Question

What would be the hexadecimal representation of the largest memory address in a memory consisting of \(4 \mathrm{MB}\) if each cell had a one-byte capacity?

Short answer

The largest memory address is 3FFFFF in hexadecimal.

Step-by-step solution

Understand Memory Size in Bytes

The memory size given is 4 megabytes (MB). Since 1 MB equals 1,024 kilobytes (KB) and 1 KB equals 1,024 bytes, you need to convert 4 MB to bytes. Therefore, 4 MB = 4 × 1,024 × 1,024 bytes, which equals 4,194,304 bytes.

Identify Largest Memory Address

The largest address in a system with 4,194,304 memory locations is one less than the total number of locations, since addressing starts at zero. Hence, the largest address is 4,194,304 - 1 = 4,194,303.

Convert the Decimal Address to Hexadecimal

To represent 4,194,303 in hexadecimal, you convert the decimal number 4,194,303. Divide the number by 16, recording remainders, until the quotient is zero. Then read the remainders in reverse order to get the hexadecimal representation. The decimal number 4,194,303 converts to 3FFFFF in hexadecimal.

Key concepts

Hexadecimal Representation

In computing, numbers are often expressed in different bases to reflect their internal use in hardware. The most common of these bases is the hexadecimal system. Hexadecimal is base-16, as opposed to the base-10 decimal system we usually use in everyday life. This means it uses sixteen distinct symbols: 0 through 9 to represent values zero to nine, and A through F for values ten to fifteen. You might be wondering why hexadecimal is so prevalent in computing. It's because it provides a shorthand way to express binary numbers, which are base-2. Each hexadecimal digit represents four binary digits (bits) because 16 is 2 to the fourth power. For example, the binary number 1010 is equivalent to the hexadecimal digit 'A'. This compactness reduces the length of binary representations, making them easier for humans to read and understand. In converting decimal to hexadecimal, you repeatedly divide the number by 16, writing down the remainder at each step. Once your quotient is zero, you read the remainders in reverse to get the hexadecimal number. Using this method, the decimal address of 4,194,303 is converted to the hexadecimal 3FFFFF.

Memory Capacity

Memory capacity is a fundamental aspect of computer systems that refers to the total amount of data that can be stored. It is usually measured in bytes. Larger units such as kilobytes (KB), megabytes (MB), gigabytes (GB), and terabytes (TB) are also commonly used, each representing powers of 1024 due to the binary nature of computer memory. To determine the total memory capacity in bytes, you multiply the number of units by the unit size in bytes. For example, in a 4 MB memory system, this is calculated as 4 × 1,024 × 1,024 bytes = 4,194,304 bytes. Each memory cell stores a byte, thus the system can address 4,194,304 separate memory cells. Understanding memory capacity helps define how much data the memory system can hold and manipulate. This measure is crucial for assessing the performance of a computer, especially in tasks that require significant data processing. Clearly comprehending memory capacity aids in memory optimization in both hardware development and software design.

Memory Architecture

The architecture of a computer's memory outlines how memory is organized and controlled. It encompasses the arrangement of memory cells, the structure of memory addressing, and how memory interacts with other computer components. Most modern computer systems have a hierarchical memory architecture. At the top of this hierarchy are the fastest, but smallest, cache memories located close to the CPU. Below that, you find the main memory (RAM), which is slower but larger. Secondary storage devices, like hard drives and SSDs, are slower and even larger, storing data persistently. Memory addresses are used to access each byte of data stored in memory. In a system with 4,194,304 bytes of memory, addresses range from 0 to 4,194,303. The largest address denotes the final memory location, and in hexadecimal, this address provides a more compact and manageable form for humans and machines alike. In our example, with 4 MB of memory and one-byte cells, the largest address is represented as 3FFFFF, demonstrating how hexadecimal simplifies referencing compared to using long decimal numbers.