Adding and Subtracting Complex Numbers

Two complex numbers a+bi and c+di can be added or subtracted by using the commutative and associative properties of real numbers. To add or subtract two complex numbers, combine their real parts and their imaginary parts separately.

Consider for example the complex numbers z_1=5+2i and z_2=3-3i. To add z_1 and z_2, the above formula can be used or, equivalently, the next three steps can be followed. In a similar way the numbers can be subtracted.

Note that the complex numbers are closed under addition and subtraction — that is, the sum or difference of two complex numbers is always a complex number. If z_1+z_2=0=0+0i, then z_2 is said to be the opposite, or additive inverse, of z_1.