When magnesium burns, it is actually reacting with oxygen in the air and not with fire. Fire is what we call the heat and light produced when things burn.
Magnesium reacts with oxygen to make a compound called magnesium oxide.
2Mg + O2 ---> 2MgO
The bright light results because this reaction produces a lot of heat. When the magnesium gets really hot, it emits energy in the form of light (heat is also how the sun produces light). Since it emits all visible wavelengths at a relatively equal rate, the light looks white-ish. Be careful when burning magnesium though, because it also produces ultra-violet (UV) light which can damage your eyes if you stare at it for too long.
The process of hot objects emitting light is known as incandescence (like an incandescent light bulb). The spectrum of light emitted by hot objects is known as black body radiation. If you want to know a little more about these topics, you can look at this website: causesofcolor
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