Question

Find the total amount of memory, in the units requested, for each of the following CPUs, given the size of the address buses: (a) 16-bit address bus (in K) (b) 24-bit address bus (in megs) (c) 32 -bit address bus (in megabytes and gigabytes) (d) 48 -bit address bus (in megabytes, gigabytes, and terabytes)

Short answer

The total amount of addressable memory is: (a) 64K for a 16-bit address bus, (b) 16,384 megs for a 24-bit address bus, (c) 4,294,967,296 megabytes or 4,096 gigabytes for a 32-bit address bus, (d) 268,435,456 megabytes, 262,144 gigabytes, or 256 terabytes for a 48-bit address bus.

Step-by-step solution

Understanding CPU Address Bus

The size of the address bus determines the maximum amount of memory a CPU can address. It is calculated using the formula: Total Memory = 2^address bus size. This calculation will give us the amount of addressable memory locations.

Calculating Memory for a 16-bit Address Bus

Apply the formula for a 16-bit address bus: Total Memory = 2^16. Since memory is usually measured in bytes and we want the answer in kilobytes (K), we divide the result by 1024 (1K = 1024 bytes).

Calculating Memory for a 24-bit Address Bus

Apply the formula for a 24-bit address bus: Total Memory = 2^24. To convert the result into megabytes (megs), divide the result by (1024)^2 (1 megabyte = 1024 * 1024 bytes).

Calculating Memory for a 32-bit Address Bus

Apply the formula for a 32-bit address bus: Total Memory = 2^32. To find the result in megabytes, divide by (1024)^2; for gigabytes, divide by (1024)^3.

Calculating Memory for a 48-bit Address Bus

Apply the formula for a 48-bit address bus: Total Memory = 2^48. To convert to megabytes, divide the result by (1024)^2; for gigabytes, divide by (1024)^3; and for terabytes, divide by (1024)^4.

Key concepts

Understanding Address Bus Size

When it comes to a CPU's ability to communicate with its memory, the size of its address bus is essential. Simply put, the address bus is a collection of wires or connections that the CPU uses to indicate the memory location it wants to access. The larger the address bus, the more memory locations it can address.

For example, a 16-bit address bus has 2^16 (65,536) unique addresses. Each address corresponds to a memory location, which is typically one byte in size. Therefore, a 16-bit address bus can address up to 64 kilobytes of memory. It's like having the ability to send a letter to any one of 65,536 different mailboxes.

Understanding Binary Systems

Our computers use a binary number system, which means that for each additional bit, the total number of addressable locations doubles. This exponential growth is what makes the calculation using the power of two a requirement when computing the total memory capacity that an address bus can handle.

Computing Memory Capacity

Computing the CPU's total memory capacity is a matter of translating the address bus size into addressable memory. By using the formula mentioned in the step-by-step solution, Total Memory = 2^address bus size, and adjusting for the units requested (kilobytes, megabytes, gigabytes, etc.), one can compute the theoretical maximum amount of memory accessible by the CPU.

As the exercise shows for different address bus sizes, the capacity increases significantly with each additional bit. It is important to remember that these values represent the total number of addressable memory locations. Because most memory comes in byte-sized chunks, we convert addresses to bytes, and then to higher units for a more comprehensible figure.

Apply Unit Conversion

Being aware that 1 kilobyte equals 1024 bytes, a megabyte equals 1024 kilobytes, and a gigabyte equals 1024 megabytes, is key to converting these massive binary numbers into an understandable format. For example, a 32-bit address bus which can theoretically address 4 gigabytes (GB) of memory is a result of dividing 2^32 addresses by 1024 three times in succession.

Binary Computation in Memory Calculations

Binary computation underpins the way that computers operate, from the logic at the core of the CPU to the way memory capacity is calculated. Each '1' or '0' in a binary system represents an on or off state, corresponding to the presence or absence of an electrical charge. In the context of our memory calculation, each bit added to the address bus doubles the potential number of states, due to the binary nature of computation.

For beginners, grasping the binary computation can be a little abstract, but a helpful analogy is to think of binary digits (bits) as 'yes' or 'no' answers to a series of questions. With only one question, there are two possible answers (2^1). With two questions, there are four possible combinations of answers (2^2), and so on. In this way, a 24-bit address bus could be likened to having 24 questions and a staggering 16,777,216 combinations of answers, each representing a unique memory location.

Binary computation is not only fundamental to understanding how memory capacity is calculated but also crucial in appreciating the potential and limitations of computer system design.