Representing numbers

• Transistors


  Element of a CPU that acts like a switch
• Binary system

  Has 2 states (ON or OFF)
• Bit-depth of operating systems always doubles over time (e.g. 16 bit, 32 bit, 64 bit)
• Bit


  Single binary digit
• With four binary digits, 16 different values can be represented
• 2^2 = 2x2
• 2^3 = 2x2x2
• Humans have evolved to use a base-10 counting system
• The image created has a bit-depth of 1 bit (black and white)
• Any number to the power of 0 is equal to 1
• Place values for decimal system

  • 100,000
  • 10,000
  • 1,000
  • 100
  • 10
  • 1
• Place values for binary system

  • 32
  • 16
  • 8
  • 4
  • 2
  • 1
• Byte

  8 bits
• In 8-bit binary, the maximum number of values is 256 (0 through 255)
• Place values for 8-bit binary

  • 128
  • 64
  • 32
  • 16
  • 8
  • 4
  • 2
  • 1
• The decimal value represented in the given 8-bit binary is 115
• Converting 8-bit binary to decimal


  • 11001101 = 205
  • 01000100 = 68
  • 10101010 = 170
  • 11110000 = 240
  • 10111100 = 188
• You can't count up to 256 in 8-bit binary because it would require 9 bits
• Method 1 for converting decimal to binary

  1. Start at the left and ask if the number is equal to or greater than the place value

  2. If less, place a 0

  3. If greater, place a 1 and subtract the place value to find the remainder
• Method 2 for converting decimal to binary


  Divide the number by 2 repeatedly, recording the remainders
• Number of bits in units

  • Nibble: 4
  • Byte: 8
  • Kilobyte: 8,000
  • Megabyte: 8,000,000
  • Gigabyte: 8,000,000,000
  • Terabyte: 8,000,000,000,000
• An image file of 213,624,133 bits is 26.7 MB
• Kibibyte


  Equal to 1024 bytes, used in binary systems instead of kilobyte (1000 bytes)
• There are 8,000,000,000 bits in a gigabyte
• 2000 MB is equal to 2 GB
• 1 terabyte is equal to 1,000,000,000 kilobytes
• Converting binary to decimal

  • 11010101 = 213
  • 01101110 = 110
  • 10101010 = 170
• Converting decimal to binary

  • 233 = 11101001
  • 153 = 10011001
• If there are 8 bits in a byte, one megabyte is equal to 8 megabits
• Bits are binary digits with values of 0 or 1
• Bits can be combined into groups like letters into words
• All computer commands and data consist of groups of bits
• In any number system, the value of a digit is determined by its place value
• Binary is a base 2 number system
• Binary numbers can be converted to decimal and vice versa
• Binary additions

  • 1001111 + 1011101 = 10101100

  0111111 + 1011101 = 10001100 (overflow)
• How to multiply in binary

  Shift to the left by the power
• Binary left shifts

  • 00101101 left shift by 3 = 01010101000

  10001101 left shift by 4 = 10011010000

  10110111 left shift by 2 = 11011100
• How to divide in binary

  Look at first two digits of dividend, compare to divisor, add 1 to quotient, subtract divisor, repeat until remainder is 0
• Binary right shifts

  • 00101101 right shift by 3 = 00001010

  10001101 right shift by 4 = 00001000

  10110111 right shift by 2 = 00101101