Summary

Complex Number Field

Definition

Definition

A complex number field is a field that is formed by tuples of real numbers $(a,b)$, where $a$ and $b$ are real numbers. $\mathbb{C}=\mathbb{R}\times \mathbb{R}$. The field operations are defined as follows:

• Addition: $(a,b)+(c,d)=(a+c,b+d)$
• Multiplication: $(a,b)\cdot (c,d)=(ac-bd,ad+bc)$
• The additive identity is $(0,0)$ and the multiplicative identity is $(1,0)$.
• Inverse of $(a,b)$ is $\left(\frac{a}{a^2+b^2},-\frac{b}{a^2+b^2}\right)$ if $(a,b)\ne (0,0)$.

Imaginary (i) Real

(3, 4i) r

3 (Real) 4i (Imaginary)

definition for imaginary unit

Definition

The imaginary unit $i$ is defined as $i=\left(0,1\right)$ in the complex number field. It satisfies the property $i^2=(0,1)\cdot (0,1)=(-1,0)$.