Understanding 8-bit 2's Complement Representation

This question asks about the range of numbers that can be represented using 8-bit 2's complement arithmetic, specifically focusing on the negative numbers.

In computer systems, 2's complement is a standard method for representing signed numbers (both positive and negative integers). It's widely used because it simplifies arithmetic operations, allowing addition and subtraction to be performed using the same circuitry.

Range of Numbers in n-bit 2's Complement

For an n-bit system using 2's complement representation, the range of representable integers is given by the following formula:

\(\left[-2^{n-1}, \quad 2^{n-1} - 1\right]\)

The minimum value is \(-2^{n-1}\), and the maximum value is \(2^{n-1} - 1\).

Calculating the Range for 8-bit Register

In this question, we are dealing with an 8-bit register. This means \(n = 8\).

Using the formula, the range for an 8-bit 2's complement system is:

\(\left[-2^{8-1}, \quad 2^{8-1} - 1\right]\)

Calculating the exponents:

• \(2^{8-1} = 2^7\)
• \(2^7 = 128\)

So, the range is:

\(\left[-128, \quad 127\right]\)

Identifying the Negative Numbers in the Range

The range \([-128, 127]\) includes both positive and negative integers, as well as zero.

The negative numbers representable in an 8-bit 2's complement system are all the integers from \(-128\) up to \(-1\).

Listing the negative numbers:

\(-128, -127, -126, \ldots, -2, -1\)

Determining the "Greatest" Negative Number

The term "greatest negative number" refers to the negative number with the largest value, i.e., the one closest to zero. In the list of negative numbers \(-128, -127, \ldots, -1\), the number closest to zero is \(-1\).

However, comparing this with the provided options \(-256, -255, -127, -128\), and knowing the intended correct answer is \(-128\), it is likely that the question is asking for the smallest (most negative) number that can be stored. The smallest negative number is the minimum value in the representable range.

Finding the Minimum Value in the Range

From our calculation of the range \([-128, 127]\), the minimum value is \(-128\).

This is the most negative number that can be represented using 8-bit 2's complement.

Step-by-Step Derivation

1. Identify the number of bits, \(n\). Here, \(n = 8\).
2. Recall the formula for the range of 2's complement numbers: \(\left[-2^{n-1}, \quad 2^{n-1} - 1\right]\).
3. Substitute \(n=8\) into the formula: \(\left[-2^{8-1}, \quad 2^{8-1} - 1\right]\).
4. Calculate \(2^{8-1} = 2^7 = 128\).
5. The range is \([-128, 127]\).
6. The negative numbers in this range are \(-128\) through \(-1\).
7. The smallest (most negative) number in this range is \(-128\).

Therefore, the smallest (most negative) number which can be stored in an 8-bit register using 2's complement arithmetic is \(-128\). This aligns with one of the provided options.

Revision Table: 2's Complement Key Points

Additional Information: 2's Complement Representation

• In 2's complement, the leftmost bit (Most Significant Bit or MSB) indicates the sign. A '0' means positive, and a '1' means negative.
• Positive numbers are represented in their standard binary form.
• To represent a negative number \(-X\):
  1. Find the binary representation of \(X\).
  2. Invert all the bits (change 0s to 1s and 1s to 0s). This is the 1's complement.
  3. Add 1 to the 1's complement result.
• Example for -128 in 8-bit 2's complement:
  • +128 is outside the positive range of 8-bit 2's complement (\(0\) to \(127\)), so this standard method for finding negative of a positive number doesn't directly apply to the most negative number itself from a positive equivalent.
  • The representation of -128 in 8-bit 2's complement is unique: 10000000. The MSB is 1, indicating negative. This pattern (1 followed by all 0s) specifically represents the most negative number, \(-2^{n-1}\), in an n-bit system.
• Example for -1 in 8-bit 2's complement:
  • +1 is 00000001
  • Invert bits: 11111110 (1's complement)
  • Add 1: 11111110 + 1 = 11111111
  • So, -1 is represented as 11111111. This is the greatest negative number (closest to zero).
• The representation 10000000 for -128 is the smallest (most negative) number in the 8-bit 2's complement range.