Complex Numbers

A number containing the term \(i\), i.e., \(\sqrt{-1}\) is known as a complex number.

It is of the form \(a+ib,\) where \(a,b\in \R\) and \(a\) is known as the real part and \(ib\) is known as the imaginary part.

Addition of Complex Numbers

Adding complex numbers gives a new complex number, whose real part is the sum of the real parts of the orignal complex numbers and the imaginary part is the sum of the imaginary parts of the orignal complex numbers.

\((a+ib) + (c+id) = (a+b) + i(b+d)\)

Multiplication in complex numbers

\((a+ib)(c+id)=a.c+i(a.d)+i(b.c)-b.d\newline\to(a+ib)(c+id)=(ac-bd)+i(ad+bc)\)

Complex Conjugate

In complex number, a term called complex conjugate is defined.

The complex conjugate of a complex number \(a+ib\) is valued as \(a-ib.\)

Length of \(a+ib\)

Length or distance of \(a+ib\) from origin \(=\sqrt{a^2+b^2}\)