Binary Number Arithmetic

Meta-Description: 

Explore the fascinating world of Binary Number Arithmetic and learn how to perform calculations in base 2 with ease. Discover the fundamental concepts, techniques, and common operations involved in manipulating binary numbers. Enhance your understanding and fluency in this essential aspect of computer science.

Introduction

The digital realm is built upon the foundation of binary numbers, which form the language of computers. Understanding binary number arithmetic is crucial for anyone delving into computer science, programming, or information technology. In this comprehensive guide, we will unravel the intricacies of binary number arithmetic, exploring its fundamental concepts, operations, and applications.

What are Binary Numbers?

Binary numbers are a numerical system that employs only two symbols: 0 and 1. Unlike our familiar decimal system, which uses ten symbols (0-9), binary numbers rely on the concept of powers of two. In this base-2 system, each digit represents a power of 2, starting from the rightmost digit as 2^0, then 2^1, 2^2, and so on.

Why is Binary Number Arithmetic Important?

Binary number arithmetic forms the backbone of digital systems, enabling the representation and manipulation of data in computers. By mastering binary number arithmetic, you gain a deeper understanding of how computers process information, laying the groundwork for more advanced concepts in computer science.

Binary Arithmetic

The four basic arithmetic operations are performed inside a computer using binary numbers. Binary arithmetic include only 0 and 1 therefore it is simple to learn.  So all binary numbers are made up of only 0’s and 1’s and when arithmetic operations are performed on these numbers, the results are also in 0’s and 1’s only.

1) Binary Addition:

Rules:

Truth table of binary addition
A	B	A+B
0	0	0
0	1	1
1	0	1
1	1	10 (0 with carry over 10)

 

Example: (101011)₂ + (10101)=(?)₂

 

                        1          1          1          1          1   (Carry)

                        1          0          1          0          1          1

                        +          1          0          1          0          1

            ------------------------------------------------------------------------

            1          0          0          0          0          0          0

            ------------------------------------------------------------------------

            (101011)2 + (10101)=(1000000)2

 

2) Binary Subtraction

Rules:

 

Truth table of binary subtraction
A	B	A-B
0	0	0
0	1	1 (borrowing 1 from left)
1	0	1
1	1	0

 

 Note: Borrowing is necessary when we have to subtract a larger number from a smaller one.  A borrow will cause a 1 in the minuend to become 0 and a 0 to become 1.  In the second case the preceding 0s in the minuend are changed to 1s until a 1 can be changed to a 0.

Example  (101010)2 – (11001)2 = (?)2

Minuend                    1          0          1          0          1          0

Subtrahend                -           1          1          0          0          1

                        ----------------------------------------------------------------------

Difference                 0          1          0          0          0          1

                        ---------------------------------------------------------------------

3) Binary Multiplication

Rules:

Truth table of binary multiplication
A	B	A X B
0	0	0
0	1	1
1	0	1
1	1	1

 

Example:  Multiply (1110)₂ by (110)₂

Multiplicand                         1          1          1          0

Multiplier                              X         1          1          0

                        -----------------------------------------------------------

                                                            0          0          0          0

                                                1          1          1          0          X

                                    1          1          1          0          X

                        -------------------------------------------------------------------

Product:         1          0          1          0          1          0          0

 

4) Binary Division

Rule:

Truth table of binary division
A	B	A+B
0	0	0
0	1	1
1	0	Not defined
1	1	1

 

Binary division can be performed using the following guidelines.

i) Subtract the divisor from the dividend.

ii) If subtraction is possible, put a 1 in the quotient and subtract the divisor from the corresponding digits of the dividend.

iii) If subtraction is not possible, place a 0 in the quotient

iv) Bring down the next digits from the dividend to add to the remaining digits.

v) The process is going on until the divisor is smaller than dividend.

Example:  Divide (1101)₂ by (111)₂

 

                        111 )  1  1  0  1  ( 0 1

                                - 0  0  0

                               ----------------

                                  1  1  0  1

                                    -1  1  1

                              ------------------

                                    1  1  0

Divisor = 111

Dividend = 1101

Quotient = 01

Remainder = 110