1.4 Addition and Subtraction of Binary Numbers

Binary arithmetic forms the bedrock of all operations within digital computers. While the concept might seem unfamiliar at first, adding and subtracting binary numbers follows rules remarkably similar to those of decimal arithmetic, with the key difference being the limited set of digits (0 and 1). This section will explore the fundamental principles and techniques for performing addition and subtraction directly on binary numbers, including understanding carries and borrows, which are crucial for multi-bit operations. Mastery of these basic operations is essential for comprehending how computers perform more complex calculations.

Part 1.4.1 Addition of Binary Numbers

To add binary numbers, we must understand how to add single binary bits.

Basic Rules of Addition

0 + 0 = 1

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 plus a carry-over 1

Carry-over operations in binary addition are analogous to those in decimal arithmetic. Given that 1 is the maximum digit in the binary system, any sum exceeding 1 necessitates a carry-over.

Example No. 1

Add 11110101 to 11101011

Solution:

	9^th Bit	8^th Bit	7^th Bit	6^th Bit	5^th Bit	4^th Bit	3^rd Bit	2^nd Bit	1^st Bit
Carry Over		1	1	1	1	1	1	1
Augend		1	1	1	1	0	1	0	1
Addend		1	1	1	0	1	0	1	1
Sum	1	1	1	1	0	0	0	0	0

Let’s read it from left to right, the answer is: 11110000₂

Example No. 2

Add 11101010101110 to 1110011001011

Solution:

	15^th Bit	14^th Bit	13^th Bit	12^th Bit	11^th Bit	10^th Bit	9^th Bit	8^th Bit	7^th Bit	6^th Bit	5^th Bit	4^th Bit	3^rd Bit	2^nd Bit	1^st Bit
Carry Over	1	1	1	0	0	0	1	0	0	0	1	1	1	0
Augend		1	1	1	0	1	0	1	0	1	0	1	1	1	0
Addend			1	1	1	0	0	1	1	0	0	1	0	1	1
Sum	1	0	1	0	1	1	1	0	1	1	1	1	0	0	1

Let’s read it from left to right, the answer is: 101011101111001₂

Part 1.4.2 Subtraction of Binary Numbers

To subtract binary numbers, we must understand how to subtract single binary bits.

Basic Rules of Subtraction

0 - 0 = 0

0 - 1 = 1 with a borrow of 1

1 - 0 = 1

1 - 1 = 0

Example No. 3

Subtract 1000 from 1011

Solution:

	5^th Bit	4^th Bit	3^rd Bit	2^nd Bit	1^st Bit
Borrow
Minuend		1	0	1	1
Subtrahend		1	0	0	0
Difference		0	0	1	1

Let’s read it from left to right, the answer is: 0011₂

Example No. 4

Subtract 101110 from 11100101

Solution:

	8^th Bit	7^th Bit	6^th Bit	5^th Bit	4^th Bit	3^rd Bit	2^nd Bit	1^st Bit
Borrow		0	0	1	1	0
Minuend	1	1	1	0	0	1	0	1
Subtrahend	0	0	1	0	1	1	1	0
Difference	1	0	1	1	0	1	1	1

Let’s read it from left to right, the answer is: 10110111₂