Maybe a little too comprehensive but here goes:
The number system today is very simple, but up to and including 1995, it was a very free-based system. The world championship winning driver (regardless of which team won the constructor's championship or which team he drove for) was awarded the number 1, whether he stayed with the same team or moved to another team. Whoever his teammate was would be number 2. After that, teams were free to choose their own number pairings.
This is what the numbers looked like in 1988:
1, 2 - Lotus
3, 4 - Tyrell
5, 6 - Williams
7, 8 - (Brabham) - team did not compete in the 1988 season
9, 10 - Zakspeed
11, 12 - McLaren
14 - AGS
15, 16 - March
17, 18 - Arrows
19, 20 - Benetton
21 - Osella
22 - Rial
23, 24 - Minardi
25, 26 - Ligier
27, 28 - Ferrari
29, 30 - Larousse
31 - Coloni
32, 33 - EuroBrun
34, 35 - open
36 - BMS
Lotus had numbers 1 and 2 because Nelson Piquet had won the previous world championship and was now their driver. Piquet had won with Williams and even though Williams clearly won both the driver's championship and the constructor's championship, Piquet left for Lotus and took the number with him since it is always awarded to the winning driver, not the winning team.
Lotus normally took numbers 11 and 12, as they had during Senna's stay with them while Senna was 12 in the 2nd Lotus. Before the 1985 season he was the unofficial number 2 driver of Lotus behind Elio De Angelis who, as their number 1 driver well before Senna ever joined the team, took the number 11 many years before and kept it. Senna still kept the number 12 when he became the clear number 1 of Lotus and even after De Angelis left, and this was very common at this time as drivers tended to keep their numbers from year to year when they stayed with the same teams (unless they won the world championship).
Tyrell had numbers 3 and 4 by pure choice as they had it the previous seasons, as did most of the other teams on the list. For example, Williams had numbers 5 and 6 and those were their typical numbers, with the 5 strongly associated with Nigel Mansell and the number 6 to whoever his teammate was (as long as he was in the team). Numbers 7 and 8 were previously with Brabham the year before but they decided to take a one year hiatus. Also, traditionally numbers 7 and 8 belonged to McLaren and were used by the team before they won their first world championship with Lauda in 1984 (Prost #7, Lauda #8). McLaren had #1 and #2 for the next three years because Prost won two more championships.
Now with Lotus taking the number 1, McLaren had to choose which numbers they wanted for 1988. Normally they may have chosen the numbers 7 and 8 but Brabham occupying those numbers, even though they were skipping the season, may have caused them to choose another pair. Thus they swapped with the team that took their numbers, Lotus, and took numbers 11 and 12, perhaps coincidentally keeping Senna's number 12 as he moved to McLaren from Lotus. Also this could have been the reason why they didn't take 7 and 8 but perhaps not.
Anyway, Prost was McLaren's unofficial number 1 driver as he was their world champion and a veteran of the team and he took 11, although at the same time Senna probably would have taken the 12 either way as he'd had it for the previous three years and likely had some attachment to it. There was also no rule as to who would drive under which number (outside of 1 and 2), but teams usually gave the lower number to the more senior and/or better driver (but not always, especially as some drivers chose certain numbers).
Some teams also ran only one car and thus had only one number. Additionally, teams did not run the number 13 due to bad luck.
After Gilles Villeneuve's death in 1982, while driving with the number 27, Ferrari decided to keep the number pairing of 27 and 28 effectively permanently (with these number rules) due to Villeneuve's strong association with the number 27. 27 now became the number usually appointed to Ferrari's designated number one and/or senior driver and 28 to his teammate. And while on the topic of 27 for Villeneuve, 28 became strongly associated with Berger and he kept the number from 1987-89 at Ferrari, 1990 at McLaren (when Ferrari had the number 1 on their car), and 1993-95 again with Ferrari.
Anyway, Senna won the 1988 world championship so he took the number 1 for 1989. Prost won it in 1989 and left for Ferrari, taking 1 and 2 away from McLaren and to Ferrari. Thus again, McLaren had to choose another number pairing. They couldn't choose 11 and 12 again since Lotus retook those numbers after they lost the numbers 1 and 2 when McLaren won in 1988, and they couldn't choose 7 and 8 since Brabham had those numbers and were competing again. Ultimately they took Ferrari's traditional 27 and 28, since they weren't using them that year as they had the number 1 car with Prost. Fittingly, Berger was able to take his familiar 28 and Senna also fittingly took the usual lower number of the pairing with 27. Senna won that year and retook the number 1 and 2 for McLaren and Ferrari went back to 27 and 28.
Nigel Mansell won the world championship for Williams in 1992 and left for IndyCar. Since there was no world champion, no car or team would get to run the number 1, but instead the number 0 was given to accompany number 2. And since a Williams driver won the championship, they were given these numbers for 1993. Prost chose 2 because he didn't want to drive a car with the number 0, just as Senna did the following year after Prost won the championship and retired.
Eventually came 1996 and Formula One changed the number system and teams were no longer free to choose their own numbers. Instead, numbers were awarded based on the winning driver and the order of the teams in the constructor's championship.
The driver who wins the world championship and competes in the next season has the #1 on his car no matter who he drives for, the same team or a different one. His teammate obviously takes #2. After determining #1 and #2, the championship order of the drivers becomes irrelevant and we turn to the order of the constructors championship.
The highest placing team, discounting the team of the winning driver if he stays (already awarded numbers 1 and 2), receives numbers 3 and 4. Usually this is the team that finished 2nd place in the constructor's championship but can sometimes be the team that won the constructor's championship in the case that the winning driver's team did not finish first. Example: Hakkinen won the championship for McLaren in 1999 and kept the number 1 for 2000, even though Ferrari won the constructor's championship.
After these numbers are determined, the rest is simple. The next best team in the points, NOT drivers, are awarded the next numbers in pairs, skipping the number 13. So one team will have numbers 11 and 12 and the next placed team will take numbers 14 and 15. With a 24 car field as there is today, the last placed team (determined by the team with the worst "best" finish in the case that these teams didn't get any points, as often happens) gets numbers 24 and 25.
Normally the numbers are given out within the teams in the order their drivers finished in the championship the previous year, but this is not set in stone and sometimes the drivers are able to choose. Case in point, Schumacher in 2010 would have been given the number 4 for Mercedes behind teammate Rosberg, however Schumacher wanted number 3 because he had never had an even number before and saw it as bad luck. Rosberg was fine with it and thus they switched numbers. This is likely due to Schumacher's seniority and perhaps even more so his legendary status, however. Normally the championship order does take precedence though. Example: McLaren had numbers 3 and 4 for 2011 and has them again for 2012. In 2011 Hamilton had 3 and Button had 4, since Hamilton finished higher up the year before. However this year, 2012, Button now has the 3 and Hamilton has the 4 since Button finished better last year.
And if you're wondering how Mercedes ended up with numbers 3 and 4 as a new team in 2010 -- Jenson won the championship with Brawn in 2009. He moved to McLaren, thus taking numbers 1 and 2 to that team. Brawn then became Mercedes, and since Brawn was the best finishing team in the constructor's championship (remembering their driver winning is irrelevant since he's now gone), they took the next numbers in line.
Maybe this has all been a bit too much in detail, but while it is a simple system on the surface there are a lot of little things to account for.
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