Lab 1 - Number Representations

CSSE2010

Negative Numbers in Binary Operations

Computers do not store positive (+) and negative (-) signs, so they must use binary digits (0,1). There are 4 different formats to try and represent negative numbers:

• Signed magnitude
• Ones’ complement
• Two’s complement (This is most widely used.)
• Excess $2^{m-1}$

Signed Magnitude Representation

In the sign–magnitude representation, also called sign-and-magnitude or signed magnitude, a signed number is represented by the bit pattern corresponding to the sign of the number for the sign bit (often the most significant bit, set to 0 for a positive number and to 1 for a negative number), and the magnitude of the number (or absolute value) for the remaining bits. For example, in an eight-bit byte, only seven bits represent the magnitude, which can range from 0000000 (0) to 1111111 (127). Thus numbers ranging from $-127_{10}$ to $+127_{10}$ can be represented once the sign bit (the eighth bit) is added. For example, $-43_{10}$ encoded in an eight-bit byte is 10101011 while $43_{10}$ is 00101011.

Ones’ Complement

In the ones’ complement representation, a negative number is represented by the bit pattern corresponding to the bitwise, opposite of the positive number.

As an example, the ones’ complement form of 00101011 $(43_{10})$ becomes 11010100 ($-43_{10}$).

Adding two numbers in this systems requires to do a necessary end-around carry. ![[Pasted image 20240229144325.png| An example of an “end-around carry”.]]

Two’s Complement

In the two’s complement