Signed and Unsigned Binary Numbers

 Understanding Number Representation Techniques

1. Integers can be represented in signed and unsigned ways.
2. Signed numbers use a sign flag to distinguish between positive and negative values.
3. Unsigned numbers store only positive numbers.
4. Techniques include Binary, Octal, Decimal, and Hexadecimal.
5. Binary Number System is a popular technique used in digital systems.
6. Binary System represents binary quantities with two possible states.
7. Binary numbers are indicated by an 0b prefix or a 2 suffix.
8. Unsigned binary numbers lack a sign bit, while signed binary numbers use a sign bit to distinguish between positive and negative numbers.

 1.Sign-Magnitude form

 

Sign-magnitude is one way to represent signed numbers in digital logic. In this form, a fixed number of bits are dedicated to representing the sign and the remaining bits represent the magnitude (absolute value) of the number. Here's a breakdown:

Key points:

• Sign bit: The most significant bit (MSB) is used to represent the sign. 0 indicates positive, and 1 indicates negative.
• Magnitude representation: Remaining bits represent the absolute value of the number, using the same format as unsigned numbers.
• Range: For n bits, the representable range is - (2^(n-1) - 1) to + (2^(n-1) - 1), meaning both positive and negative numbers can be represented within the same format.

Example (8-bit representation):

• +43: 00101011
• -43: 10101011

Limitations:

• Inefficient: Two representations exist for zero (positive 0 and negative 0), wasting space.
• Complex arithmetic: Addition and subtraction require different logic depending on the signs, making them more complex than other methods like 2's complement.
• Overflow detection: Detecting overflow conditions is more challenging compared to other representations.

Comparison with other forms:

• 1's complement: Similar to sign-magnitude but uses an inverted version of the magnitude for negative numbers. Less complex addition/subtraction but suffers from negative zero and overflow issues.
• 2's complement: Adds 1 to the 1's complement representation of negative numbers. Eliminates negative zero, simplifies arithmetic, and offers efficient overflow detection. This is the most common representation in modern digital systems.

Applications:

While not widely used in modern digital logic due to its limitations, sign-magnitude has some historical significance and niche applications:

• Simple educational tool to understand signed number representation.
• Specialized applications where simplicity is valued over efficiency (e.g., low-power systems).

Addition

A number is represented inside a computer with the purpose of performing some calculations using that number. The most basic arithmetic operation in a computer is the addition operation. That’s why a computer can also be called as an adder.

When adding two numbers with the same signs, add the values and keep the common sign. 

Example 1

Add the numbers (+5) and (+3) using a computer. The numbers are assumed to be represented using 4-bit SM notation.

             111  <- carry generated during addition
             0101 <- (+5) First Number
           + 0011 <- (+3) Second Number
             1000 <- (+8) Sum                 

Let’s take another example of two numbers with unlike signs.

Example  2

Add the numbers (-4) and (+2) using a computer. The numbers are assumed to be represented using 4-bit SM notation.

              000 <- carry generated during addition
              1100 <- (-4) First number
           +  0010 <-(+2) Second Number
              1110 <- (-2) Sum

Here, the computer has given the wrong answer of -6 = 1110, instead of giving the correct answer of -2 = 1010.

 

1's Complement

 By inverting each bit of a number, we can obtain the 1's complement of a number. The negative numbers can be represented in the form of 1's complement. In this form, the binary number also has an extra bit for sign representation as a sign-magnitude form.

2's Complement

 By inverting each bit of a number and adding plus 1 to its least significant bit, we can obtain the 2's complement of a number. The negative numbers can also be represented in the form of 2's complement. In this form, the binary number also has an extra bit for sign representation as a sign-magnitude form