# How to use the IMABS function

The IMABS function calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.

A complex number consists of an imaginary number and a real number, complex numbers let you solve polynomial equations using imaginary numbers if no solution is found with real numbers. It has applications in engineering such as electronics, electromagnetism, signal analysis, and more.

### Table of Contents

## 1. IMABS Function Syntax

IMABS(*inumber*)

## 2. IMABS Function Arguments

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

## 3. IMABS function example

The IMABS function calculates the modulus of a complex number, IMABS probably stands for imaginary absolute. The absolute value is the same as the modulus.

The modulus of a complex number is the distance of the complex number from the origin in the complex plane. It is the square root of the sum of the squares of the real part and the imaginary part of the complex number. If the complex number is Z then the modulus is denoted |Z|.

The image above shows a chart of the complex plane, complex number 9+12i is the blue line ending with an arrow. The complex plane has an imaginary axis and a real axis, the dashed circle shows the modulus value when it crosses both the imaginary axis and the real axis.

Formula:

The formula calculates the modulus based on the value in cell C26 which is "9+12i" in this example. The formula returns 15 which the dashed circle also shows when it crosses the imaginary and real axis. Section 4 below explains in greater detail how the IMABS function calculates the modulus.

The modulus is needed when you want to:

- convert complex numbers from rectangular form to polar form or vice versa.
- compare sizes or magnitudes of different complex numbers.
- calculate the distance between two complex numbers

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMABS(*inumber*)

becomes

IMABS(C26)

#### Step 2 - Evaluate IMABS function

IMABS(C26)

becomes

IMABS("9+12i")

and returns 15.

## 4. How is the modulus calculated in detail?

The formula to calculate the absolute value or modulus z from a complex number is based on the Pythagorean theorem.

z^{2} = x^{2}+y^{2}

The IMABS function calculates the absolute value using this formula which is based on the Pythagorean theorem:

IMABS(z) = |z| =âˆš(x^{2}+y^{2})

x is the real coefficient and y is the imaginary coefficient of the complex number.

The modulus of a complex number is how far it is from the point where the real and imaginary axes cross (0,0). It is the square root of the real part squared plus the imaginary part squared.

Here is how the modulus is calculated for complex number 9+12i:

z = x + yi

z = 9 + 12i

IMABS(z) = |z| =âˆš(x^{2}+y^{2})

IMABS(z) = |z| =âˆš(9^{2}+12^{2})

IMABS(z) = |z| =âˆš(81+144)

IMABS(z) = |z| =âˆš225

IMABS(z) = |z| =15

## 5. How imaginary numbers were invented

## 6. How to convert complex numbers to polar form?

Formula in cell D4:

Complex numbers are usually presented in this form

z = x + yi

or

z = x + yj

However, complex numbers can also be represented in polar form:

z = r*(cos Î¸ + isin Î¸)

In other words, theta Î¸ in the polar form is calculated using the IMARGUMENT based on complex numbers.

Pythagorean Theorem

r^{2Â }= x^{2}Â + y^{2}

To calculate the absolute value we can use this formula:

r = âˆš(x^{2}+y^{2})

Excel has a function that does this for you, the IMABS function calculates the absolute value based on complex numbers.

### Explaining formula

#### Step 1 - Calculate theta Î¸

The IMARGUMENT function calculates theta Î¸ which is an angle displayed in radians based on complex numbers in rectangular form.

Function syntax: IMARGUMENT(inumber)

IMARGUMENT(C4)

becomes

IMARGUMENT("12+9i")

and returns

0.643501108793284

#### Step 2 - Calculate the absolute value

The IMABS function calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.

Function syntax: IMABS(inumber)

IMABS(C4)

becomes

IMABS("12+9i")

and returns

15

#### Step 3 - Join calculations with text

The ampersand character lets you concatenate values in an Excel Formula.

IMABS(C4)&"(cos "&IMARGUMENT(C4)&" + isin "&IMARGUMENT(C4)&")"

#### Step 4 - Evaluate the formula

IMABS(C4)&"(cos "&IMARGUMENT(C4)&" + isin "&IMARGUMENT(C4)&")"

becomes

15&"(cos "&0.643501108793284&" + isin "&0.643501108793284&")"

and returns

15(cos 0.643501108793284 + isin 0.643501108793284)

### Useful links

### 'IMABS' function examples

The following article has a formula that contains the IMABS function.

### Functions in 'Engineering' category

The IMABS function function is one of 42 functions in the 'Engineering' category.

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