### Video Transcript

If 𝑟 equals five plus two 𝑖 and 𝑠 equals five minus two 𝑖, what is the modulus of 𝑟 plus 𝑠?

In order to add two complex numbers of the form 𝑥 plus 𝑖𝑦, we need to add the real parts and, separately, the imaginary parts. In this question, 𝑟 plus 𝑠 is equal to five plus two 𝑖 plus five minus two 𝑖. Grouping the real parts gives us 10, as five plus five equals 10. Grouping the imaginary parts gives us zero 𝑖, as two 𝑖 minus two 𝑖 is zero 𝑖. This is the same as zero. Therefore, 𝑟 plus 𝑠 is equal to 10. If we have any complex number in the form 𝑧 equals 𝑥 plus 𝑖𝑦, then the modulus of 𝑧 is equal to the square root of 𝑥 squared plus 𝑦 squared. We calculate the modulus by finding the sum of the squares of the real and imaginary parts and then square rooting the answer.

The real part of 𝑟 plus 𝑠 is equal to 10, and the imaginary part is equal to zero. This means that the modulus of 𝑟 plus 𝑠 is equal to the square root of 10 squared plus zero squared. 10 squared equals 100 and zero squared is zero. So we are left with the square root of 100. This is equal to 10. Therefore, the modulus of 𝑟 plus 𝑠 is 10. It is important to note that in most cases, the modulus of 𝑟 plus 𝑠 is not equal to the modulus of 𝑟 plus the modulus of 𝑠. We would not be able to calculate the modulus of 𝑟, the modulus of 𝑠 and then add them to calculate the modulus of 𝑟 plus 𝑠.