Understanding Hexadecimal Representation in Memory Addressing

As we progress in the realm of computer architecture and system design, it becomes crucial to comprehend the underlying principles that govern how memory addresses are represented. One common scenario involves a PC system utilizing a 20-bit address code. In this article, we explore the number of hexadecimal (hex) digits required to represent such memory addresses, breaking down the process step-by-step for clarity and ease of understanding.

Step-by-Step Guide to Hexadecimal Representation

To determine the number of hex digits needed to represent a memory address using a 20-bit address code, we follow a systematic approach:

Understanding the Relationship Between Bits and Hex Digits

Hexadecimal digits play a vital role in representing binary data concisely. Each hex digit can represent four bits, as shown below:

2^4 16

This equation highlights that 16 possible combinations (0-9 and A-F) can be represented by a single hex digit. This relationship is fundamental to the encoding process in memory addressing.

Calculating the Number of Hex Digits

Given a 20-bit address, the calculation to find the number of hex digits required can be outlined as follows: Divide the total number of bits (20) by the number of bits each hex digit represents (4):

Number of hex digits frac{20}{4} 5

Thus, 5 hex digits are needed to represent a 20-bit memory address. This calculation ensures that the memory address is accurately represented, with no information loss.

Hexadecimal Representation of 20-Bit Address

To illustrate this concept further, we can break down the 20-bit address into its hexadecimal format. The range for a 20-bit address can be represented as:

20 bits: 0b00000000000000000000 - 11111111111111111111

This converts to the hexadecimal representation:

Hex: 00000 - FFFFF

The range from 00000 to FFFFF represents 1,048,576 (2^20) possible addresses, each requiring 5 hex digits for representation.

Legacy Architectures and Segmented Addressing

For those interested in the evolution of computer architecture, the 20-bit address code finds its roots in older systems, such as the PC/XT/AT series. In these systems, the memory addressing scheme implemented a 'segmented addressing scheme,' a fundamental approach to managing larger address spaces on smaller processors.

Segmented Addressing in Old x86 Systems

In older x86 systems, the main registers were 16 bits, addressing a 64kb (2^16) segment. This was complemented by 'Segment Registers,' which when combined with the segment addresses, allowed the PC to address a total of 1 megabyte (2^20).

The effect of the segment register was to shift the address left by 4 bits (one hex digit), thus adding 1 to the segment register would add 16 to the final address.

Here's an example of how this works in a practical scenario: Consider an address with DS:DX 1000:0800 DX 0800 DS 1000 The final address would then be

10800

This illustrates that the segment register (DS) value 1000 actually represents 10000, as the extra digit is implied in the system architecture. This segmented addressing model was particularly useful in the era of the 20-bit address bus, allowing systems to manage large memory spans efficiently. However, with the introduction of the 386 series, the flat non-segmented architecture became more prevalent, although the segmented architecture remains supported for backward compatibility.

Conclusion

In summary, when a PC system uses a 20-bit address code, the memory address can be represented using 5 hex digits. This concept is rooted in the fundamental relationship between bits and hex digits, and its practical applications extend to legacy systems with segmented addressing, demonstrating the ongoing relevance of understanding memory addresses in computer architecture.