2.4 Data representation: integers, floating-point numbers, and characters
4 min read•august 13, 2024
Data representation is the backbone of computer systems. It's how computers store and manipulate information. In this section, we'll explore how integers, floating-point numbers, and characters are represented in .
We'll dive into binary representation of integers, floating-point standard, and methods like and Unicode. Understanding these concepts is crucial for grasping how computers process and store data.
Binary Representation of Integers
Unsigned Integers
Top images from around the web for Unsigned Integers
Converting from Decimal to Binary View original
Is this image relevant?
ICND1 | Binary Basics | Decimal to Binary Explained | Examples View original
Is this image relevant?
Binary Basics | Decimal to Binary Explained | Examples | ICND1 100-105 View original
Is this image relevant?
Converting from Decimal to Binary View original
Is this image relevant?
ICND1 | Binary Basics | Decimal to Binary Explained | Examples View original
Is this image relevant?
1 of 3
Top images from around the web for Unsigned Integers
Converting from Decimal to Binary View original
Is this image relevant?
ICND1 | Binary Basics | Decimal to Binary Explained | Examples View original
Is this image relevant?
Binary Basics | Decimal to Binary Explained | Examples | ICND1 100-105 View original
Is this image relevant?
Converting from Decimal to Binary View original
Is this image relevant?
ICND1 | Binary Basics | Decimal to Binary Explained | Examples View original
Is this image relevant?
1 of 3
- integers are represented in binary format using a fixed number of bits
- Each represents a power of 2
- The value is calculated by summing the products of each bit and its corresponding power of 2
- Example: The unsigned 8-bit binary number
represents the value 170 (128 + 32 + 8 + 2)10101010
Signed Integers and Two's Complement
- integers can be represented using various methods, including sign-magnitude, one's complement, and two's complement
- Two's complement is the most commonly used method in modern computer systems
- In two's complement representation, the leftmost bit (most significant bit) is used as the sign bit
- 0 indicates a positive number
- 1 indicates a negative number
- The remaining bits represent the magnitude of the number
- Converting between positive integers and their two's complement representation:
- For positive integers, simply express the integer in binary format
- For negative integers:
- Express the absolute value of the integer in binary
- Invert all the bits
- Add 1 to the result
- The range of values that can be represented using a fixed number of bits depends on the representation method and the number of bits used
- Example: An 8-bit unsigned integer can represent values from 0 to 255
- Example: An 8-bit two's complement signed integer can represent values from -128 to 127
IEEE 754 Floating-Point Standard
Components of IEEE 754 Floating-Point Numbers
- The IEEE 754 standard defines a format for representing floating-point numbers in binary, consisting of three parts:
- Sign bit: Indicates whether the number is positive (0) or negative (1)
- Exponent: An unsigned integer that represents the power of 2 by which the mantissa is multiplied
- The exponent is biased by adding a fixed value to the actual exponent, allowing for the representation of both positive and negative exponents
- Mantissa (or significand): A fractional binary number that represents the significant digits of the floating-point number
- The mantissa is normalized, meaning that the most significant bit is always assumed to be 1 and is not stored explicitly (implied bit)
Single-Precision and Double-Precision Formats
- The IEEE 754 standard defines two main formats for floating-point numbers:
- Single-precision (32 bits):
- Sign bit: 1 bit
- Exponent: 8 bits
- Mantissa: 23 bits
- Double-precision (64 bits):
- Sign bit: 1 bit
- Exponent: 11 bits
- Mantissa: 52 bits
- Single-precision (32 bits):
- The standard also defines special values:
- Positive and negative infinity
- NaN (Not a Number) for representing the results of undefined or invalid operations
Decimal vs Binary Floating-Point Conversion
Decimal to Binary Conversion
To convert a decimal floating-point number to its binary representation using the IEEE 754 standard:
- Determine the sign of the number and set the sign bit accordingly
- Convert the absolute value of the number to binary, separating the integer and fractional parts
- Normalize the binary number by shifting the radix point until there is only one non-zero digit to the left of the radix point
- Adjust the exponent accordingly
- Encode the exponent by adding the bias to the actual exponent
- For single-precision, the bias is 127
- For double-precision, the bias is 1023
- Combine the sign bit, encoded exponent, and mantissa (without the implied leading 1) to form the final binary representation
Binary to Decimal Conversion
To convert a binary floating-point number to its decimal representation:
- Extract the sign bit, exponent, and mantissa from the binary representation
- Subtract the bias from the encoded exponent to obtain the actual exponent
- Reconstruct the normalized mantissa by prepending the implied leading 1
- Calculate the decimal value by multiplying the mantissa by 2 raised to the power of the actual exponent
- Apply the sign to the result based on the sign bit
ASCII vs Unicode Character Encoding
ASCII Character Encoding
- ASCII (American Standard Code for Information Interchange) is a 7-bit character encoding standard
