Question

How many cells can be in a computer's main memory if each cell's address can be represented by two hexadecimal digits? What if four hexadecimal digits are used?

Short answer

256 cells for two digits; 65,536 cells for four digits.

Step-by-step solution

Understand the Hexadecimal System

Hexadecimal is a base-16 number system, meaning it uses 16 different symbols: the numbers 0 to 9 and the letters A to F. Each hexadecimal digit represents four binary bits.

Calculate Cells for Two Hexadecimal Digits

With two hexadecimal digits, each digit can have 16 possible values. Thus, the total number of different addresses is given by multiplying the possibilities for each digit: \(16 \times 16 = 256\). Therefore, there can be 256 unique addresses.

Calculate Cells for Four Hexadecimal Digits

With four hexadecimal digits, similarly, each digit can have 16 possible values. The total number of different addresses is: \(16 \times 16 \times 16 \times 16 = 16^4 = 65,536\). Thus, there can be 65,536 unique addresses.

Key concepts

The Hexadecimal System

Hexadecimal is a fascinating and useful numeral system used in computers. It's a base-16 system, meaning it has 16 different symbols to represent values. These symbols range from 0 to 9 for the numbers zero to nine. Beyond nine, the letters A to F are used to stand for the values ten to fifteen. This unique setup allows for compact representations of binary numbers.
Each hexadecimal digit corresponds to a four-bit binary sequence. For example, the hexadecimal digit 'A' translates to the binary sequence '1010'. This characteristic makes it simpler to express binary terms in a shorter form. When dealing with large binary numbers, such as those in memory addresses, hexadecimal can make things more manageable.
In computing, hexadecimal is often used because it can represent large numbers in a small space and is easier for humans to read and understand compared to long strings of 0s and 1s. Every day, programmers use it to simplify code and manage data efficiently.

Binary Representation

In the realm of computing, binary representation forms the foundational language that computers speak. It's a simple system based on only two digits: 0 and 1. This is called base-2. Each digit in this system is a "bit", short for binary digit. Every bit can have one of two states, represented by these two numbers.
Everything stored and processed by a computer is ultimately represented in binary. This includes numbers, text, images, and even instructions. For example, the binary number 1101 can represent an integer, a command, or a part of data, depending on the context.

• Binary is flexible and efficient for computer operations.
• It's the closest representation to how data physically exists in hardware.

In another example, each time we see a hexadecimal digit, we can immediately understand which bit pattern it corresponds to, thus making binary to hexadecimal conversion straightforward. By manipulating binary data, computers perform operations quickly and efficiently, serving the needs of modern applications.

Addressing in Computers

Addressing is a critical concept in computer architecture. It refers to how data is stored and retrieved from memory. Every byte of data stored in a computer's RAM (Random Access Memory) has a specific address. Just like a house address, this unique identifier locates where data lives.
When discussing addressing with hexadecimal numbers, the beauty lies in its ability to handle large amounts of data without cumbersome notation. For example, when two hexadecimal digits are used for addressing, we have 256 unique addresses. This is because each digit in the hexadecimal system can represent 16 values, leading to a total of 16 multiplied by 16.
With four hexadecimal digits, the number of addressable units jumps significantly to 65,536. This multiplication of possibilities allows computers to efficiently manage and occupy larger spaces of memory.
Utilizing hexadecimal in addressing not only saves space but also simplifies the management of the system, enhancing both performance and readability.