LO1 – Understand Computer Hardware 1.8 Number Systems 1.9 Number Conversions

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 64+16+8+1 = 89 Tens Ones 1 6 Hundreds Tens Ones 1 2 8 128 64 32 16 8 4 2 1

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 128 Put a 1 in the 128 column Then take the 128 off the 178 so 178-128= 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: https://www.youtube.com/watch?v=q7nZbAUTSC4 1 1 Denary to Binary: https://www.youtube.com/watch?v=70lM1qAD5u4 1 1 1 1 1 1 1

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 100000010 (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 00010001 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 16^1=16 16^2=256 1 1

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

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

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

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: https://www.youtube.com/watch?v=NTiBqXm9u8Q 100 10 1 The following videos shows Hex to Denary: https://www.youtube.com/watch?v=bjpMvXd1TTQ 235 ÷ 16=14 remainder 11 This equals 14 (E) lots of 16 (32) and 11 (B) lots of 1