Sistem Digital : Sistem Bilangan Ii ( Representasi Bilangan Biner Negatif, Aturan Two's Complement, Dan Aturan Overflow )

Sistem Digital : Sistem Bilangan II ( Representasi Bilangan Biner Negatif, Hukum Two's Complement, dan Hukum Overflow )

Integer

In the binary number system arbitrary numbers can be represented with :

• The digits zero and one
• The minus sign (for negative numbers)
• The period, or radix point (for numbers with a fractional component)

For purposes of computer storage and processing we do not have the benefit of special symbols for the minus sign and radix point. Only binary digits (0,1) may be used to represent numbers.

Sign-Magnitude Representation

Contoh :

+18 = 00010010

-18 = 10010010

Representasi 0

+0 = 00000000

-0 = 10000000

Two's Complement Representation

Uses the most significant bit as a sign bit. It differs from sign-magnitude representation in the way that the other bits are interpreted.

Complements of Binary Numbers

1. 1’s complements
2. 2’s complements

1’s complement

Change all 1s to 0s and all 0s to 1s

2’s complement

Find 1’s complement and then add 1

Alternative Representations for 4-Bit Integers

Range Extension

Range of numbers that can be expressed is extended by increasing the bit length. In sign-magnitude notation this is accomplished by moving the sign bit to the new leftmost position and fill in with zeros. 

This procedure will not work for twos complement negative integers

• Rule is to move the sign bit to the new leftmost position and fill in with copies of the sign bit
• For positive numbers, fill in with zeros, and for negative numbers, fill in with ones
• This is called sign extension

Negation

Twos complement operation

Take the Boolean complement of each bit of the integer (including the sign bit)

Treating the result as an unsigned binary integer, add 1. The negative of the negative of that number is itself :

Negation Special Case 1

Negation Special Case 2

Addition

OVERFLOW RULE

If two numbers are added, and they are both positive or both negative, then overflow occurs if and only if the result has the opposite sign.

SUBTRACTION RULE

To subtract one number (subtrahend) from another (minuend), take the twos complement (negation) of the subtrahend and add it to the minuend.

Subtraction

Geometric Depiction of Twos Complement Integers

Multiplication Unsigned Binary Integers

Twos Complement Multiplication

Comparison

Sumber

http://informatika.unpar.ac.id/

Slide AOK : Representasi Bilangan

Sumber http://wikiwoh.blogspot.com