Adding single bits

Today we'll look at what it's like to add two single-digit binary numbers together. Each of the two single-digit binary numbers can either have the value 0b0 or 0b1. Considering both bits together, this means there are 4 different possible combinations of input for single-bit addition ((0b0+0b0), (0b0+0b1), (0b1+0b0), and (0b1+0b1)). Here are the results for all of those additions:

0b0+0b0 = 0b0 = 0

0b0+0b1 = 0b1 = 1

0b1+0b0 = 0b1 = 1

0b1+0b1 = 0b10 = 2

The result for each of those additions can be represented by a single result bit EXCEPT 0b1+0b1, which results in the 2-bit number, 0b10 (the decimal number 2). This leads to a very important observation. When adding two binary numbers, it is possible that the resulting number will require 1 more bit to store the result than either of the inputs. Possible, but not guaranteed to need it. It's exactly the same as in the decimal number system. Adding two single-digit decimal numbers might result in a single-digit answer, as in the case of 2+2=4, or it might result in a two-digit answer, as in the case of 7+8=15. You can never require 3 result digits when adding two single-digit numbers, as 9+9=18 (the result of 9+9 is the largest possible number you can get from adding two single-digit values).

So we know that the addition of two single-digit binary numbers may or may not produce a two-bit result, but what if it does? What does a real computer do with that? In a real computer binary adder, the second bit of the answer is called the carry-out bit. The first bit is called the result bit. When learning arithmetic in elementary school, we are taught about the carry-out bit, or rather the carry digit. When you are adding two numbers you might get "8+6=four carry the one, for a total of fourteen." It's the exact same thing with single-bit binary addition. Let's look again at previous additions in the context of result bits and carry bits.

0b0+0b0, result bit = 0, carry-out bit = 0

0b0+0b1, result bit = 1, carry-out bit = 0

0b1+0b0, result bit = 1, carry-out bit = 0

0b1+0b1, result bit = 0, carry-out bit = 1

So if there's a carry-out bit, does that mean there's a carry-in bit? There is, actually, and it is instrumental to multi-bit addition, which we'll talk about next time.