JOURNAL OF COMPUTERS (JCP)

ISSN : 1796-203X

Volume : 3 Issue : 2 Date : February 2008

**Impact of Shift Operations on (-1+j)-Base Complex Binary Numbers**

Tariq Jamil

Page(s): 63-71

Full Text: PDF (622 KB)

**Abstract**

Complex numbers play a very important role in various applications of electrical and computer

engineering. These days, arithmetic operations dealing with these numbers rely on a “divide-and-

conquer” technique wherein a complex number is broken into its real and imaginary parts and then,

after representing each part in binary number system, operation is carried out on each part as if part

of the real arithmetic. Thus, addition of two complex numbers requires two separate additions, and

their multiplication requires four individual multiplications, one subtraction and one addition. In

an effort to reduce the number of arithmetic operations within the realm of complex arithmetic, binary

number system with base (−1+j), called complex binary number system, has been proposed in the

literature which allows a complex number to be represented as single-unit instead of two separate

units as in the base-2 binary number system. In this paper, the effects of shift operations on

complex binary numbers have been examined and mathematical equations describing their

behavior have been obtained. Analysis of these equations leads to the conclusion that the impact of

shift operations on a complex binary number is, to a large extent, similar to typical multiply-by-2 (for

per-bit shift-left) and divide-by-2 (for per-bit shift-right) operations of traditional base-2 binary number.

**Index Terms**

complex number, complex binary number, shift-left, shift-right.

ISSN : 1796-203X

Volume : 3 Issue : 2 Date : February 2008

Page(s): 63-71

Full Text: PDF (622 KB)

engineering. These days, arithmetic operations dealing with these numbers rely on a “divide-and-

conquer” technique wherein a complex number is broken into its real and imaginary parts and then,

after representing each part in binary number system, operation is carried out on each part as if part

of the real arithmetic. Thus, addition of two complex numbers requires two separate additions, and

their multiplication requires four individual multiplications, one subtraction and one addition. In

an effort to reduce the number of arithmetic operations within the realm of complex arithmetic, binary

number system with base (−1+j), called complex binary number system, has been proposed in the

literature which allows a complex number to be represented as single-unit instead of two separate

units as in the base-2 binary number system. In this paper, the effects of shift operations on

complex binary numbers have been examined and mathematical equations describing their

behavior have been obtained. Analysis of these equations leads to the conclusion that the impact of

shift operations on a complex binary number is, to a large extent, similar to typical multiply-by-2 (for

per-bit shift-left) and divide-by-2 (for per-bit shift-right) operations of traditional base-2 binary number.