Ek=1/2mv^2 is a classical formula. The famous result of E0=mc^2 falls out of relativistic dynamics. This follows from the formula in special relativity: E^2=p^2*c^2+m^2*c^4, where p is the relativistic momentum.
As for your other question, the exponential function is defined for complex values in general. Euler’s formula gives e^(i*z)=cos(z) + i*sin(z). (One derivation uses the taylor series expansions for e^x, cos(x), and sin(x). If you haven’t seen those before, there are also more heuristic demonstrations through geometry, and how numbers are represented in the complex plane.) Euler’s identity is a special case of this formula: e^(iπ)+1=0.
Anyhow, the human body is generally not described well as a closed system. Weight loss is not a consequence of releasing energy stored in virtue of the human body’s mass. Rather, exercising increases the forward rate of certain chemical processes that still remain in the classical limit of conserving mass, and some of the products of those processes are expelled from the body (taking their associated mass with them). The energy dissipated through the course of a workout is that released as certain chemicals are reconfigured from higher to lower energy states. This released energy does not result from a “mass/energy conversion.”
Ek=1/2mv^2 is a classical formula. The famous result of E0=mc^2 falls out of relativistic dynamics. This follows from the formula in special relativity: E^2=p^2*c^2+m^2*c^4, where p is the relativistic momentum.
As for your other question, the exponential function is defined for complex values in general. Euler’s formula gives e^(i*z)=cos(z) + i*sin(z). (One derivation uses the taylor series expansions for e^x, cos(x), and sin(x). If you haven’t seen those before, there are also more heuristic demonstrations through geometry, and how numbers are represented in the complex plane.) Euler’s identity is a special case of this formula: e^(iπ)+1=0.
Anyhow, the human body is generally not described well as a closed system. Weight loss is not a consequence of releasing energy stored in virtue of the human body’s mass. Rather, exercising increases the forward rate of certain chemical processes that still remain in the classical limit of conserving mass, and some of the products of those processes are expelled from the body (taking their associated mass with them). The energy dissipated through the course of a workout is that released as certain chemicals are reconfigured from higher to lower energy states. This released energy does not result from a “mass/energy conversion.”