Question

What are the advantages and disadvantages of using a very large memory cell size, say, \(W\) \(=64\) instead of the standard size \(W=8\) ? If each integer occupies one 64-bit memory cell and is stored using sign/magnitude notation, what are the largest (in terms of absolute value) positive and negative integers that can be stored? What if two cells are used to store integers?

Short answer

A larger memory cell allows storing larger numbers with fewer accesses but uses more memory. Largest positive is 9,223,372,036,854,775,807 for 64-bit; two cells allow far larger numbers.

Step-by-step solution

Understanding Memory Cell Size

The standard memory cell size is typically 8 bits, denoted by W=8. When using a larger memory cell size of W=64, it means each memory cell can hold 64 bits of data. An advantage of using a larger memory cell size is the ability to store larger integers or more information in a single memory cell, reducing the overall number of memory accesses needed for data retrieval. The disadvantage is that it can lead to increased memory usage, as not all applications require such large storage per cell, potentially wasting space if only smaller numbers are frequently stored.

Calculating Largest Positive Integer (W=64)

Using the sign/magnitude notation for a 64-bit memory cell, the first bit is used to indicate the sign (0 for positive, 1 for negative). Therefore, the largest positive integer is determined by all remaining 63 bits being 1. This can be expressed in binary as 0111...111 (63 ones). In decimal, this is equivalent to \[2^{63} - 1\]which equals 9,223,372,036,854,775,807.

Calculating Largest Negative Integer (W=64)

Similarly, using sign/magnitude notation, the largest negative integer would be represented by a 1 in the first bit followed by all zeros. Thus, it is \[-0\ldots0\] This effectively means the integer is -0 which isn't applicable as an integer value. However, the integer with the second most negative magnitude is \[-2^{63} + 1 \]which equals -9,223,372,036,854,775,807.

Utilizing Two Memory Cells

When using two 64-bit cells, we effectively have 128 bits to store an integer in sign/magnitude notation. The first bit still indicates the sign, while the remaining 127 bits contribute to the magnitude. The largest positive integer is obtained by filling 127 bits with 1s, yielding:\[2^{127} - 1\]The largest negative integer is:\[-2^{127} + 1 \]This setup allows encoding even larger numbers, expanding the range of values significantly.

Key concepts

Sign/Magnitude Notation

In computing, sign/magnitude notation is a method used to represent integers, involving a special bit reserved for indicating the integer's sign. The first bit in this scheme, known as the sign bit, designates whether the number is positive or negative.

• A '0' in the sign bit means the number is positive.
• A '1' in the sign bit means the number is negative.

This becomes particularly useful when differentiating between positive and negative numbers stored in memory.
The remaining bits express the magnitude or absolute value of the number. For instance, utilizing 64 bits in total, where the first bit denotes the sign, allows the remaining 63 bits to specify the number's size. Sign/magnitude notation is relatively simple but sometimes inefficient in handling zero, since it permits both +0 and -0 representations, which can be unnecessary.

Integer Representation

Integer representation in digital computers is crucial for performing accurate calculations and data storage. The sign/magnitude method is one of several approaches, important for its straightforward interpretation of positive and negative numbers.
In a 64-bit system, the sign/magnitude representation allows a wide range of integers to be expressed: the number of bits directly relates to the size of the integers that can be stored.

• The largest positive integer is given by having a sign bit of 0 followed by 63 ones. This translates to: \[2^{63} - 1 = 9,223,372,036,854,775,807 \]
• The largest negative integer, though, hits a limitation: because of symmetric distribution around zero in sign/magnitude, representing negative numbers perfectly isn't now possible, which is why there's a -0.

To utilize a broader range, two cells (128 bits) can be implemented, enabling extensive integer values but naturally requiring more memory.

Memory Access Efficiency

When talking about memory cell size, it's important to consider the effects on memory access efficiency. Increasing the memory cell size (for example, from 8 bits to 64 bits), allows more data to be fetched in a single memory access.
This increases efficiency because each access delivers substantial data, potentially reducing the frequency and number of accesses needed. This is advantageous in several scenarios:

• Data throughput: Larger chunks allow faster processing of large data sets.
• Reduced latency: Fewer accesses mean less time waiting for data retrieval.

However, the downside includes increased memory bandwidth usage and potential inefficiency when dealing with smaller values that do not fully utilize the entire cell's capacity. Cost and complexity could also rise, as larger cells need more sophisticated management and technology.

Data Storage

Data storage plays an integral role in computer architecture, determining how effectively data is kept and accessed. The size of a memory cell, such as a standard 8-bit vs. a 64-bit, dictates not just the volume, but also the efficiency and granularity of data handling.
When a large memory cell size is selected:

• More substantial integers or image data blocks can be stored directly in one cell.
• It supports applications like high-performance computing and large-scale simulations needing extensive data.

However, this approach might lead to memory wastage in routine applications that do not need such high storage features per cell. As both hardware and software need to accommodate this data size, it can also result in increased overall cost and complexity. Optimizing for particular application needs and, striking a balance between size, speed, and cost is crucial in practical data storage scenarios.