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Why are typical baud rates not in multiples of 1000 instead of being in multiples of 300 i.e. 4800, 9600, 19200, 115200, etc...? Is there a mathematical or electrical reason behind this?

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This goes way, way back to the Bell 103 modem from 1962. It operated at 300 bits per second for reasons deriving from the use of audio frequency-shift keying to modulate data; the frequencies available were limited by the quality of the phone lines. All common serial data rates since then are integer multiples of this rate.

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