Signed Number Representation

Why Do We Need Signed Number Representations?

In the decimal system, we use a simple minus sign (–) to denote a negative number. In the binary world of 1s and 0s, we need a different approach. Digital circuits use a specific bit, typically the Most Significant Bit (MSB), as a sign bit.

There are three primary methods for representing signed binary numbers. Let's begin by watching an overview of these methods.

As the video explained, while the representation of positive numbers is the same across all three methods, negative numbers are handled very differently. Let's examine each method in detail.

1. Sign-Magnitude Form

This is the most straightforward method, conceptually similar to how we use '+' and '–' signs in decimal.

• The Most Significant Bit (MSB) is the sign bit.
  • 0 indicates a positive number.
  • 1 indicates a negative number.
• The remaining $n-1$ bits represent the magnitude of the number in true binary form.

To see this in action, continue with the video.

Key Points for Sign-Magnitude:

• Example (8 bits):
  • +34 = 00100010 (Sign bit 0, magnitude 34)
  • -34 = 10100010 (Sign bit 1, magnitude 34)
• Range: For an n-bit number, the range is from $-(2^{n-1}-1)$ to $+(2^{n-1}-1)$.
• Disadvantage: There are two representations for zero: 00000000 (+0) and 10000000 (-0). This redundancy and the complex hardware required for arithmetic make it impractical for modern computers.

2. 1's Complement Form

This method provides a different way to represent negative numbers by using the concept of a complement.

• Positive numbers are represented the same way as in sign-magnitude (MSB=0, followed by the true binary magnitude).
• Negative numbers are represented by taking the 1's complement of the corresponding positive number. To find the 1's complement, you simply invert all the bits (change 1s to 0s and 0s to 1s).

Let's see how this works.

Key Points for 1's Complement:

• Example (8 bits):
  • +34 = 00100010
  • To find -34, we take the 1's complement of +34: 11011101. The MSB of '1' automatically signifies a negative number.
• Range: Same as sign-magnitude: $-(2^{n-1}-1)$ to $+(2^{n-1}-1)$.
• Disadvantage: Like sign-magnitude, it has two representations for zero: 00000000 (+0) and its 1's complement 11111111 (-0). This still complicates arithmetic logic.

3. 2's Complement Form

This is the most important method and the one universally used in modern computing for signed integer representation. It overcomes the issues of the previous two methods.

• Positive numbers are represented in the same way (MSB=0, followed by true binary).
• Negative numbers are represented by taking the 2's complement of the corresponding positive number.

There are two common ways to calculate the 2's complement:

1. Method 1 (Definition): Take the 1's complement and add 1.
2. Method 2 (Shortcut): Starting from the Least Significant Bit (LSB), copy all bits until the first '1' is encountered. Then, invert all the remaining bits to the left. This is a very fast method for manual calculations, perfect for the GATE exam.

The next segment of the video covers this crucial topic.

Key Points for 2's Complement:

• Example (8 bits):
  • +34 = 00100010
  • To find -34:
    • 1's complement: 11011101
    • Add 1: 11011101 + 1 = 11011110
• Range: For an n-bit number, the range is from $-2^{n-1}$ to $+(2^{n-1}-1)$. Notice it can represent one more negative number than positive numbers.
• Advantage: It has only one representation for zero (00000000). Taking the 2's complement of zero yields zero (with carry ignored). This, along with making subtraction as simple as addition, is why it is the standard.

Comparing the Three Systems

Seeing all three representations side-by-side solidifies the differences. The following table compares 4-bit signed numbers.

Converting from Signed Binary to Decimal

GATE questions can also ask you to do the reverse: given a signed binary number, find its decimal value. The method depends on the representation used.

For a summary of these conversion techniques, please review the following document.

Here's a summary of the fastest ways to convert a negative number (MSB=1) back to decimal:

• Sign-Magnitude: Ignore the sign bit, convert the remaining magnitude bits to decimal, and add a minus sign.
• 1's Complement: Find the 1's complement of the number (invert all bits). The result is the positive magnitude. Add a minus sign.
• 2's Complement (Two Methods):
  1. Complement Back: Find the 2's complement of the number. The result is the positive magnitude. Add a minus sign. (Example: 2's complement of -6 (1010) is 0110, which is +6).
  2. Negative Weight Method (Highly Recommended): Assign a negative weight to the MSB and positive weights to all other bits. Sum the weights where there is a '1'. This is extremely fast.
     • Example: For the 8-bit 2's complement number 11101010:
       The weights are $-128,64,32,16,8,4,2,1$.
       Sum = $-128+64+32+8+2=-22$.

Practice Problem

Let's tackle a typical GATE-style question that tests your ability to interpret numbers in all three forms.

Question: The 4-bit binary number is 1011. What is its decimal equivalent if it is represented in:
a) Sign-Magnitude form?
b) 1's Complement form?
c) 2's Complement form?

Try to solve this on your own first using the methods we've just discussed.

Solution Walkthrough

Now, let's watch a video that solves a very similar problem and also demonstrates representing a final result in 2's complement.

Let's quickly verify the answer to our question (1011):

• a) Sign-Magnitude: Sign bit is 1 (-). Magnitude is 011 (3). Result: -3.
• b) 1's Complement: MSB is 1, so it's negative. Take 1's complement: 0100. This is 4. Result: -4.
• c) 2's Complement: MSB is 1, so it's negative. Using the negative weight method for 4 bits (weights -8, 4, 2, 1): $-8+2+1=-5$. Result: -5.

Conclusion

In this lesson, you have learned the three fundamental ways digital systems represent signed numbers. Understanding these representations, especially 2's complement, is non-negotiable for success in the Digital Electronics section of the GATE exam.

Key Takeaways:

• Signed numbers use the MSB as a sign bit (0 for positive, 1 for negative).
• Sign-Magnitude: Simple concept but has two zeros and complex arithmetic.
• 1's Complement: Negative numbers are the bit-wise inverse of positive ones. Also has two zeros.
• 2's Complement: The industry standard. It has a unique zero and an asymmetric range ($-2^{n-1}$ to $+2^{n-1}-1$). Its structure simplifies arithmetic hardware significantly.
• You should be able to convert both ways: from a decimal number to its signed binary form, and from a signed binary number back to its decimal value for all three systems.

Next Lesson Preview:

Now that we know how to represent negative numbers, we are ready to perform binary arithmetic. In the next lesson, we will focus on binary addition and subtraction using 2's complement. We will see firsthand why this representation is so powerful and learn how to detect overflow, a critical error condition in computer arithmetic that is a favorite topic for GATE questions.