Question

At a minimum, how many bits are needed in the MAR with each of the following memory sizes? a. 1 million bytes b. 10 million bytes c. 100 million bytes d. 1 billion bytes

Short answer

1 million: 20 bits, 10 million: 24 bits, 100 million: 27 bits, 1 billion: 30 bits.

Step-by-step solution

Understanding Concept: Memory Address Register

The Memory Address Register (MAR) is used in computer architecture to hold the memory location of data that needs to be accessed. The number of bits in the MAR determines how many distinct memory addresses can be accessed by the system. In binary systems, each addressable memory unit requires its own unique binary address.

Converting Memory Size to Addresses (Part 1)

Each memory byte requires a unique address. Therefore, for memory sizes given in bytes, we need to find the corresponding number of bytes in powers of 2. For example, 1 million bytes is equivalent to approximately \(2^{20}\) bytes, because \(2^{20} = 1,048,576\).

Calculating for 1 Million Bytes

For 1 million bytes:- Convert 1 million to powers of 2. Since \(2^{20}\) is approximately 1 million, the number of bits needed is 20. Thus, the MAR needs 20 bits to address each byte uniquely.

Calculating for 10 Million Bytes

For 10 million bytes:- Convert 10 million to powers of 2. Since \(2^{23.22}\) is approximately 10 million, we round up to the next whole number. Therefore, the MAR needs 24 bits to address 10 million bytes uniquely.

Calculating for 100 Million Bytes

For 100 million bytes:- Convert 100 million to powers of 2. 100 million is approximately \(2^{26.57}\). Rounding up gives us 27, so the MAR needs 27 bits.

Calculating for 1 Billion Bytes

For 1 billion bytes:- Convert 1 billion to powers of 2. 1 billion bytes is about \(2^{29.9}\). Rounding up gives us 30. Thus, the MAR needs 30 bits.

Key concepts

Binary Systems

Binary systems are the backbone of modern computers. Everything in a computer is represented using binary numbers, which are combinations of 0s and 1s. A single binary digit is called a "bit," and it is the smallest unit of data in a computer.
Binary systems are based on powers of 2, unlike the decimal system which is based on powers of 10. This means that every piece of information, whether it be numbers, text, or data addresses, is encoded in binary. For example, decimal numbers like 5, 10, and 256 must be converted to binary as 101, 1010, and 100000000, respectively.

• Binary is efficient for computers because it only requires two states, often represented as electrical signals (on or off).
• Every data and instruction used by the computer is translated into a binary code.
• Understanding binary systems is crucial for comprehending how computers process and store information.

Memory Architecture

Memory architecture in computers refers to how memory is organized and managed in a computing system. It includes components such as cache, RAM, and storage drives. These components work together to ensure the system operates efficiently and at high speed.
At the core of this architecture is how memory is structured in layers, allowing quick access to frequently used data through cache while having larger storage for less frequently needed information.

• Cache provides the fastest access to data, as it stores copies of frequently accessed data from main memory to speed up processing time.
• RAM, which stands for Random Access Memory, is the main volatile memory it is used for running applications and operating systems.
• The structure of memory directly influences how fast a computer can operate, as it determines the latency and speed of data retrieval.

Addressable Memory

Addressable memory allows a computer system to find specific locations of data within its memory structures. Each piece of data is stored at a particular address, and that address is what the central processing unit (CPU) uses to retrieve or write data.
The concept of addressable memory is crucial because it facilitates systematic organization, accessible locations for storing and retrieving data. Memory addresses are typically expressed in binary numbers to match the machine's operation system.

• Memory addresses are vital because they allow the CPU to access data and instructions efficiently.
• Each address in a memory corresponds to a specific data byte, making organization efficient and methodical.
• Without addressable memory, a computer would have no way of knowing where specific data or instructions are stored.

Computer Organization

Computer organization refers to the operational structure of a computer system. It defines how various components of a computer—such as the processor, memory, and storage—interact and coordinate with each other.
A key part of computer organization is how data flows through the system, including how it's processed, stored, and retrieved. It determines the computer's performance, efficiency, and ability to execute different tasks.

• Good computer organization maximizes the use of hardware, enabling the processing of tasks efficiently and swiftly.
• Understanding computer organization helps in optimizing software and hardware configurations.
• It covers aspects such as instruction sets, memory hierarchy, input/output operations, and interconnections among components.