: Energy can't be converted into mass and mass can't be converted into energy, even though most books including textbooks say that they can.

Then please tell me how one electron and one positron of equal masses become a photon after annhilation. Mass+mass ! = no mass according to you.

What really happens is the total masses are converted to energy, the energy of the photon that has no rest mass. The photon can then disassociate back into an electron/positron pair.

:E2 = m2c4 +p2 c2 is a statement about how mass, energy, and momentum ("p") are interrelated, but it doesn't imply that one quantity can be converted into another.

It is a statement that the total energy is invariant under a lorentz transform. That is, the total energy is not increased if we change frames. Which is to say the energy is independant of the relative frames of the observors.

What it also says is that if v=0 then E=Mc2. In english, if momentum is zero, mass has an inherent energy.

:What it is says is that if energy has zero momentum then energy itself has rest mass. As I said above this is somewhat of a nitpick.

Please explain how energy can have momentum if energy can not be converted to mass.

Your point is not a nitpick it is just plain wrong.

To quote from my copy of the Berkely Physics Course, Volume 1, Mechanics, P358. which is of course wrong as you want it to be.

E2-p2c2=M2p2 (12.11)

Which is a lorentz invariant. If we transform from one reference frame to another, with p -> p', then the invariance of (12.11) means that,

E'2-p'2c'2=E2-p2c2=M2p2

This is what we mean when we say Equation 12.11 is a Lorentz invariant. We emphasize that M denotes the rest mass of the particle and is a number invariant under a Lorentz transformation. Note from Eq. 12.11

E= sqrt (E2-p2c2) 12.12

If pc > Mc2, then

E=pc

This is an approximation made often by high-energy particle physicists. We shall see later that is valid for light quanta where M=0.

In between these two limits, there is no simple relation between E and p or between the kinetic energy K and p or v. Note that the K, as given in Eq 12.7 {K=Mc2 * (gamma - 1), where gamma is the lorentz transform}, now becomes, with the use of E=gamma*Mc2,

K=E-Mc2

it is important here to note that if v=0, E=Mc2. In other words, the mass M has energy even when at rest. This energy is natuarally called the rest energy and we shall see some examples of its importance. The difference between the energy E (in case V>0) and the rest energy is the kinetic energy K.

Note also that E=gamma * M2 and gamma*M is just the relativistic mass, so that E is just the relativistic mass times c2.Mass and energy are just particular names for the same quantity. It makes no no particular sense to ask: Does a particle have more mass becuase it has kinetic energy or does it have more kinetic energy becuase it has more mass? "More Mass" and "kinetic energy" must go together.

RFL

Which part of the above statement does not make sense to you? Where is your evidence that mass != energy.