Good book that explains the features of electronics with math

There is no single rule that drives you to just one analog chip manufacturer's parts. (Not in microcontrollers, either, despite your claim that only Atmel makes sense for you. But that's a separate question.)

A really good component engineer could probably fill books about how to choose a chip and/or manufacturer. I am not a component engineer, but I can lay out a bunch of cases where I've been forced to one manufacturer or another:

• Unique chip: TI's TLE2426 is truly unique. There are numerous ways to build something similar out of generic components, but if you need that function and it has to fit in a TO-92 footprint, you're stuck with TI's chip.

• No standard pinout: Unlike with op-amps, analog buffers never really settled on a common pinout, and there are few enough of them available that it's rare for more than a few to be really suitable for a given circuit. Say you pick NatSemi's LMH6321 for some reason. Even though another chip may be an acceptable alternative, no other analog buffer on the market shares that particular pinout, so as soon as you make a PCB with the LMH6321 in mind, you cannot change to another chip without re-spinning the board.

• Subjective matters: In high-end audio circuits, a good ear can distinguish the sound of op-amps, or at least, op-amp families. If you happen to like the Burr-Brown sound, you will be looking only at their chips, while another will swear by Analog Devices chips, and you may both wish to ignore the Linear fanbois.

• Price: Maybe you need an LM324 but aren't too picky about quality. In that case, you're going to be looking at one of the second-source suppliers for this chip like Fairchild or ST, rather than the chip's creator, NatSemi.

• Durability: This is the inverse of the previous point. I'll bet anyone in the industry for more than a few years could tell you stories where the design engineer chose something widely second-sourced (an LM317 for example), did all their prototyping with a high-quality implementation of that chip design, sent it off to the manufacturing folk who substituted a far cheaper generic, and then had to deal with field failures because the replacement, though claiming equivalent performance in its datasheet, turned out less durable when abused in ways you cannot control from your clean lab. (ESD in winter in North Dakota, idiot end users unplugging connectors the manual says not to unplug while power is applied, etc.) It is often cheaper to go back to using the more expensive first-source version than re-spin the board with protection components scattered around the cheap knockoff.

• Obsolescence: You're tasked with stuffing boards you can't re-spin for a design made 20 years ago, and some of the chips aren't available from their original manufacturers any more, or at least aren't available in the packages the design engineer chose 20 years ago. You might be forced into the arms of a company like NTE or Central Semiconductor, who specialize in offering old designs that all the tier 1 companies have abandoned.

• Higher performance: A highly popular IC like the LM317 often inspires pin-compatible follow-ons with higher performance. Linear Technology specializes in this. If you find yourself needing better performance, it is often worth paying the higher price for a less common pin-compatible replacement than redesigning your circuit to give better performance with the original chip you selected.

• Availability: Once you've run the gauntlet above, you might still seem to have a few choices of sources for a given part, but one of the sources has burned you in the past on availability. That sort of thing can force you to another manufacturer all by itself.

  A more extreme variant of this situation is that you've whittled away your choices until you end up looking at just one part, but it's made by a manufacturer that doesn't always have ready stock. That might force you to abandon that choice, desirable though it may be on paper, simply to switch to a more reliable source. You might even end up reinventing that perfectly good wheel with generics just to get better control of parts availability.

I'll start first with definition of amplification. In the most general way amplification is just a ratio between two values. It does not imply that the output value is greater than the input value (although that's the way it's most commonly used). It is also not important if the current change is big or small.

Now let's move to some common amplification values used:

The most important (and the one your question talks about) is \$ \beta\$. It is defined as \$ \beta= \frac {I_c} {I_b} \$, where \$I_c\$ is the current going into the collector and \$I_b\$ is the current into the base. If we rearrange the formula a bit, we'll get \$I_c=\beta I_b\$ which is the most commonly used formula. Because of that formula, some people say that the transistor "amplifies" the base current.

Now how does that relate to the emitter current? Well we also have the formula \$I_c+I_b+I_e=0\$ When we combine that formula with the second formula, we get \$\beta I_b + I_b + I_e=0\$. From that we can get the emitter current as \$-I_e=\beta I_b + I_b= I_b (\beta + 1)\$ (note that \$ I_e\$ is current going into the emitter, so it's negative).

From that you can see that using the \$ \beta \$ as a handy tool in calculations, we can see the relationship between the base current of the transistor and the emitter current of the transistor. Since in practice the \$ \beta \$ is in the hundreds to thousands range, we can say that the "small" base current is "amplified" into "large" collector current (which in turn makes "large" emitter current). Note that I didn't speak about any deltas until now. That's because the transistor as an element does not require current to change. You can simply connect the base to a constant DC current and the transistor will work fine. If the change in current is required, it's not because of the transistor but because of the rest of the circuit which could be blocking the DC part of the input current.

There is another value also used and it's name is \$ \alpha\$. Here's what it is: \$ \alpha = \frac {I_c} {I_e} \$. When we rearrange that, we can see that \$I_c= \alpha I_e\$. So \$ \alpha\$ is the value by which the emitter current is amplified in order to produce collector current. In this case, the amplification actually gives us a smaller output (although in practice \$ \alpha \$ is close to 1, something like 0.98 or higher), because as we know, the emitter current going out of the transistor is the sum of the base current and collector current which are going into the transistor.

Now I'll talk a bit about how transistor amplifies the voltage and current. The secret is: It doesn't. The voltage or current amplifier does! The amplifier itself is a bit more complex circuit which is exploiting properties of a transistor. It also has input node and output node. The voltage amplification is the ratio of voltage between those nodes \$A_v = \frac {V_{out}}{V_{in}}\$. The current amplification is ratio of currents between those two nodes: \$ A_i=\frac {I_{out}}{I_{in}}\$. We also have power amplification which is the product of current and voltage amplification. Do note that the amplification can change depending on the nodes we chose to be input node and output node!

There are few more interesting values related to transistors which you can find here

So to sum this up: We have transistor which is doing something. In order to safely use transistor, we need to be able to represent what transistor is doing. One of the ways of representing processes happening in the transistor is to use the term "amplification". So using amplification, we can avoid actually understanding what is happening in transistor (if you have any semiconductor physics classes, you'll learn that there) and just have few equations which will be useful for a large number of practical problems.