Binary Numbers

• As computers only understand 1s and 0s, all data must be converted into binary to be processed. Binary can be used to represent all numbers in our standard number system
• In our standard number system we have 10 different digits. This is called denary, decimal or base-10
• Binary only uses 2 different digits (1 and 0) - we call this base-2
• Counting in binary is similar to counting in denary, but the place values from right to left increase by powers of 2, instead of powers of 10
• Denary to binary

  0 = 0
• Denary to binary

  1 = 1
• Denary to binary

  2 = 10
• Denary to binary

  3 = 11
• Denary to binary

  4 = 100
• Denary to binary

  5 = 101
• Denary to binary

  6 = 110
• Denary to binary

  7 = 111
• Denary to binary

  8 = 1000
• Denary to binary

  9 = 1001
• Denary to binary

  10 = 1010
• Denary to binary

  11 = 1011
• Denary to binary

  12 = 1100
• Denary to binary

  13 = 1101
• Denary to binary

  14 = 1110
• Denary to binary

  15 = 1111
• Most binary numbers are given as 8-bit numbers, e.g. 01101011, which can represent the denary numbers 0-255
• Thr bit with the largest value is called the most significant bit, and the but with the smallest value is called the least significant bit
• Drawing a table with binary place values in the first row makes binary to denary conversion easier
• Example

  Convert the 8-bit binary number 10100110 to a denary conversion easier

  • Draw up a table with binary place values in the top row. Start with 2 at the right, then move left, doubling each time
  • Write the binary number 10100110 into your table
  • Add up all the numbers with a 1 in their column:

  > 128 + 32 + 4 + 2 = 166

  So 10100110 is 166 in binary

  
• Binary numbers are easier to convert using tables
• Convert denary to binary by subtracting
• When converting from denary to binary, it's easier to draw a table of binary place values, then subtract them from largest to smallest
• Example

  Convert the denary number 166 into an 8-bit binary number

  • Draw an 8-bit table
  • Move along the table, only subtracting the number in each column from your running total if it gives a positive answer
  • Put a 0 in every column that gives a negative answer, and a 1 in the rest

  >So 166 converted to an 8-bit binary number is 10100110

  
• Example

  Add the following 8-bit binary numbers together: 10001101 and 01001000

  • First, put the binary numbers into columns
  • Starting from the right, add the numbers into columns
  • When doing 1 + 1 = 10, carry the 1 into the next column

  > So 10001101 + 01001000 = 11010101
• Example

  Add the following 8-bit numbers together: 00110011 + 01111001

  • Start at the right-hand side and add each column
  • Sometimes you'll get something like 1 + 1 + 1 = 11, so you need to write 1, then carry 1 to the next column

  > So 00110011 + 01111001 = 10101100
• Overflow errors can occur when a number has too many bits
• Sometimes, during binary arithmetic you will get a result that requires more bits than the CPU is expecting - this is called overflow
• In binary the 8-bit calculation 1111 1111 + 0000 0001 gives the 9-bit answer 10000 0000. Computers will see the 1 as an overflow error and just output 0000 0000
• Computers usually deal with overflow errors by storing the extra bits elsewhere
• Overflow flags are used to show that an overflow error has happened
• In an overflow error, the most significant bit will be the first to overflow
• Example

  a)Add the 8-bit binary numbers 01000110 and 11100010, giving your answer as an 8-bit number

  • Add the binary numbers in the usual way
  • The final calculation is 1 + 1 = 10, so carry the 1
  • You are left with a 9-bit answer - this is an overflow error
  • Ignore the overflow to give your 8-answer = 00101000

  b) Identify any problems that could be caused by giving your answer as an 8-bit number

  > There is an overflow error which can lead to a loss of data and a loss of accuracy in your answer. It could also cause software to crash if it doesn't have a way of dealing with the extra bit

  
• The original 8-bit Pac-Man arcade game had an infamous overflow error. At level 256, half of the screen was replaced with glitchy colourful code
• Binary shifts can be used to multiply or divide by 2
• A binary shift (also known as a logical shift) moves every bit in a binary number left or right a certain number of places