Definition:Addition/Complex Numbers

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Definition

The addition operation in the domain of complex numbers $\C$ is written $+$.

Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$.


Then $z + w$ is defined as:

$\paren {a + i b} + \paren {c + i d} = \paren {a + c} + i \paren {b + d}$


Complex Addition

When the formal definition of complex numbers is used, complex addition is defined thus:


Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $\tuple {x_1, y_1} + \tuple {x_2, y_2}$ is defined as:

$\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$


Examples

Example: $\paren {3 + 2 i} + \paren {5 + 6 i}$

$\paren {3 + 2 i} + \paren {5 + 6 i} = 8 + 8 i$


Example: $\paren {-1 + 4 i} + \paren {2 + \paren {-7} i}$

$\paren {-1 + 4 i} + \paren {2 + \paren {-7} i} = 1 + \paren {-3} i$


Also see


Sources