Question Video: Finding the Conjugate of a Complex Number and Their Sum Mathematics • 12th Grade

Video Transcript

Find the complex conjugate of negative seven minus 𝑖 and the sum of this number with its complex conjugate.

The complex conjugate of a complex number 𝑍, which is equal to 𝑎 plus 𝑏𝑖, is the number 𝑍 star, which is equal to 𝑎 minus 𝑏𝑖. The difference between a number and its complex conjugate is that the sign of the imaginary part of the number is changed. It’s multiplied by negative one.

So to find the complex conjugate of the complex number negative seven minus 𝑖, we just change the sign of the imaginary part. The complex conjugate is equal to negative seven plus 𝑖.

Next, we need to find the sum of the number and its complex conjugate. The sum is equal to negative seven minus 𝑖 plus negative seven plus 𝑖. To find this sum, we just need to add the real parts and add the imaginary parts.

For the real part, negative seven plus negative seven is equal to negative 14. For the imaginary part, we have negative 𝑖 plus positive 𝑖. The imaginary parts therefore cancel out. And we have an imaginary part of zero, so no imaginary part in the sum.

This isn’t a coincidence. And in fact, it’s an illustration of a general rule, which is that if we add a number and its complex conjugate together, then the result is a real number. In fact, it’s equal to twice the real part of the complex number.

You can see this clearly if you consider adding the complex number 𝑎 plus 𝑏𝑖 to its conjugate 𝑎 minus 𝑏𝑖. The imaginary parts cancel each other out. And we’re left with two 𝑎, twice the real part of the complex number. The complex conjugate of negative seven minus 𝑖 is negative seven plus 𝑖. And the sum of this number with its complex conjugate is negative 14.