One milliampere-hour is exactly 3.6 coulombs. The pair connects the unit printed on every battery to the unit every physics equation expects, and the factor is the 3,600 seconds in an hour scaled by the milli prefix.
About the units
The milliampere-hour is the charge delivered by a thousandth of an ampere for an hour. The coulomb is one ampere-second. Dividing the first by the second gives 3,600 divided by 1,000, which is 3.6. The convenience of the milliampere-hour is that it matches the scale of small cells; the convenience of the coulomb is that it fits into equations without a conversion constant. Neither is more correct, and the 3.6 is the bridge between the product label and the physics.
The exact factor
The factor is exactly 3.6. A 4,000 mAh phone battery holds 14,400 coulombs, which is about 9 × 10²² electrons. In energy terms, the same battery at 3.85 V holds 15.4 Wh or 55,400 joules — roughly the chemical energy in three grams of sugar, which is a useful reminder of how modest battery storage still is compared with chemical fuels. A litre of petrol holds about 34 megajoules, over six hundred times as much.
Where you meet this conversion
Battery modelling and electronics design, where a datasheet gives capacity in milliampere-hours and a simulation works in coulombs. Coulomb counting is the standard method of estimating state of charge in battery management systems: the controller integrates current over time and tracks the charge that has entered and left the pack. Supercapacitor specifications mix the two, giving capacitance in farads and stored charge in coulombs while the application compares them against batteries rated in mAh.