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If you’ve ever wondered how electronic devices like computers can count, this article gives a simple introduction to binary and logic and shows how they are tied together with electronics to make both simple and complex computers.

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== Fundamentals of binary counting ==
To begin with let’s have a look at some fundamentals.  Since digital computers can only represent two states, on and off or zero and one, there are only two numbers available; therefore they have to count in base 2, not base 10 as we would do.  However, it’s very similar, instead of ones, tens, hundreds and thousands, base 2 counts in ones, twos, fours, eights and so on.  So, for example 2 in base 10 is one- zero in base 2.

[[Image:Slide1.PNG|400px|Slide 1 - Counting in Binary]]

When we add numbers in base ten we carry over any digits which are greater than 9 into the next magnitude of units, so nine plus one equals zero carry one, or ten.

[[Image:Slide2.PNG|400px|Slide 2 - Addition in Base 10]]

Binary works exactly the same, however you carry over if the result is greater than 1, so one plus one equals one-zero.

[[Image:Slide3.PNG|400px|Slide 3 - Addition in Base 2]]

This means that if you have two single-figure binary numbers and you add them together there are only 4 possible results.

[[Image:Slide4.PNG|400px|Slide 4 - 4 Possible results]]

(Note: There is an error in this slide: It should read 1 + 1 = 0 carry 1, I will correct the graphic shortly.)

It is useful to represent this in what’s known as a truth table. Here you can see the 4 possible values of the inputs A and B, and the four possible outputs represented by Sum and Carry.

[[Image:Slide5.PNG|400px|Slide 5 - 4 Possible results]]

In order to represent the ‘logic’ required to get from the possible range of inputs to the desired outputs we use Boolean operations, or as they are more commonly called in electronics, logic gates.

== Logic Gates ==
Here are the three basic types of logic gates which I’ve chosen because they are the simplest gates to make from transistors.  You can make all other types of gates by combining these three.

[[Image:Slide6.PNG|400px|Slide 6 - Basic Boolean operations]]

An AND gate outputs one only when both its inputs are one.  An OR gate outputs one when either input is one.  Finally a NOT gate (or inverter as it is sometimes called) outputs the opposite of its input, so if the input is one the output is zero and vice-versa.

So, going back to our truth table, let’s look at the logic required to get ‘Sum’ based on the inputs A and B.  Here we want the logic to output one only when one input is one and not the other, this is known as an exclusive OR gate.  We can do this by simply using two AND gates with NOT gates on opposing inputs.  If either gate outputs a 1 the result is 1 via the final OR gate.

[[Image:Slide7.PNG|400px|Slide 7 - 4 possible results]]

The carry output is even simpler, we want the carry to be 1 if both A and B is one, so we use an AND gate.

[[Image:Slide8.PNG|400px|Slide 8 - 4 possible results]]

To get the whole truth table we simply add the two logic circuits together.  This logic is called a ‘half-adder’ due to the fact that it is only capable of working on single bit numbers, since you cannot input the carry bit, you can’t cascade them together to work on larger binary numbers.

[[Image:Slide9.PNG|400px|Slide 9 - The half-adder]]

To solve this we combine two half-adders together to make a full-adder.  This logic takes A, B and a carry as input and outputs the sum and carry.  If you followed along with the half adder it’s pretty easy to see how this works from the logic diagram.

[[Image:Slide10.PNG|400px|Slide 10 - The full-adder]]

Now the simple full-adder logic circuits can be combined to allow bigger binary numbers to be added together.  This picture shows a four bit adder, in fact, due to the way the carry bit ‘ripples’ down, this is known as a ripple carry adder.  Since both the A and B inputs are now 4 bits we can add together 1111 and 1111 or 15 plus 15 in base 10 to get a five bit result.

[[Image:Slide11.PNG|400px|Slide 11 - A 4-bit adder]]

== Building logic gates from transistors ==
Now let’s take a quick look at how we build logic gates using transistors.  First up is the NOT gate.  Here if the input is 1 it causes the electricity to flow from the collector to the emitter (top to bottom).  Since the electricity will always follow the path of least resistance the output will be zero.  If the input is zero, the transistor prevents the flow from collector to emitter, so the electricity flows out of the output causing it to be one. 

[[Image:Slide12.PNG|400px|Slide 12 - Transistor NOT gate]]

Next up is the AND gate.  This requires two transistors, the inputs are on the bases and only if both inputs are one can the electricity flow to the output, making it a one also.

[[Image:Slide13.PNG|400px|Slide 13 - Transistor AND gate]]

The OR gate is similar but has two possible paths for the electricity, so if one base or the other is one, electricity flows to the output.

[[Image:Slide14.PNG|400px|Slide 14 - Transistor OR gate]]

== Building a full-adder ==
Once you have these basic building blocks you can combine them together using the logic diagram for a full adder, which gives us this circuit diagram:

[[Image:Logic design PNG.png|400px|Full-adder circuit schematic]]

Here you can see a picture of a completed full-adder, the A, B and carry inputs are on the left and the sum and carry outputs are on the right.  By making more of these it is possible to build adders capable of dealing with bigger and bigger numbers.

[[Image:Full Adder circuit wiki.jpg|400px|Picture of a full-adder circuit]]

== Building a computer ==
So with all the theory out of the way, let’s look at a real 4-bit computer built from discrete transistor gates.  This circuit has 4 switches for each input (A and B) and a simple five LED output showing the result.  Note that both the inputs and the output are little-endian meaning the smallest binary value is on the right, just like in base 10.  You can clearly see the 4 full-adder circuits which perform the processing.

[[Image:4-bit computer wiki.jpg|400px|A 4-bit computer made from transistors]]

The computer is made by combining 4 full adder circuits (as shown above) and some extra circuitry which drives the inputs to the adders and displays the output:

[[Image:Logic design driving circuit PNG.png|400px|4-bit computer input and output circuit schematics]]

== Conclusion ==
And there you have it, how a simple transistor can be made to count.  Whilst this ‘computer’ is a very basic one, you can easily see why modern processors contain hundreds of millions of transistors which enable modern computers to perform so much logic at such amazing speeds.

== Build your own computer ==
If you're crazy enough to want to build you own computer from scratch (well, I was!) you can download the schematics and the PCB artwork used in this article from [[Media:4bit Computer schematics.zip|here]].
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