stylistic:tone, stylistic:style_transfer, linguistic:voice_change, information:filtering
Complex numbers

A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane. The absolute value (or modulus) of z may be thought of as the distance of P from the origin of that space. The formula for the absolute value of z = a + bi is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:

where the real numbers a and b are the real part and the imaginary part of z, respectively. For instance, the modulus of −3 + 4 i is (− 3) 2 + 4 2 = 5 {\displaystyle {\sqrt {(-3)^{2}+4^{2}}}=5}. Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, z ¯ {\displaystyle {\bar {z}}}, where for any complex number z = a + b i {\displaystyle z=a+bi}, its complex conjugate is z ¯ = a − b i {\displaystyle {\bar {z}}=a-bi}.

(where i 2 = − 1 {\displaystyle i^{2}=-1}).