COMP-1200: Integer Representation Notes

At the fundamental level, a computer represents data using electronic signals, which can be understood as a system of two distinct states. These states are characterized by the presence of voltage (represented as 1) and the absence of voltage (represented as 0). This binary foundation underlies all data representation in computing.

How Data is Represented

Data in computers is primarily represented using the Binary System, where the basic unit is known as a bit. Each bit can hold a value of either 0 or 1, and it is the cornerstone of digital computing. The use of only these two states allows for stability and efficiency in processing and storage, as transitions between states are not analog but discrete, enhancing reliability.

The number of bits used directly affects the number of possible states that can be represented. For instance, with n bits, the total number of possible states can be calculated using the formula $2^n$. As such, increasing the number of bits exponentially increases the state capacity, making it fundamental for not just data storage but also for the complexity of operations executed by computers.

Types of Data Represented

Various types of data can be represented using this binary system, notably:

• Numbers: This includes integers and rational numbers, which can be represented in various forms such as whole numbers and fractions.

• Text: Characters and strings can be encoded in binary using systems like ASCII or UTF-8, where each character is assigned a unique binary value.

• Images: Images are represented through pixel information, where each pixel can denote color values, typically using a color model (such as RGB). This allows for the representation of complex visual information.

• Logical Data: This includes true or false values, essential for decision-making processes in computing.

• Instructions: Binary code can also represent commands for the computer, which dictate operations within applications and the operating system.

Unsigned Integers

The representation of numbers in computing often starts with unsigned integers. The decimal system (base-10) comprises digits ranging from 0 to 9 and employs a weighted positional notation based on powers of ten. For example, in the number 329:
329 = 3 \times 10^2 + 2 \times 10^1 + 9 \times 10^0

Conversely, the binary system (base-2) comprises only the digits 0 and 1, maintaining a similar weighted positional notation, though based on powers of two. For instance, the binary number 101 can be evaluated as:
101 (binary) = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 5

Binary Notation Standards

In practice, binary can be represented in a standard format known as the 0b format (e.g., 0b1001). To enhance readability in longer binary sequences, underscores can be utilized (e.g., 0b11_0010), making it easier for humans to interpret complex binary numbers.

Conversion Processes

Binary to Decimal Conversion: To convert binary numbers to decimal, begin from the least significant bit (LSB) to the most significant bit (MSB), summing the values of the powers of 2 at positions where there are 1s. For example, in $0b110111$: 0b110111 = 1 \times 2^0 + 1 \times 2^1 + 1 \times 2^2 + 0 \times 2^3 + 1 \times 2^4 + 1 \times 2^5 = 55

Decimal to Binary Conversion: This process requires dividing the decimal number by 2 and recording the remainders. For instance, to convert the decimal number 13, the remainder sequence will yield:
13 \rightarrow 0b1101 (obtained from the remainders).

Unsigned Binary Integer Ranges

For a given N bits, the range of unsigned integers is established from 0 to $2^N - 1$. As an instance, for a 3-bit unsigned integer, the possible values range from 0 to 7, reflecting the basic principle of binary representation in practical application.

Overflow in Unsigned Integers

Overflow occurs in unsigned integers when calculations exceed the maximum representable value. For example, when adding 1 to the binary representation of 111 (which stands for 7 in decimal), the operation leads to an overflow, resulting in 000 (the binary representation of 0).

Detection of Unsigned Overflow

Detection of overflow in unsigned integers is performed by monitoring if the most significant bit (MSB) carries over during addition. If a carry is detected from the MSB, it indicates an overflow has occurred, necessitating attention in calculations.

Signed Integers

To accommodate negative numbers, several methods are employed:

• Signed Magnitude: The first bit indicates the sign of the number; this allows representation within the range of $-2^{N-1}+1$ to $2^{N-1}-1$.

• One’s Complement: This method involves inverting bits to represent negative numbers, leading to two representations of zero (+0 and -0), which can complicate arithmetic operations.

• Two’s Complement: By inverting bits and then adding 1, this method eliminates the representation of -0 and streamlines arithmetic processes.

Signed Integer Operations

Operations for signed integers, including addition, mimic those used for unsigned integers; however, careful attention must be paid to potential overflow scenarios. Overflow detection is analogous to unsigned integers, again emphasizing the importance of monitoring the MSB carries.

Practical Applications

Understanding conversion patterns between positive and negative representations and arithmetic calculations is vital for performance across various applications, ranging from system designs in hardware to software algorithms.

Summary

Different number systems—binary, decimal, and hexadecimal— inform various operational processes such as addition, zero-extension, and overflow detection. Mastery of methodologies for handling integers is essential for effective programming and computer science practices, impacting both performance and accuracy in digital systems.