Binary Arithmetic: Simple Addition

Binary addition works just like the decimal addition you learned in primary school, but it is much simpler because there are only two digits: $0$ and $1$. There are four fundamental rules to memorize: 1. $0 + 0 = 0$ 2. $0 + 1 = 1$ 3. $1 + 0 = 1$ 4. $1 + 1 = 10$ (This is $0$ with a carry of $1$ to the next column).

Think of $10_2$ in binary as the number $2$ in decimal. Since we cannot write '2' in binary, we move to the next place value (the twos column), just like we move to the tens column when we hit $9 + 1$ in decimal.

When a column adds up to more than $1$, we must carry the extra value to the left. This is exactly like adding $7 + 5$ in decimal; you write $2$ and carry the $1$. In binary, if a column has three $1$s (one from a previous carry and two from the numbers being added), the rule is: $1 + 1 + 1 = 11_2$. You write $1$ in the current column and carry $1$ to the next. This simple mechanism allows computers to process massive numbers by chaining simple logic gates together.

Computers store numbers in fixed-size containers called registers. A common size for beginners to study is 8-bit (one byte). But what happens if the result of your addition is 9 bits long? This is called an overflow error. The computer 'drops' the extra bit because it has no place to store it. This can lead to massive calculation errors, like a game character's health resetting to zero or a spacecraft's navigation system failing. Understanding the limits of your bit-width is crucial for stable programming.