Real Number Multiplied by Complex Number

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Let $a \in \R$ be a real number.

Let $c + d i \in \C$ be a complex number.


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


$a$ can be expressed as a wholly real complex number $a + 0 i$.

Then we have:

\(\ds a \times \paren {c + d i}\) \(=\) \(\ds \paren {a + 0 i} \times \paren {c + d i}\) Definition of Wholly Real
\(\ds \) \(=\) \(\ds \paren {a c - 0 d} + \paren {a d + 0 c} i\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds a c + i a d\) simplification

The result for $\paren {c + d i} \times a$ follows from Complex Multiplication is Commutative.