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.
`11 and 12 Grade Math
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