- Represents 128 characters, including uppercase and lowercase English letters, digits, punctuation marks, and control characters
- ASCII assigns each character a unique binary code ranging from 0 to 127
- Example: The ASCII code for the character 'A' is 65 (binary:
)1000001 - Example: The ASCII code for the character 'a' is 97 (binary:
)1100001
- Example: The ASCII code for the character 'A' is 65 (binary:
Unicode Character Encoding
- Unicode is a more comprehensive character encoding standard that aims to represent characters from various scripts and languages worldwide
- Provides a unique for each character, allowing for the representation of a much larger set of characters compared to ASCII
- The most common Unicode encoding is (Unicode Transformation Format - 8 bits)
- Uses a variable number of bytes to represent characters
- Backward-compatible with ASCII, as the first 128 Unicode code points are identical to the ASCII character set
- In UTF-8:
- Characters with code points from 0 to 127 are encoded using a single
- Characters with higher code points are encoded using multiple bytes, depending on the code point range
- Other Unicode encoding formats include:
- UTF-16: Uses a fixed number of 2 bytes to represent characters
- UTF-32: Uses a fixed number of 4 bytes to represent characters
Key Terms to Review (23)
Ascii: ASCII, which stands for American Standard Code for Information Interchange, is a character encoding standard used to represent text in computers and other devices that use text. It assigns a unique numerical value to each character, including letters, digits, punctuation marks, and control characters, enabling computers to process and store textual data effectively. ASCII primarily focuses on representing characters for the English language but forms a foundation for more complex encoding systems that handle a wider array of characters across different languages.
Binary: Binary is a numerical system that uses only two digits, 0 and 1, to represent all possible values. This base-2 system is fundamental in computer architecture and digital electronics as it aligns perfectly with the on/off states of electronic circuits. Binary serves as the foundation for data representation in computers, allowing complex information to be encoded efficiently in the form of integers, floating-point numbers, and characters.
Bit: A bit is the smallest unit of data in computing, representing a binary value of either 0 or 1. This fundamental concept forms the basis of the binary number system, which underlies all digital systems, including data representation and instruction processing. Bits are essential in encoding information, allowing integers, floating-point numbers, and characters to be represented in a format that computers can process. They also play a crucial role in determining how instructions are formatted and how data is addressed in memory.
Byte: A byte is a unit of digital information that typically consists of 8 bits and is used to represent a single character or value in computing. This fundamental unit plays a critical role in data representation, storage, and processing, linking binary and hexadecimal systems to various types of data such as integers, floating-point numbers, and characters. Understanding bytes is essential for grasping how data is organized and manipulated in computer architecture.
Character Encoding: Character encoding is a system that assigns a numerical value to each character in a set, allowing computers to store and manipulate text. It bridges the gap between human-readable characters and machine-readable binary data, ensuring consistent representation of characters across different devices and platforms. This concept is crucial for the representation of not only characters but also integers and floating-point numbers, as it establishes how these values are stored and processed in digital formats.
Code point: A code point is a numerical value that represents a specific character in a character encoding system. This value allows computers to store and manipulate characters, including letters, symbols, and control characters, in a consistent manner. Code points are essential for data representation as they provide a unique identifier for each character across various encoding formats, ensuring accurate communication and processing of textual data.
Data structures: Data structures are organized formats for storing, managing, and retrieving data in a computer system. They allow efficient access and modification of data, making it easier to perform operations such as searching and sorting. Data structures are crucial for representing different types of data like integers, floating-point numbers, and characters in a way that maximizes performance and minimizes resource usage.
Decimal: Decimal refers to the base-10 number system, which uses ten digits (0 through 9) to represent values. This system is the most commonly used numerical system in daily life and is crucial for understanding data representation, as it provides a foundation for integers, floating-point numbers, and characters. The decimal system is inherently linked to how computers process and display numerical information, especially when converting between different bases.
Fixed-point: Fixed-point representation is a method of storing real numbers in a way that maintains a fixed number of digits before and after the decimal point. This allows for precise calculations involving integers and decimal fractions, making it particularly useful for applications where performance and memory efficiency are critical, such as in embedded systems or digital signal processing.
