# 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

• 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.

• Convert:

1000 1000

To denary.

• Convert:

1100 1111

To denary.

• 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.

• Convert:

28

To binary

• Convert:

32

To binary

## Example 3

• Convert:

254

To binary

Transform to binary:

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

Transform to denary:

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

Work through the calculations on the worksheet.

5 Computer Systems : 1.1 Binary systems