Algebraic Form of Complex Numbers

Complex number z=(a;b) can be represented in the form z=a+bi. This is called algebraic form of complex number.

In the representation z=a+bi, `i^2=-1` and `i` is called imaginary unit. a is real part (denoted by Re z) and b is imaginary part (denoted by Im z) of complex number.

If imaginary part of complex number a+bi is not 0 (`b!=0`) then such number is called imaginary, for example 1+bi. If a=0 and `b!=0` then complex number is called purely imaginary.

For example, 3i is purely imaginary.

If b=0 then we obtain real number a.

If we are given complex number z=a+bi then number a-bi is called complex conjugate of z.

It is denoted by `bar(z)` or `z^(**)`. For example complex conjugate of 2+3i is 2-3i.

Sum and product of two complex conjugates are real numbers.

Indeed, if z=a+bi then `bar(z)=a-bi` , so `z+bar(z)=a+bi+a-bi=2a`.

Also, `zbar(z)=(a+bi)(a-bi)=a^2-(bi)^2=a^2-b^2i^2=a^2-b^2(-1)=a^2+b^2`.

Absolute value (or modulus) of complex number z is real number `r=|z|=sqrt(zbar(z))=sqrt(a^2+b^2)`.

Conjugate complex numbers have same absolute values, i.e. `|a+bi|=|a-bi|`.