Lesson 3 of 12 · 8 min
Adding and subtracting complex numbers
NCERT §4.3.1
The rover starts at 3 + 4i and is sent on a second leg of 5 m west and 2 m north, which is −5 + 2i. Where does it finish?
The lesson in notes
In short
To add, add the real parts and the imaginary parts separately: (a + ib) + (c + id) = (a + c) + i(b + d). For example, (3 + 4i) + (−5 + 2i) = −2 + 6i.
Addition obeys the closure law (the sum is again complex), the commutative law z₁ + z₂ = z₂ + z₁ and the associative law (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃).
The additive identity is 0 = 0 + i0, since z + 0 = z for every z.
Each z = a + ib has an additive inverse −z = −a + i(−b), with z + (−z) = 0. For z = 4 − 9i, −z = −4 + 9i.
Subtraction is adding the negative: z₁ − z₂ = z₁ + (−z₂). So (7 + 2i) − (3 − 5i) = (7 + 2i) + (−3 + 5i) = 4 + 7i, while (3 − 5i) − (7 + 2i) = −4 − 7i; the two answers are negatives of each other.
On the Argand plane the sum is found by placing the second arrow at the tip of the first, as with displacements: a move of 3 + 4i followed by a move of −5 + 2i ends at −2 + 6i.