Real parts with real parts
To add 3 + 2i and 1 + 4i, add the real parts, 3 + 1 = 4, and add the imaginary parts, 2 + 4 = 6. The sum is 4 + 6i.
This is ordinary addition, regrouped. Addition can be done in any order, so 3 + 2i + 1 + 4i = (3 + 1) + (2i + 4i) = 4 + 6i. The 3 and the 1 count ones; the 2i and the 4i count multiples of i.
Each column is added down on its own: 3 + 1 = 4 and 2 + 4 = 6, so the sum is 4 + 6i.
Two counts that never mix
The real parts and the imaginary parts are two separate counts. Adding 1 + 4i moves the real part from 3 to 4 and, separately, moves the imaginary part from 2 to 6. Nothing passes from one count to the other.
The real parts: from 3, adding 1 lands on 4.
The imaginary parts, counted in multiples of i: from 2i, adding 4i lands on 6i.
Like terms
Adding complex numbers is collecting like terms. (3 + 2x) + (1 + 4x) = 4 + 6x, and (3 + 2i) + (1 + 4i) = 4 + 6i in exactly the same way, with i standing where x stood.
A real number and a multiple of i are unlike terms, as 4 and 6x are. So 4 + 6i is as far as the sum goes: it is not 10, and it is not 10i.
Subtracting
To subtract, take away the real part from the real part and the imaginary part from the imaginary part: (5 + 3i) − (2 + i) = (5 − 2) + (3 − 1)i = 3 + 2i.
The minus sign applies to the whole bracket, so −(2 + i) = −2 − i, and both parts of 2 + i are taken away. Check by adding back: (3 + 2i) + (2 + i) = 5 + 3i.
Subtracting down each column: 5 − 2 = 3 and 3 − 1 = 2, so (5 + 3i) − (2 + i) = 3 + 2i. The imaginary part of 2 + i is 1.
Negative parts, and parts that cancel
Negative parts add as negative numbers do. (3 + 2i) + (1 − 5i) = (3 + 1) + (2 − 5)i = 4 − 3i. (2 − 3i) + (−5 + i) = (2 − 5) + (−3 + 1)i = −3 − 2i.
A real number is a complex number with imaginary part 0, so 7 + (2 + 3i) = 9 + 3i: only the real part changes.
When the imaginary parts cancel, the sum is real: (4 + 3i) + (2 − 3i) = 6 + 0i = 6. When both parts cancel, the sum is 0: (3 + 2i) + (−3 − 2i) = 0.
The usual mistakes
Lumping all four numbers into one i-term. (3 + 2i) + (1 + 4i) is not 10i: 3 and 1 are real numbers, not multiples of i.
Adding each bracket’s own parts. 3 + 2i is not 5, and the sum is not 5 + 5i. A real part and an imaginary part are unlike terms and stay separate.
Taking away only the real part. (5 + 3i) − (2 + i) is not 3 + 4i: the minus sign reaches the i in the second bracket too.