Binary Subtraction: A Step-by-Step Guide with Examples

Binary subtraction is a fundamental operation in digital electronics and computer science. Understanding how to subtract binary numbers is crucial for anyone working with digital systems, from engineers to programmers. This article will walk you through the process of binary subtraction using a specific example, including alignment, borrowing, and conversions to decimal notation.

The Example: 101010102 - 1111012

Let's delve into the binary subtraction process of 101010102 (170 in decimal) and 1111012 (61 in decimal).

Step 1: Align the Numbers

First, let's align the numbers for easy subtraction.

10101010
- 111101

Step 2: Subtract from Right to Left

We start from the rightmost column and move left, borrowing when necessary to perform the subtraction.

Column 1 (Least Significant Bit)

0 - 1

Claim a borrow, making the 0 into 2 and the 1 into 0.

2 - 1 1

Column 2

1 - 0 1

Column 3

0 - 1

Claim a borrow, making the 0 into 2 and the 1 into 0.

2 - 1 1

Column 4

1 - 1 0

Column 5

0 - 1

Claim a borrow, making the 0 into 2 and the 1 into 0.

2 - 1 1

Column 6

1 - 1 0

Column 7

0 - 0 0

Column 8 (Most Significant Bit)

1 - 0 1

Step 3: Write the Result

The result of the binary subtraction is:

10101010
- 111101
00100101

In decimal, this is:

170 - 61  109

Comparison with Decimal System

To match our binary result with the decimal system, let's perform the subtraction in a similar manner:

0 1 2
  0 1 2
    0 2 0 2          6 A
10101010-        170-
  111101         61
_________        ____
01101101         109
                 Note: A  10
0–1 claims a borrow and 2–1 is 1

1 with a borrow is 0. 0–0 is 0.
0–1 claims a borrow and 2–1 is 1.
1 with a borrow is 0. 0–1 claims a borrow and 2–1 is 1.
0 with a borrow claims a borrow and 2 with a borrow is 1 and 1-1  0.
1 with a borrow is 0. 0–1 claims a borrow and 2–1 is 1.
0 with a borrow claim a borrow and 2 with a borrowa is 1 and 1–0 is 1.
1 with a orrow is 0 and 0–0 is 0.
The result is 01101101 (109 in decimal).
Converting Binary to Decimal
To convert the binary result (011011012) back to decimal:
0 - 1 claims a borrow, 10 - 1 is 9,
7 with a borrow is 6 minus 6 is 0,
1 minus 0 is 1.
Thus, the result is 109.
Conclusion
Mastering binary subtraction and understanding its relationship with the decimal system is essential for digital electronics and computer science. By following the steps outlined in this article, you can perform binary subtraction with ease and convert the results back and forth between binary and decimal systems.