Sums of Complex Numbers

Complex numbers can be added and subtracted to produce other complex numbers. This is done by adding the corresponding real parts and the corresponding imaginary parts. 

For example, the sum of $2+3i$ and $5+6i$ can be calculated by adding the two real parts $(2+5)$ and the two imaginary parts $(3+6)$ to produce the complex number $7+9i$. Note that this is always possible since the real and imaginary parts are real numbers, and real number addition is defined and understood. 

As another example, consider the sum of $1-3i$ and $4+2i$. In this case, we would add $1$ and $4$ to produce $5$ and also would add $-3$ and $2$ to produce $-1$. Thus we would write:

 $(1-3i)+(4+2i)=5-i$

Differences (Subtraction) of Complex Numbers

In a similar fashion, complex numbers can be subtracted. The key again is to combine the real parts together and the imaginary parts together, this time by subtracting them. 

Thus to compute: 

$(4-3i)-(2+4i)$ 

we would compute $4-2$ obtaining $2$ for the real part, and calculate $-3-4=-7$ for the imaginary part. 

We would thus write $(4-3i)-(2+4i) = 2-7i$. 

Note that the same thing can be accomplished by imagining that you are distributing the subtraction sign over the sum $2+4i$ and then adding as defined above. Thus you could write:

 $(4-3i)-(2+4i) = (4-3i)+(-2-4i) = 2-7i.$

Note that it is possible for two non-real complex numbers to add to a real number. For example, $(1-3i)+(1+3i)=2+0i=2$. However, two real numbers can never add to be a non-real complex number.