This video talks through the process of binary addition and how is it can be implemented using logic gates and half/full adders.
Overflow
Binary Overflow
When we add two binary numbers together we might end up with a number larger than the space available in which to store it. This is known as binary overflow and will result in binary overflow and the left most digit(the most significant bit) will usually be discarded.
Example
Image we have two 4-bit binary numbers and we want to add them together and store the result in a 4 bit register.
1111
+1000
= 10111
Notice that the result is 5 bits long and therefore will not fit within a 4-bit register and as a result an overflow will occur. Depending on the system the left hand bit may be discarded.
Domino computer
Domino Adder
Although computers use electricity and semiconductors to perform binary addition, you can make adders using other media too, like water logic gates or even using tens of thousands of dominoes!