and -i if y is negative. This makes the two square roots have opposite signs for both parts.

For example, Z = 3+4i

|Z| = (3^2+4^2)^0.5 = (9+16)^0.5 = 25^0.5 = 5

Z^0.5 = ((5+3)/2)+i((5-3)/2)^0.5 = 4^0.5+i1^0.5 = 2+i

### Polar form

IMSQRT(C) = âˆš(x + yi)Â = âˆšrÂ cos (Î¸/2) + iâˆšrÂ sin (Î¸/2)

Î¸ = tan^{-1}Â (x/y)

r = âˆš(x^{2} + y^{2})

### Useful links

IMSQRT function - Microsoft

How do I get the square root of a complex number? - Stack Exchange

### Functions in 'Engineering' category

The IMSQRT function function is one of many functions in the 'Engineering' category.

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