The last few day's I had the main breaker triped a few times, I checked the amps with a multimeter and when the central air conditioner turns on it draws 150! amps for 1 second and then goes down to 20, its breaker is a double 50A breaker and the main is 125A. So is this the cause or its normal that when turning on it should go up so high and maybe the breaker is designed so?
If a breaker is tripping regularly, that's a serious problem you should look at. That is not normal or acceptable. It means something is wrong with your overcurrent protection, or a defective device actually drawing too much current.
It is normal for circuit breakers to allow overcurrent for a short amount of time. This is needed by motors to start, incandescent bulbs to light, and many electronic power supplies to initially start up. The term is called "inrush current".
Ultimately the purpose of breakers is to protect wiring and receptacles. Overcurrent makes them excessively warm, but small overcurrent makes them warm slowly. The breaker is designed to trip before this heat would endanger the wires.
Many breakers also have an instantaneous trip, where they will trip immediately for very high current (i.e. a dead short). This is many times the nameplate rating of the breaker.
There is a chart for your breaker of how much overcurrent it will permit for how long. Here is one example, there should be a vaguely similar chart for your breaker. I'd get and see if it explains your breaker's tripping behavior. If it doesn't, then either your measuring method is not precise enough, or you have a bad breaker. I would be surprised at a breaker instantaneously tripping at 125% of rating.
Breakers do not trip right away. Whether they trip depends on how far over their rated capacity the current is and how long the current has been going for.
This behavior is documented in a circuit breaker trip curve for your specific breakers, which gives a range of times for a specific current amount. E.g. in that example, a current of 3x the rated current should take between 10 and 35 seconds to trip. At 100x the rated current (e.g. a short circuit) it should trip in less than 1 cycle (1/60 second).
Motor starting loads (its the compressor motor in your A/C drawing all the current) can be very high and it is normal for them to briefly exceed the continuous rating of their circuit including the breaker. If breakers were designed to open with more sensitively, with that kind of load they would nuisance trip all the time.