It is generally accepted that Auschwitz doctors originally tattooed the Soviets to determine who was suitable for reeducation. But towards the latter half of the war, why were prisoners, primarily the Jewish ones, tattooed? I have read that it was to identify corpses, but wouldn't it make more sense in the prevailing doctrine to say, 1 more person has died, and not that AXXXXX has died?
Ok, let me start with clearing a few things up: Auschwitz (all three major sub-camp, Stammlager, Birkenau, and Monowitz) was the only camp that tattooed people with identification numbers. However, all camps had them. In other camps though, prisoner identification was done mostly via dog tags prisoners wore around their neck. The camp personnel would only address prisoners by their number and the most immediate use of these identification numbers came from counting the prisoners daily. Everyday, the Appell (role call) took place where the camp guards called out all the numbers of prisoners in the camps and saw if they matched the list they had for today. This was to determine if people had died during the night or if anybody had attempted to escape as well as to assign prisoners their work detail.
The last mentioned factor is the most important: In almost all camps, the prisoners worked (save those deemed unfit to work and sent to the gas chambers or towards other execution matters right on arrival; the famous selections in Auschwitz e.g. but those prisoners were also not registered). And with work came the necessity to know who was to be in which work detail. From the economic standpoint of the SS, it made little sense to, say, use a carpenter in a stone cutter work detail or a shoe maker in a construction detail. You have to imagine concentration camps such as Auschwitz as really big prisons and economic juggernauts when it came to provide labor force to companies and for that reason you needed to administer the camp. In order to administer it, you needed to know which prisoner was which and for that purpose, identification numbers were introduced.
Now, the reason Auschwitz starting tattooing prisoners rather than using dog tags like other camps came from Auschwitz being an incredibly big camp. In January 1945 at last roll call the total number of prisoners in the Auschwitz camp complex numbered 54.000 with about one tenth of that number as prison guards (around 5.000).
Initially, Auschwitz had used dog tags as well as prisoner numbers sewn to clothing together with other categories of identification: the famous triangles and letters where the color denoted which kind of prisoner you were (red for political, green for criminals, pink for homosexual, black for "asocial" and so on) together with a further triangle to denote a prisoner was Jewish (forming a Star of David) and a letter denoting a prisoner's nationality (e.g. P for Poles, B for Beligum, none for German). This system of identification was also kept once tattoos were introduced because it also made it easier for the camp guards to treat prisoners differently according to their status.
Usually, when prisoners arrived they received their number. When a prisoner was executed or died, his clothing was removed and their number was written across their chest with indelible ink. This system however became impractical in the autumn of 1941 when huge numbers of starved and ill Soviet POWs started arriving at the Auschwitz I camp (around the same time when the construction of Birkenau began). Due to their ill health and poor physical state, they started dying in huge numbers and during that phase mix-ups started to occur because the people applying the numbers with ink were overwhelmed or the ink didn't prove as indelible as was advertised. Thus the practice of tattooing the corpses of prisoners with their identification number before their clothing was removed was institutionalized.
One reason among others, the Nazi camp administration cited as a reason for why knowing who was dead and who was alive was so important is because they needed to know how much workers they had and how much food (if you want to call it food) they had to prepare everyday. Another reason was because they needed to know how many prisoners worked for various companies they were rented too (e.g. the IG Farben requesting a carpenter work detail) in order to bill said companies properly for the prisoner labor.
In March 1942 the practice was adopted in Birkenau, where it also started that still living prisoners would be tattooed on arrival in order to streamline the process and to not have to tattoo them after they were dead and in 1943 the practice was institutionalized in the whole camp complex, including Monowitz.
All prisoners who were not selected to be gassed immediately were tattooed upon arrival with two exceptions: Prisoners of German nationality and the work education prisoners. The latter were forced laborers as well as German workers who their employers had accused of working too slow or of being lazy and who were sent to a separate compound at Auschwitz to be held there for a maximum of 56 days in order to be "educated to work through work", i.e. to be subjected to an even harsher labor regime than before.
Ok, so now for the really confusing part: The numbers. There were several series of numbers running parallel to each other but all starting with "1". The first one is the very first series of numbers introduced even before tattooing in May 1940 upon the camp opening. It ran continuously until 1945 and ended with 202,499. In October 1941 with the influx of Soviet POWs a second series was begun, this one with "R" (for Russian) at the beginning. This number ran up to 12.000 and was in continuous use until 1944. A third series began in March 1942 with the arrival of the first female prisoners and was exclusively used for women until May 1944. In 1943 four additional series were begun: Two for male and female "gyspies" respectively, starting with a "Z" (for Zigeuner) when the so-called Zigeuner camp was opened in February 1943. The numbers went up to 10,094 for men and 10,888 for women. And another two series in 1943 that identified the male and female work education prisoners retroactively, tracing those who had dies, transferred or released. The work education numbers were the only ones that were recycled for new prisoners.
In May 1944 huge numbers (about 500.000) of Hungarian Jews arrived at the camp. In order to avoid excessively high numbers another set of series were begun for male and female prisoners: The "A" and "B" series according to the letter they had the beginning. A went up to 20.000 when B was started (except for women where the A series went to 30.000 for unknown reasons). The B series ran up until about 15.000.
All in all camp authorities in Auschwitz gave out about 400.000 prisoner identification numbers we know off. The prisoners who received their tattoos did so on their left forearm (sometimes with a Star of David if they were Jewish). The only exception here being new born and babies who were born in the camp. They received their numbers on the left upper leg.
Sources:
Nicholaus Wachsmann: KL. A history of the Concentration Camps.
Franciszek Piper: Auschwitz Nazi Death Camp.
Serge Klarsfeld (ed.): Les matricules tatoues des camps d'Auschwitz Birkenau.