Binary Arithmetic: Addition and Subtraction Explained

Addition and Subtraction in Base Two: A Guide to Binary Arithmetic

Binary arithmetic forms the foundation of how computers represent data and perform calculations. In this guide, we'll explore the principles behind adding and subtracting numbers using base-two notation, also known as binary. We will focus on four primary aspects — binary addition, binary subtraction, carry in binary addition, and borrow in binary subtraction.

Binary Representation

Before diving into specific operations, let's review the basics. Each digit in a binary number is called a bit, which can have one of two values: zero (0) or one (1). For instance, 101 represents three bits with respective values of 1, 0, and 1. To work with larger integers, these individual bits are grouped together by their positions from right to left, each position representing twice the value of its predecessor. So 101_2 equals seven when converted back to decimal format.

Binary Addition

In base ten, we use carrying over from place to place. Similarly, during binary addition, if the sum of any pair of bits exceeds the highest possible bit value (one), we carry over to the next column, just like in our familiar decimal system. Using the example below, we see that binary addition follows the same logic used in conventional arithmetic:

        +
    1  0  1    <-- Bit 1
+ 1  1  0      <-- Bit 2
---------
| 0 | 1 | 1  <-- Sum
-------
Carry = 1

The result is 1011, which corresponds to decimal twelve when translated back to base ten.

Note that unlike decimal systems, a carryover does not affect the original digits being added. It only impacts the higher-order bit(s) further to the left.

Carry in Binary Addition

To summarize, whenever the sum of two bits plus any potential carries exceeds one, it results in a carryout to the next position. This concept is crucial in understanding the mechanics underlying binary addition.

Binary Subtraction

Performing subtractions in binary involves more complex concepts compared to addition due to the lack of negative numbers in digital circuits. Instead, we borrow from adjacent bits following a similar approach to decimal subtraction. To illustrate, based upon the rules governing borrowing in base two:

         -
    0  1  1    <-- Minuend
- 1  0  0      <-- Subtrahend
-------------------
Borrow = 1
----------
| 1 | 1 | 1  <-- Difference
----
      1
------
Result = 111 - 100 = 111_2 = 7

If the minuend's corresponding bit is less than or equal to the subtrahend's bit while considering a borrow, set it to zero; otherwise, set it to one and borrow once again to the previous bit. If there isn't a precedent bit, leave it unchanged.

Borrow in Binary Subtraction

A borrow in binary subtraction signifies taking away one unit from the neighboring bit to make up for the excess resulting after performing the operation between the current bit(s) and the subtrahend. Again, it works similarly to decimal subtraction but without explicitly reflecting negativity within the binary system itself.