I am trying to get into electronics. I'm planning on getting several general-use transistors for arduino projects, I know that a 2N5551 will do what I need. I just want to know what the assosiated numbers mean. For a 2N5551, the Collector − Emitter Voltage is 160, the Collector − Base Voltage is 180 and the Emitter − Base Voltage is 6. Are these the max voltages that may pass through? The Collector Current − Continuous is 600 mA, it has a gain of 80. Wouldn't that mean that I'd need a current of 7.5 mA (600/80) going through the Base to turn the transistor all the way (I'd get 600 mA going from the collector to the emitter)? The datasheet in question
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A voltage doesn't pass through anything. But semantic details aside, yes, I suspect these numbers are the absolute maximum ratings for your transistor. You do not want to design with those alone.
Yes, at least approximately. The gain is usually used for small signals, so getting to the max. might require a (very) slightly different base current. Do read the fine print; the maximum continuous current likely requires keeping the ambient temperature below some value.