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