Data Representation
Introduction to Data Representation:
 Data representation is the process of encoding data for storage and manipulation in a computer system.
 In computer arithmetic, data representation is crucial for performing numerical calculations.
Number Systems:

Binary Number System:
 Fundamental to digital computers, consisting of only two symbols, 0 and 1.
 Each digit is a power of 2 (e.g., 2^0, 2^1, 2^2, ...).

Decimal Number System:
 The base10 system used in everyday life.
 Comprises digits from 0 to 9, with each digit representing a power of 10.
Binary to Decimal Conversion:
 To convert a binary number to decimal, multiply each digit by the corresponding power of 2 and sum the results.
Decimal to Binary Conversion:
 To convert a decimal number to binary, repeatedly divide the number by 2, keeping track of the remainders.
Hexadecimal Number System:
 Base16 system using digits 09 and letters AF.
 Convenient for representing binary data in a more compact form.
Bit, Byte, and Word:

Bit (Binary Digit):
 The smallest unit of data, representing a 0 or 1.

Byte:
 Consists of 8 bits and is a common unit of data manipulation.
 Can represent 256 different values (2^8).

Word:
 A computer's natural unit of data, often defined by the CPU's architecture.
 Typically 32 or 64 bits in modern systems.
FixedPoint Representation:
 Fixedpoint representation is used to store integers or fixed precision numbers.
 The position of the binary point is predetermined.
 Arithmetic operations can be performed directly.
FloatingPoint Representation:
 Used for real numbers with variable precision.
 Comprises three components: sign bit, exponent, and mantissa (fraction).
 Conforms to the IEEE 754 standard for binary floatingpoint arithmetic.
Overflow and Underflow:
 In fixedpoint and floatingpoint arithmetic, operations can result in values exceeding the representable range (overflow) or becoming too small to represent (underflow).
Two's Complement:
 A binary representation of signed integers where the most significant bit (MSB) indicates the sign.
 Negative numbers are represented by taking the two's complement of their absolute values.
Endianness:
 Refers to the byte order of multibyte data types in memory.
 Bigendian stores the most significant byte at the lowest memory address, while littleendian stores it at the highest address.
Binary Coded Decimal (BCD):
 A way to represent decimal numbers in binary form.
 Each decimal digit is represented by a 4bit binary code.
Conclusion: Data representation in computer arithmetic is fundamental to how computers store and manipulate numbers. Understanding binary, decimal, and hexadecimal number systems, as well as fixedpoint and floatingpoint representations, is essential for working with numerical data in computer science and engineering. Proper handling of data representation is critical to prevent errors like overflow and underflow in computations.