** Understanding Number Representation Techniques**

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

- 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 Number1110 <- (-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

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