1.5. Arithmetic Operations on Number Systems

Binary Arithmetic

Addition: The process of adding two binary numbers follows these rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (carry 1)

Example: Add 1011 and 1101.

   1011
 + 1101
 ------
  11000

In binary addition, we add two binary numbers using simple rules similar to decimal addition. For each column starting from the right, we add the corresponding bits together. For example, if both bits are 1, we write down 0 and carry over 1 to the next column (just like 9 + 9 = 18 in decimal). Thus, when adding 1011 (11 in decimal) and 1101 (13 in decimal), you work from the rightmost bit to the left.

1. Starting with the rightmost bits: 1 + 1 = 10, write 0, carry 1.
2. Next, second column: 1 + 0 (plus the carry of 1) = 10, again write 0, carry 1.
3. Moving to the third column: 0 + 1 (plus carry of 1) = 10, write 0, carry 1.
4. Last column: 1 + 1 (plus carry of 1) = 11, write 1 and the carry of 1 goes beyond the leftmost column, resulting in 1 at the start. The final result is 11000.

Think of binary addition like counting with your fingers. If you hold up 1 finger on each hand, that’s like saying 1 + 1 = 2, but in binary, it's more like '1 finger from each hand means I have 2 fingers in total, which I represent as 10 in binary, and I need to remember to carry over the extra finger to the next count.' So, each time you get a total beyond 1, you just 'carry the one'.

Subtraction: Similar to binary addition, binary subtraction involves borrowing when needed:

• 0 - 1 = 1 (borrow from the next higher bit)

Binary subtraction works similarly to decimal subtraction, except we need to borrow when subtracting a larger digit from a smaller digit. For example, if you need to subtract 0 (no fingers up) from 1 (one finger up), you would typically say you can't do that without borrowing. You'd look for another binary digit pair that can provide you a finger to borrow from. If we have 0 - 1, we go back to the next higher order (just like if you had 0-1 in decimal, you would borrow from the next position). So, the digit we borrow from has to be decreased by one. This can turn a situation of needing to subtract into a situation where addition will occur afterward.

Consider a scenario where you have a pocket with 0 dollars, but you want to buy something that costs 1 dollar. To make the purchase, you would need to borrow a dollar from a friend. If your friend says they can lend you 1 dollar but expects you to give them back 1 dollar from your ‘next earnings’. This scenario illustrates how borrowing works in subtraction: you can 'create' the money you need temporarily until you are able to repay, just like you would find a higher bit to borrow from in binary.

Multiplication and Division: Binary multiplication and division work similarly to decimal, but follow the rules of binary digits.

In binary multiplication, you multiply each digit of the first binary number by each digit of the second binary number, just like in decimal multiplication. For instance, multiplying 1 by 1 gives 1, while 0 multiplied by any number always gives 0. When doing binary division, you divide one binary number by another and keep track of the binary digits as quotients and remainders, much like how you would do it with decimal numbers. It involves assessing if one number fits into another and systematically determining how many times that number fits.

If you picture multiplication like an assembly line: you take a single item from the first batch and see how many times you can apply it to create parts in the second batch. Similarly, with division, imagine splitting a cake into equal pieces; you need to know how many times you can divide it into portions until nothing is left. Each binary operation follows similar patterns to this assembly process, just with only two options (0 or 1).

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