What is negative nine plus seven plus four 𝑖 plus negative four minus four 𝑖 minus one plus three 𝑖?
Remember, we can add or subtract complex numbers by adding their real parts and separately adding their imaginary parts. Here we have four complex numbers. Now it might not look like it, but we could say that negative nine is actually a complex number. It’s negative nine plus zero 𝑖.
So to solve this problem, we’re going to work out the real part first. That’s negative nine plus seven minus four minus one, which is negative seven. Similarly, the imaginary parts are zero, four, negative four, and negative three.
Remember, we distribute this final set of parentheses. And a negative multiplied by a positive is a negative. That gives us negative three. So the real part of our solution is negative seven, and the imaginary part is negative three. So negative nine plus seven plus four 𝑖 plus negative four minus four 𝑖 minus one plus three 𝑖 is negative seven minus three 𝑖.
Now we also know that many algebraic properties can be extended into the concept of negative numbers. We could actually have collected like terms. Distributing that final set of parentheses and then rearranging slightly, we get negative nine plus seven plus negative four minus one plus four 𝑖 plus negative four 𝑖 minus three 𝑖, which once again gives us negative seven minus three 𝑖. This latter method, the one of collecting like terms, is generally the one that we use when adding and subtracting complex numbers.