Binary Systems

Learning Intentions

  • Understand how to convert decimal/denary numbers to binary

  • Understand how to convert binary numbers to decimal/denary

     

Success Criteria

  • I am able to convert between binary and denary

Denary numbers

  • Denary numbers are the numbers we work with every day. Den = 10, in other words we count in tens and have the following numbers available:
    0, 1, 2, 3, 4, 5, 6, 7, 8, 9

  • We use units that we call tens, hundreds, thousands, tens of thousands and so on.

  • We call denary base 10.

     

Denary numbers

  • For example, the number of 2385 can be broken down like this:
    1000s   100s   10s    Units
    2           3        8          5

  • Each of these units can also be represented by:

    103   102    101    100
    2       3        8        5

Binary numbers

  • Binary numbers (bi meaning two, e.g. bicycle means two wheels) use just 0s and 1s.
  • They also use placeholders, but not in the same way. Because binary only has a 0 or a 1, binary is what we call base 2.
  • Placeholders are as follows:

    23     22   21    20
    1        0     1      0

An easier way to figure it out

  • 23 = 8
    22 = 4
    21 = 2
    20 = 1

Figuring out denary from a binary number

  • Let’s figure out what the binary number 1001 means in denary:

    8   4    2   1
    1   0    0   1

    To do this, add each number with a 1 under it together, so 8 + 1. Total is 9. So 1001 = 9.

Example 1

  • Convert:

    1000 1000

    To denary.

Example 2

  • Convert:

    1100 1111

    To denary.

Example 3

  • Convert:

    0010 0100

    To denary.

Figuring out binary from a denary number

  • Reversing binary to give us a decimal number is a bit trickier. The placeholders are the same and should be put at the top. Hint: the leftmost placeholder should be no bigger than the denary number.

  • Let’s calculate the number 17:
    16    8     4     2     1
    1      0     0     0     1

     

Denary to binary

  • Now let’s calculate 35:
    32  16   8    4    2    1
    1    0     0    0    1    1
    Double check it by adding all the numbers with 1s under them together.

     

Example 1

  • Convert:

    28

    To binary

Example 2

  • Convert:

    32

    To binary

Example 3

  • Convert:

    254

    To binary

Transform to binary:

  1. 33
  2. 19
  3. 28
  4. 73
  5. 109

Task

Transform to denary:

  1. 1011
  2. 10001000
  3. 00011111
  4. 1111
  5. 0110

Work through the calculations on the worksheet.

JB
5 Computer Systems : 1.1 Binary systems
© J Balfour
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