A to Z of Excel Functions: The IMSQRT Function
4 January 2021
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the IMSQRT function.
The IMSQRT function
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/image1.png/e774d10cbbb9450fc45efbe51abdf434.jpg)
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i (sometimes denoted j) which is defined by its property i2 = −1. In general, the square of an imaginary number bi is −b2. For example, 9i is an imaginary number, and its square is −81. Zero is considered to be both real and imaginary.
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.
The polar form of a complex number is another way to represent the number. The form z = a + bi is called the rectangular form of a complex number.
![](http://sumproduct-4634.kxcdn.com/assets/blog-pictures/2020/a-to-z/237/image2.gif)
The horizontal axis is the real axis and the vertical axis is the imaginary axis. You can find the real and imaginary components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis.
From the Pythagorean Theorem,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/f1.1.png/c2252580a5d243da597e57f2abba8508.jpg)
By using the basic trigonometric ratios,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f2.png/a62f6fb677ff4000d756979eb0ebdcd2.jpg)
Therefore, multiplying each side by r:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f3.png/20989b8738b96b49fae50c4825b1e49c.jpg)
Therefore,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f4.png/4b16297c45de4f462633df9cd7be2a2d.jpg)
In the case of a complex number, r represents the absolute value, or modulus,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f5.png/e26cb93271653fb859f28e0bc673668c.jpg)
and the angle θ is called the argument of the complex number,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f6.png/45f49acc02e7f78f392f1b158362ffb2.jpg)
Using Euler’s Formula,
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/image3.png/f1140ff857fc3b6f5f97a6a24f4a6fc7.jpg)
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f7.jpg/ca03e84aba2ce451efad821811824a0a.jpg)
It is using this approach that the square root of the complex number x + yi may be determined:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/f8.jpg/30963f00b4730f8307426766c7e8b6b6.jpg)
where:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/236/capture.png/e983a7b17fc1ef61166f94c0a09170d4.jpg)
The IMSQRT function returns the square root of a complex number in x + yi or x + yj text format. It employs the following syntax to operate:
IMSQRT(inumber)
The IMSQRT function has the following argument:
- inumber: this is required and represents the complex number for which you want to calculate the square root.
It should be further noted that:
- you should use COMPLEX to convert real and imaginary coefficients into a complex number
- IMSQRT recognises either the i or j notation
- if inumber is a value that is not in the x + yi or x + yj text format, IMSQRT returns the #NUM! error value
- if inumber is a logical value, IMSQRT returns the #VALUE! error value
- if the complex number ends in +i or -i (or j), i.e. there is no coefficient between the operator and the imaginary unit, there must be no space, otherwise IMSQRT will return an #NUM! error.
Please see my example below:
![](http://sumproduct-4634.kxcdn.com/img/containers/main/blog-pictures/2020/a-to-z/237/image5.png/36776d1da4d05b45bb5a5d09375f407c.jpg)
We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every business day.
A full page of the function articles can be found here.