Edexcel GCSE Computer Science: Binary Systems

Edexcel GCSE Computer Science: Binary Systems

Introduction

Computers use a system called binary to represent data. Unlike humans who use decimal numbers (base 10) with digits 0-9, computers use only two digits: 0 and 1. This is because computers operate using electronic circuits with two states: on (1) or off (0). This tutorial will help you understand binary and its significance in computer science.

Binary Numbers

A binary number is a sequence of 0s and 1s. Each position in a binary number represents a power of 2, starting from the rightmost digit as 20, then 21, 22, and so on.

Example:

• 10112 (The subscript '2' indicates binary)
• 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 8 + 0 + 2 + 1 = 1110 (in decimal)

Converting between Decimal, Binary, and Hexadecimal

• Decimal to Binary: Repeatedly divide the decimal number by 2, writing down the remainders. The remainders, read from bottom to top, form the binary equivalent.

  Example: 1310 to binary: * 13 ÷ 2 = 6 (remainder 1) * 6 ÷ 2 = 3 (remainder 0) * 3 ÷ 2 = 1 (remainder 1) * 1 ÷ 2 = 0 (remainder 1) * Therefore, 1310 = 11012

• Binary to Decimal: Multiply each binary digit by its corresponding power of 2, then sum the results.

  Example: 10112 to decimal: * 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 8 + 0 + 2 + 1 = 1110

• Decimal to Hexadecimal: Repeatedly divide the decimal number by 16, writing down the remainders. Convert any remainders greater than 9 to their hexadecimal equivalent (A=10, B=11, C=12, D=13, E=14, F=15). The remainders, read from bottom to top, form the hexadecimal equivalent.

  Example: 2610 to hexadecimal: * 26 ÷ 16 = 1 (remainder 10 = A) * 1 ÷ 16 = 0 (remainder 1) * Therefore, 2610 = 1A16

• Hexadecimal to Decimal: Multiply each hexadecimal digit by its corresponding power of 16, then sum the results.

  Example: 1A16 to decimal: * 1 x 161 + 10 x 160 = 16 + 10 = 2610

Two's Complement

Two's complement is a method used to represent negative numbers in binary. It's essential for calculations and representing signed integers in computers.

• To find the two's complement of a binary number:
  1. Invert all the bits (0 becomes 1, 1 becomes 0).
  2. Add 1 to the result.

Example: * Find the two's complement of 01012: 1. Invert the bits: 10102 2. Add 1: 10112 * Therefore, the two's complement of 01012 is 10112, representing -5 in decimal.

Binary Addition

Binary addition follows the same principles as decimal addition, but with only two digits:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (carry-over 1)

Example: * 1012 + 112 = 10002

Binary Shifts

Shifting bits left or right is a fundamental operation used for multiplication and division by powers of 2.

• Left Shift: Shifting a binary number left by one position multiplies the number by 2.
• Right Shift: Shifting a binary number right by one position divides the number by 2.

Example: * 1012 (left shift by 1) = 10102 (2 x 1012 = 10102)

Overflow Errors

Overflow errors occur when a binary calculation results in a number that is too large to be represented within the available number of bits. This can lead to unexpected results in programs.

Example: * Using 8-bit representation, 12710 (011111112) is the largest positive number. * Adding 1 to 12710 would result in 12810 (100000002) which is outside the range of 8-bit representation, causing an overflow error.

Conclusion

Understanding binary systems is crucial for anyone working with computers. By mastering the concepts of binary representation, conversions, two's complement, addition, shifts, and overflow errors, you can effectively handle data, understand how programs work, and troubleshoot problems in your code.