# Do Math With Imaginary Numbers in Excel & Google Sheets

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*This tutorial demonstrates how to do math with imaginary numbers in Excel and Google Sheets.*

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## Real, Imaginary, and Complex Numbers

**Real** numbers are what we are all familiar with and use in everyday math. More complicated mathematical principles, such as differential equations, involve the use of **complex** numbers. Complex numbers combine **real** and **imaginary** numbers. Imaginary numbers are written as a real number multiplied by the imaginary unit ** i**, where

**is the square root of**

*i***negative 1**. Complex numbers are shown as

**a+bi**, where

*a*is the real part of the number, and

*b*is the coefficient for the imaginary part of the number.

So for example, in the complex number **5+3i**, **5** is the real part of the number, and **3i** is the imaginary part.

### Express Complex Numbers in Excel

Consider the following table of complex numbers. Column B has the real (*a*) part of each number, and Column C is the coefficient (*b*) for the imaginary part.

- Type the following formula in the column to the right of the first imaginary number to create a complex number.
=COMPLEX(B3,C3)

- If you wish to change the suffix from
**i**to**j**, you can specify this by selecting it in the formula. As it is an**optional**argument, if you leave it blank, then**i**is assumed.

- Copy the formula down to populate the rest of the rows with the formula.

### Extract Real and Imaginary Numbers From Complex Numbers

Starting with complex numbers, you can extract the real and imaginary parts with formulas. Consider the following data:

To **extract the real part** of the complex number, use the IMREAL Function.

=IMREAL(B3)

To **extract the imaginary part** of the complex number, use the IMAGINARY Function.

=IMAGINARY(B3)

## Math With Real and Imaginary Numbers

Complex numbers are stored as **text** in Excel, so you cannot add, subtract, multiply, and divide with complex numbers in the same way you would with ordinary rational (**real**) numbers.

### Addition and Subtraction

If you try to add an ordinary number to a complex number with the standard Excel formula, you get a #VALUE error.

To add **5** to the real part of the complex number, first extract the real part of the complex number. Then add **5** to the real part and recreate the complex number.

=COMPLEX(IMREAL(J2)+J3,IMAGINARY(J2))

For subtraction, replace the addition sign with the subtraction sign.

=COMPLEX(IMREAL(J2)-J3,IMAGINARY(J2))

The mathematical operation (addition or subtraction) is performed **on the real number only** and then a new complex number is created.

### Multiplication and Division

For multiplication, we use the multiply sign:

=COMPLEX(IMREAL(J2)*J3,IMAGINARY(J2)*J3)

and for division, the division sign:

=COMPLEX(IMREAL(J2)/J3,IMAGINARY(J2)/J3)

For multiplication and division, the mathematical operation is performed **both** on the real number and the coefficient of the imaginary number.

## Add Two Complex Numbers Together

The simplest way to add complex numbers together is to use the IMSUM Function.

=IMSUM(D3:D7)

You could, alternatively, add up the real column and add up the imaginary column, and then create a complex number sum from the totals of those columns.

- Use the SUM Function to add up Columns B and C

- Then use the
**COMPLEX**function to get the total amount.

This converts the 2 individual sum’s back to a COMPLEX number. You will see that this is the same value as calculated using the IMSUM function.

## Math With Imaginary Numbers in Google Sheets

Google Sheets handles real, complex, and imaginary numbers just like Excel.

Use the COMPLEX Function to **create a complex number**.

**Extract real and imaginary numbers** from a complex number using the IMREAL and IMAGINARY Functions. Then **add, subtract, multiply, or divide** the same way you would in Excel.

Use the IMSUM Function to **add several complex numbers**.