Adding and Subtracting

A complex number has a real part and an imaginary part, written $a+bi$. In $3+4i$ the real part is 3 and the imaginary part is 4. A real number like 5 is a complex number with imaginary part 0, and 5i is one with real part 0.

To add, combine real parts with real parts and imaginary parts with imaginary parts, as you collect like terms: $\left(3+4i\right)+\left(2-7i\right) = 5-3i$, since 3 and 2 make 5, and 4i and -7i make -3i.

To subtract, the minus belongs to the whole second number, so it flips both of its signs: $\left(8+3i\right)-\left(2-5i\right)$ becomes $8+3i-2+5i$, which is $6+8i$. The answer keeps two parts because a real number and a multiple of i cannot combine, just as $6+8x$ stays as it is.