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Definition and Algebra

In this section we will define complex numbers as an extension of \(\mathbb{R}\) as the following

\[\mathbb{C} \cong \mathbb{R}[x] / \langle x^2+1 \rangle = \{a+bx+\langle x^2+1 \rangle\ | ~ a,b \in \mathbb{R} \}\]

Algebra of Complex Number

Addition of two complex numbers are done as follows

\(\displaystyle a + b x + c + d x = a + c + x \left(b + d\right)\)

Multiplication is done as follows

\(\displaystyle \left(a + b x\right) \left(c + d x\right) = a c + a d x + b c x + b d x^{2}\)

\(\displaystyle a c - b d + x \left(a d + b c\right)\)

Last modified on: 2023-01-04 23:20:05