OCR A-level Computer Science: Data Representation – Binary, Hexadecimal, and Character Encoding

OCR A-Level Computer Science: Data Representation

This tutorial explores the fundamental concepts of data representation in computers, including binary, hexadecimal, character encoding, and bitwise operations.

1. Binary and Hexadecimal Number Systems

Computers operate on a binary system, using only two digits: 0 and 1. Each digit represents a bit (binary digit).

a) Binary Representation:

• Each position in a binary number represents a power of 2.
• Example:
• 10112 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) = 8 + 0 + 2 + 1 = 1110

b) Hexadecimal Representation:

• Hexadecimal uses 16 digits: 0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, and F=15.
• It is often used for representing binary data in a more compact format.
• Each position represents a power of 16.
• Example:
• 2A516 = (2 x 162) + (10 x 161) + (5 x 160) = 512 + 160 + 5 = 67710

c) Conversion:

• Binary to Decimal: Sum the values of each bit based on its position (power of 2).
• Decimal to Binary: Repeatedly divide the decimal number by 2, noting the remainder (0 or 1) at each step. The remainders, read from bottom to top, form the binary representation.
• Binary to Hexadecimal: Group binary digits into sets of 4, starting from the rightmost digit. Convert each group into its hexadecimal equivalent.
• Hexadecimal to Binary: Convert each hexadecimal digit into its 4-bit binary representation.

2. Representing Integers: Sign-and-Magnitude and Two's Complement

a) Sign-and-Magnitude:

• The leftmost bit represents the sign (0 for positive, 1 for negative).
• The remaining bits represent the magnitude of the number.
• Example:
• 01012 = +510
• 11012 = -510
• Limitation: Two representations for zero (0000 and 1000).

b) Two's Complement:

• The most common method for representing signed integers.
• For a positive number, it is the same as its binary representation.
• For a negative number, it is calculated by:
  1. Invert all bits of the positive equivalent.
  2. Add 1 to the result.
• Example:
• +510 = 01012
• -510 = 10112 (invert 0101, add 1: 1010 + 1 = 1011)
• Advantages:
  • Single representation for zero.
  • Arithmetic operations can be performed directly without considering sign.

3. Character Encoding

Computers store text as numerical values representing characters.

a) ASCII (American Standard Code for Information Interchange):

• Uses 7 bits to represent 128 characters, including uppercase and lowercase letters, numbers, punctuation marks, and control characters.
• Example:
• 'A' = 6510 = 010000012
• 'a' = 9710 = 011000012

b) Unicode:

• Extends ASCII to represent characters from different languages and scripts.
• Uses 16 bits (UTF-16) or 32 bits (UTF-32).
• Example:
• 'é' = 23310 = 00000000 111010012 (UTF-16)

4. Bitwise Operations

These operations work on individual bits of data.

a) AND ( & ):

• Returns 1 if both corresponding bits are 1, 0 otherwise.
• Example: 1011 & 0101 = 0001

b) OR ( | ):

• Returns 1 if at least one corresponding bit is 1, 0 otherwise.
• Example: 1011 | 0101 = 1111

c) XOR ( ^ ):

• Returns 1 if exactly one corresponding bit is 1, 0 otherwise.
• Example: 1011 ^ 0101 = 1110

d) NOT ( ~ ):

• Flips the bits of a number (0 becomes 1, 1 becomes 0).
• Example: ~ 1011 = 0100

Summary

Understanding data representation is crucial for working with computers. This tutorial covered binary, hexadecimal, character encoding, and bitwise operations, providing a foundation for understanding how data is stored, manipulated, and processed within a computer system.