1. Lesson Objectives Aims You should be able to: Convert Denary to Binary
Add binary numbers
Explain the term “overflow”
Convert binary to hexadecimal
Explain the use of hexadecimal numbers

2. A lot to do
This will take more than 1 lesson
We’re going to jump straight in and look at converting between different number systems
You will need to practise these conversions – there are exam questions along the way.

3. Denary to Binary Rules Write out the column values
Start with the highest column value
Does it fit into your number?
Yes – put a 1 in that column, take the value away
No – put a 0 in that column
Move to the next column, repeat step 3 for each column

4. Step 1 – write out the values:
Denary to Binary
Convert 217 to binary
Step 1 – write out the values:
128 64 32 16 8 4 2 1

5. Denary to Binary – Convert 217
128 is less than 217. Put a 1 in that column
Take 128 away from our target number : 217 – 128 = 89
Next column is worth 64, 64 is less than 89 so put a 1 in that column
Take 64 away from our remaining total : 89 – 64 = 25 left over
Next column is worth is more than 25 so we put a zero
Move to the next column – is less than 25 so put a 1 in that column
25 – 16 = 9
8 is less than 9 so put a 1 in that column.
9-8 = 1….
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1

6. Denary to binary – another example
Convert 187 to Binary
Write out your table (just keep doubling each number from 1)
Working from LEFT to RIGHT, put a 1 under each number that is nearest to your remaining total
187 – 128 = 59
59 – 32 = 27
27 – 16 = 11
11 – 8 = 3
3 – 2 = 1
1-1 = 0
187 =
128 64 32 16 8 4 2 1

7. Binary to Denary Covnersion Rules
Too easy!
Put the column values on…
Each column that contains a 1 – add up its value.

8. Convert the following binary number to Denary
Binary to Denary
Convert the following binary number to Denary
Step 1 – add values
Step 2 – add the value of any column with a 1 in it. =
= 159 1
128 64 32 16 8 4 2 1

9. June 2013 Q5 a

10. Hexadecimal Hexadecimal: Base 16 16 digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
The letters are NOT letters, they are digits!

11. Reasons for Hex
Computers DO NOT understand Hexadecimal, it is purely for us.
Hexadecimal is used because:
Binary produces long strings / Hex is shorter
Binary is difficult to work with / Hex easier to work with
Hex can be easily converted to/from binary
Convenience of 1 hex digit per nybble
Hex is less susceptible to error than long binary numbers

12. Hex
A useful table…

13. Rules – Hex Conversions
Each hex digit is perfectly represented by 4 BITS – this is important.
You convert in the following order:
Hex – Binary – Denary
Denary – Binary – Hex
Do NOT try to jump directly from Hex to Denary and vice versa.

14. Method Denary to Hex Convert to binary as normal
Split the binary number in to two 4 bit nybbles
Convert each nybble in to a hex digit
Put the two digits together
Hex to Denary
Split the hex digits into individual digits
Convert each to a 4 bit nybble
Put the two nybbles together to form a byte
Convert the byte to Denary as normal

15. Denary to Binary to Hexadecimal
Convert 175 in to Hexadecimal
Step 1 – Binary
Step 2 – Split in to Nybbles
Step 3 – Hex Digits
1010 = A
1111 = F
Step 4 – Combine
175 Denary = AF Hexadecimal
128 64 32 16 8 4 2 1
Note the column values! 8 4 2 1 8 4 2 1

16. Hexadecimal to Binary to Denary
Convert C4 in to Denary
Step 1 – Split and convert to Nybbles
C = =
Step 2 – Combine
Step 3 – Convert =
= 196 8 4 2 1 8 4 2 1
Note the column values!
128 64 32 16 8 4 2 1

17. June 2014 Q9 a

18. June 2014 Q9 b

20. June 2013 Q5 b

22. Sum Equals 0 + 0 0 + 1 1 1 + 0 1 + 1 0 CARRY 1 1 + 1 + 1 1 CARRY 1
Binary Addition
Rules:
Sum
Equals
0 + 0
0 + 1 1
1 + 0
1 + 1
0 CARRY 1
1 CARRY 1

23. Now do some sums..
Sum
Equals
0 + 0
0 + 1 1
1 + 0
1 + 1
0 CARRY 1
1 CARRY 1 + + +

24. Overflow
The biggest number we can have in an 8 bit byte is 255
If our computer is 8 bit, this is the highest number we can handle
What happens if you add 1 to 255???
You get zero… (and an error, or it crashes)

25. Explanation of Overflow
Try it out.
What happens?
Why might this be bad?! +

26. June 2014 Q3

28. Past Questions Jun-15 Binary 8b Jan-13 8a May-12 10a Jun-14
Binary - Addition 3a 1a
Jan-11 3b
Jun-13
Binary - Conversion
5a - i
5a - ii 6a
May-11 6c
Binary - Overflow 1b
Jun-14
Hexadecimal 9a 9b
Jun-13 5b
May-12 6b
May-11 6a 6d