1. LO1 – Understand Computer Hardware
1.8 Number Systems
1.9 Number Conversions

2. Tens Ones Hundreds Tens Ones 128 64 32 16 8 4 2 1
Number systems
What is Binary?
Because humans have 10 fingers, we count using a denary number system (base 10):
We count to ten…
Then we record it by placing a 1 in the 10s column…
So the number 16 would be represented by one lot of 10 and 6 lots of one
…and when we get to 100, we make a record of it by placing a 1 in the 100s column…and so on!
So the number 128 would be represented by 1 lot of 100 plus 2 lots of 10 and 8 lots of one.
In binary the only numbers you can use are 0s and 1s
So the binary number above can be converted to denary by adding = 89
Tens
Ones 1 6
Hundreds
Tens
Ones 1 2 8
128 64 32 16 8 4 2 1

3. Binary to Denary: Denary to Binary: Number Systems continued…
To convert a denary number to binary
Convert the denary number 178 to Binary
So look at the biggest number that will go into 178 which is Put a 1 in the 128 column
Then take the 128 off the 178 so = 50
So the biggest number that will go into 50 is 32. Put a 0 in the 64 column and a 1 in the 32 column.
Then take the 32 off the 50 so 50-32=18 Put a 1 in the 1 in the 16 column.
This leaves a remainder of 2 So the biggest number that will go into 2 is 2 leaving no remainder.
Put a 0 in the 8 and 4 column, a 1 in the 2 column and a 0 in the 1 column. Always make sure all 8 columns are filled in to make a byte.
128 64 32 16 8 4 2 1
Binary to Denary: 1 1
Denary to Binary: 1 1 1 1 1 1 1

4. In binary this is represented as 00010001 1 1
Number Systems
Hexadecimal
As we know, computers can only deal with 2 numbers (0 and 1).
The problem for computer scientists is that very quickly, a fairly small denary number like 258 (3 digits long) becomes the massive binary number of (9 digits!) in binary.
To solve this issue, computer scientists came up with another number system to help them deal with base two numbers (binary) but without the long string of digits!
100 10 1
The denary number 17 7
In binary this is represented as 1 1
256 16 1
In hexadecimal this is represented as 11 (1 lot of 16 and 1 lot of 1). So the first column can represent up to 15 and the second column up to 255.
16^0= ^1= ^2=256 1 1

5. The Hexadecimal Number System
Number Systems
Hexadecimal
The Hexadecimal Number System
100 10 1

6. Number Systems 100 10 16+15=31 (16x10)+(15x1) 160+15=175 Hexadecimal
Denary
Hexadecimal 1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F
100 10
One lot of 16 plus 15 (F) lots of 1
16+15=31
Ten (A) lots of 16 plus 15 (F) lots of 1
(16x10)+(15x1)
160+15=175

7. This equals 2 lots of 16 (32) and 14 (E) lots of 1
Number Systems
Hexadecimal
Converting Denary into Hexadecimal
This is a little harder…
We use the following method:
Count how many 16s fit into the number
Place the answer in the 16s column
Place the remainder in the 1s column
100 10 1
46 ÷ 16=2 remainder 14
This equals 2 lots of 16 (32) and 14 (E) lots of 1

8. An Easier Method can be seen in this video:
Number Systems
Hexadecimal
Converting Denary into Hexadecimal
This is a little harder…
We use the following method:
Count how many 16s fit into the number
Place the answer in the 16s column
Place the remainder in the 1s column
An Easier Method can be seen in this video:
100 10 1
The following videos shows Hex to Denary:
235 ÷ 16=14 remainder 11
This equals 14 (E) lots of 16 (32) and 11 (B) lots of 1