What you see as fire is the light released by plasma, ionised gas produced by the combustion reaction. Electrons in the atoms of gas are excited to a higher energy level by the energy released in the reaction and when they fall back down they release photons, light. This is why fire can be different colours, as the frequency of light released is dependant on the amount of energy the electrons transition by. Higher energy corresponding to high frequency, shorter wavelength light (blue and purple). This is the same reason you can see hot things with an infrared camera, as warm bodies will be releasing light in the infrared range of frequency
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