Many people have seen the massive undertakings of minecrafters. A personal favorite of mine was the redstone computer, but I have not yet explored enough of electronics to attempt that undertaking (or have a great enough graphics unit). But here is a binary adder explanation, my example is eight bit (for those who are unaware, a bit is one digit of 1 or 0, a byte is eight bits) yet still massive.

Step 1: The U.I.

The first step is to make a user interface, this is where you give it commands from. My particular example is fairly simple it, I made a row of levers spaced one apart (so they do not cross feeds). These when flipped on indicate 1 and when off indicate 0. This is how you input the addends, and hence I have two rows for two addends. For the output I used redstone lamps that indicate in the same method as the levers below them. I included sixteen bits but you only really need one more the max number of bits you have in addends.

Step 2: The Interworkings

Despite how it may look, this mechanism is fairly simple. The white and orange ends represent two values being added, such as a + b, and the wonder of binary is only four responses are 0, 1, or 10. The actual significance of these going in also doesn't matter, the same as when adding in decimal. When you add 3 + 5 it will equal 8, and so long as they are in the same place that value will always be the same aka 300 + 500 = 800. This means once this mechanism is built it can be copied throughout the entire device. Everything required for its construction is shown above, except that under the piston next the purple there is a block (non-glass, or ice... it must transmit redstone current) and before it a repeater. There is also redstone feeding the repeater between purple and cyan blocks from the orange block, but it is under the cyan line. Make a structure block file of this (so it easily copied).

Note: on the white line, the block powering the piston is supposed to have a repeater before it.

Step 3: The First Row

Place a row I.W. (interworkings) with a spacing of one block and in number to be equal to the number of bits you have inputting (for each addend). Connect all the white ends to one addend (in order) and all the orange ends to the other. This is the first row.

Step 4: Continuing Rows

Next you will make another row, but feeding the lowest value's back to the lowest value on your display (U.I.). Even rows will have one less I.W. than the row previous, while odds (not one, because there is no previous & also as it is technically classified as a unit not odd) will have the same. Connect the feed cyan feeds of the values to the value above (on odd rows it will skip one level and jump to the last one of the next odd rows orange slot. Then connect the purple feeds to the orange lines. Continue making rows until you have reached only one I.W. on an odd row. The adder is now finished.

Step 5: Finsished, Fertig, Perfectus

It is now complete, should all your linkages be made. The first picture here is an example of 1 + 1 = 10 (1+1=2). The second is 1100 + 1000 = 10100 (12+8=20). The last is the max value this calculator can add 11111111 + 11111111 = 111111110 (255+255=510).

ps. the display is reversed from right to left as is regularly used in math to left to right, sorry if it throws you off when reading the examples.