Floating-point representation: Floating-point representation is a way to express real numbers in a format that can accommodate a wide range of values, including very small and very large numbers. This format uses a combination of a significand (or mantissa), an exponent, and a base to represent numbers, allowing for efficient storage and computation in digital systems. It connects closely with data representation techniques for integers and characters by enabling computers to perform calculations with fractions and handle precision in numerical data.
Font: A font is a specific style and size of text characters used to display written content visually. It encompasses the design, weight, and spacing of the characters, affecting how information is perceived and read. Different fonts can convey various moods and intentions, making them a crucial aspect of data representation in both textual and graphical formats.
Glyph: A glyph is a visual symbol or character that represents a particular idea or concept, often used in the context of writing systems and typography. In data representation, glyphs can symbolize characters or numbers, playing a key role in how information is displayed and interpreted. They are integral to rendering text in various formats, ensuring that the intended meaning of the data is conveyed through recognizable forms.
Hexadecimal: Hexadecimal is a base-16 number system that uses sixteen symbols to represent values, ranging from 0 to 9 and A to F. It is widely used in computing and digital electronics because it offers a more compact and human-friendly way to express binary data, which is inherently represented in base-2. Hexadecimal simplifies the representation of binary numbers, making it easier to read and understand large values or addresses in computer systems.
IEEE 754: IEEE 754 is a standard for floating-point arithmetic used in computers, defining how numbers are represented and manipulated. It establishes a consistent format for representing real numbers, which is crucial for ensuring accurate calculations across different computing systems. This standard covers various aspects such as precision, rounding modes, and the representation of special values like infinity and NaN (Not a Number).
Integer representation: Integer representation refers to the method by which whole numbers are encoded in a computer system. This involves converting numerical values into a format that can be easily processed and stored by digital devices, typically using binary code. Understanding integer representation is crucial for manipulating numbers in programming, performing arithmetic operations, and managing memory efficiently.
Memory allocation: Memory allocation is the process of reserving a portion of computer memory for use by programs during their execution. This process is critical as it allows programs to store data such as integers, floating-point numbers, and characters, which are fundamental for program operations. Efficient memory allocation is vital for optimal main memory organization and ensures that applications run smoothly without exhausting available resources.
Overflow: Overflow occurs when a calculation exceeds the maximum limit that can be represented within a given data type. This issue is particularly significant when dealing with integers, floating-point numbers, and characters, as it can lead to unexpected results and errors in computation. Understanding overflow helps in designing systems that manage numerical limits effectively, ensuring accuracy and reliability in data representation.
Signed: The term 'signed' refers to a representation of numerical values that can express both positive and negative integers in computing. In binary systems, signed numbers are typically represented using a method like two's complement, allowing for an effective way to encode negative values while maintaining a straightforward arithmetic operation. This characteristic is crucial for accurately performing mathematical operations on both positive and negative numbers, which are essential in many computing applications.
Type conversion: Type conversion is the process of converting a value from one data type to another, such as from an integer to a floating-point number or from a character to an integer. This process is crucial in programming and computer architecture because it ensures that data can be manipulated correctly and efficiently, especially when different data types interact in calculations or operations. Understanding type conversion helps in managing precision and memory usage, particularly when dealing with integers, floating-point numbers, and characters.
Underflow: Underflow refers to a condition in computer systems where a calculation results in a number that is too small to be represented within the available data type. This situation often occurs with floating-point numbers when the value is closer to zero than the smallest representable value, leading to inaccuracies or unexpected results. Underflow is crucial to understand, as it can impact calculations and data representation in various contexts, particularly with integers and floating-point arithmetic.
Unsigned: Unsigned refers to a type of numerical representation that can only express non-negative values. In the context of data representation, this means that the value range starts at zero and extends upwards without any negative numbers. This characteristic is particularly significant when dealing with integers and floating-point numbers, as it allows for a broader range of positive values compared to signed representations, which account for both positive and negative numbers.
Utf-8: UTF-8 is a variable-length character encoding system that can represent every character in the Unicode character set. It is designed to be backward compatible with ASCII, using one byte for standard ASCII characters and up to four bytes for other characters. This flexibility makes UTF-8 an essential format for representing text in computers, enabling the use of diverse characters from different languages and scripts.
Word size: Word size refers to the number of bits that a computer's processor can handle in a single operation, directly influencing the amount of data that can be processed at once. It determines the range of values that can be represented for data types such as integers and floating-point numbers, as well as how instructions are formatted and executed. The word size plays a critical role in memory addressing, performance capabilities, and overall system architecture.