Ordering of Complex Numbers

The ordering of complex numbers is not possible. Real numbers and other related number systems can be ordered, but complex numbers cannot be ordered. The complex numbers do not have the structure of an ordered field, and there is no ordering of the complex numbers that are compatible with addition and multiplication. Also, the non-trivial sum of squares in an ordered field is a number ≠0≠0, but in a complex number, the non-trivial sum of squares is equal to i² + 1² = 0. The complex numbers can be measured and represented in a two-dimensional argrand plane by their magnitude, which is its distance from the origin.

Euler’s Formula: As per Euler’s formula for any real value θ we have e^iθ = Cosθ + iSinθ, and it represents the complex number in the coordinate plane where Cosθ is the real part and is represented with respect to the x-axis, Sinθ is the imaginary part that is represented with respect to the y-axis, θ is the angle made with respect to the x-axis and the imaginary line, which is connecting the origin and the complex number. As per Euler’s formula and for the functional representation of x and y we have e^{x + iy} = e^x(cosy + isiny) = e^xcosy + ie^xSiny. This decomposes the exponential function into its real and imaginary parts.