Let P and Q be two complex numbers such that
P = a+bi
Q = c+di
Where a,b,c,d are real numbers and i = sqrt(-1).
This means i^2 = -1.
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Adding P and Q means
P+Q = (a+bi)+(c+di)
P+Q = a+bi + c+di
P+Q = (a+c) + (bi+di)
P+Q = (a+c) + (b+d)i
As you can see, we just add the corresponding components together.
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Subtraction is a similar story.
P-Q = (a+bi)-(c+di)
P-Q = a+bi - c-di
P-Q = (a-c) + (bi-di)
P-Q = (a-c) + (b-d)i
We subtract the corresponding components
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Multiplication is a bit more complicated.
We'll use the FOIL rule
P*Q = (a+bi)*(c+di)
P*Q = a*c + a*di + bi*c + bi*di
P*Q = a*c + ad*i + bc*i + bd*i^2
P*Q = a*c + ad*i + bc*i + bd*(-1)
P*Q = a*c + ad*i + bc*i - bd
P*Q = (ac - bd) + (ad*i + bc*i)
P*Q = (ac - bd) + (ad + bc)i
Unfortunately multiplication isn't as simple as addition or subtraction, but we can at least make a tidy formula for it. You could also use the box method to visually organize the terms into a table to help multiply out P and Q.