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Years are counted since the Hijra, that is, Mohammed's emigration to Medina in AD 622. On 16 July (Julian calendar) of that year, AH 1 started (AH = Anno Hegirae = year of the Hijra).
In the year AD 2003 we have witnessed the start of Islamic year AH 1424.
Note that although only 2003-622=1381 years have passed in the Christian calendar, 1423 years have passed in the Islamic calendar, because its year is consistently shorter (by about 11 days) than the tropical year used by the Christian calendar.