Modified Julian Date, MJD, is a modification of the Julian Date that is routinely used by astronomers, geodesists, scientists, financiers, and even some historians. This dating convention, designed to facilitate simplified chronological calculations, numbers all days in consecutive fashion, beginning at a date sufficiently far into the past so as to precede the historical period. Julian Day Number is an integer counter of the days beginning at noon on January 1, 4713 B.C. (no, even though I may look like it, I was not there...), which is Julian Day Number 0. The Julian Date (as opposed to Julian Day) is the non-integer extension of the Day Number to include a real fraction of day, allowing a continuous time unit. MJD modifies this Julian Date in two ways. The MJD begins at midnight rather than noon (the .5 in the formula), in keeping with more standard conventions and modern representation of time. Secondly, for simplicity, the first two digits of the Julian Date are removed. This is because, for some three centuries following November 17, 1858, the Julian day lies between 2400000 and 2500000. The MJD drops those first "24" digits. Thus, we have as a formula for converting from JD to MJD, MJD = JD - 2400000.5. (You may wish to use our MJD converter.)
Truncated Julian Date, TJD, is a truncated form of MJD. The starting TJD point is midnight May 24, 1968 (Ahhh, I remember that night well...). Thus there are two formulae for calculating TJD. TJD = MJD - 40000, assuming you know the MJD. If you only know the JD, TJD = JD - 2440000.5. In both cases, MJD and TJD, it is generally used in integer form, representing the day only, not the day and time, though it is fully capable of both day and time unless expressed as an integer.