Addition of Two Complex Numbers

We will discuss here about the usual mathematical operation - addition of two complex numbers.

How do you add Complex Numbers?

Let z$$_{1}$$ = p + iq and z$$_{2}$$ = r + is be any two complex numbers, then their sum z$$_{1}$$ + z$$_{2}$$ is defined as

z$$_{1}$$ + z$$_{2}$$ = (p + r) + i(q + s).

For example, let z$$_{1}$$ = 2 + 8i and z$$_{2}$$ = -7 + 5i, then

z$$_{1}$$ + z$$_{2}$$ = (2 + (-7)) + (8 + 5)i = -5 + 13i.

If z$$_{1}$$, z$$_{2}$$, z$$_{3}$$ are any complex numbers, then it is easy to see that

(i) z$$_{1}$$ + z$$_{2}$$ = z$$_{2}$$ + z$$_{1}$$             (Commutative law)

(ii) (z$$_{1}$$ + z2) + z$$_{3}$$ = z$$_{1}$$ + (z$$_{2}$$ + z$$_{3}$$), (Associative law)

(iii) z + 0 = z = 0 + z, so o acts as the additive identity for the set of complex numbers.

Negative of a complex number:

For a complex number, z = x + iy, the negative is defined as -z = (-x) + i(-y) = -x - iy.

Note that z + (-z) = (x - x) + i(y - y) = 0 + i0 = 0.

Thus, -z acts as the additive inverse of z.

Solved examples on addition of two complex numbers:

1. Find the addition of two complex numbers (2 + 3i) and (-9 - 2i).

Solution:

(2 + 3i) + (-9 - 2i)

= 2 + 3i - 9 - 2i

= 2 - 9 + 3i - 2i

= -7 + i

2. Evaluate: (2√3 + 5i) + (√3 - 7i)

Solution:

2√3 + 5i + √3 - 7i

= 2√3 + √3 + 5i - 7i

= 3√3 - 2i

3. Express the complex number (1 - i) + (-1 + 6i) in the standard form a + ib.

Solution:

(1 - i) + (-1 + 6i)

= 1 - i -1 + 6i

= 1 - 1 - i + 6i

= 0 + 5i, which is the required form.

Note: The final answer of addition of two complex numbers must be in simplest or standard form a + ib.

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