Unit complex number

Complex conjugate, absolute value, argument and division

For any complex number z = x + yi , the product $${\displaystyle z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}}$$ is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root $${\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}$$ By Pythagoras' theorem, ${\displaystyle |z|}$ is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that ⁠${\displaystyle |z|=1}$⁠, known as the unit complex numbers. If ${\displaystyle z=x=x+0i}$ is a real number, then ⁠${\displaystyle |z|=|x|}$⁠: its absolute value as a complex number and as a real number are equal.