The document covers various number systems, focusing on the hexadecimal system and its relationship with binary and denary systems, allowing conversions between these. It includes definitions, conversion methods, and practical exercises for transforming numbers across these bases. The discussion also touches on the relevance of hexadecimal in IPv6 addressing and its advantages over IPv4.

1. Mathematics for IT
Hexadecimal
Number
System
Unit 26
Lee Chadwick

2. Hexadecimal Number System
• Topic Definitions
• A Question!
• Lesson learning outcomes
• Hex Character Range
• Binary – Hex Conversion
• Hex – Binary Conversion
• Hex – Denary Conversion

3. • Hexadecimal: "Hexadecimal" means "based on 16" (From Greek hexa:
"six" and Latin decima: "a tenth part").
• Decimal: Based on 10; Example: the numbers we use in everyday life
are decimal numbers, because there are 10 of them
(0,1,2,3,4,5,6,7,8 and 9).
• Denary: Same as Decimal – Base 10
• Binary: The word binary comes from "Bi-" meaning two. We see "bi-" in
words such as "bicycle" (two wheels) or "binocular" (two eyes).
Binary only uses 2 digits; 1 & 0
• Octal: An Octal Number uses only these 8 digits: 0, 1, 2, 3, 4, 5, 6 and
7 Examples:
• 10 in Octal equals 8 in the Decimal Number System.
• 167 in Octal equals 119 in the Decimal Number System.
Also called Base 8.
Definitions:

4. • Identify the concept of Hex
• Establish the purpose of Hex
• Compare Base 2, Base 10 and Base 16
• Convert Hex to Denary and Binary – And Back

5. A Question:
When does 20 = 14?
When 2010 = 1416

6. By the end of this session:
All will be able to convert positive denary whole
numbers (0-255) into 2-digit hexadecimal
numbers and vice versa
Most will be able to convert between binary and
hexadecimal equivalents of the same number
Some will be able to explain the use of
hexadecimal numbers to represent binary
numbers

7. Hexadecimal Character Range:
(16 Values)
0 1 2 3 4 5 6 7
8 9 A B C D E F
The word "Hexadecimal" means "based on 16"
(From Greek hexa: "six" and Latin decima: "a
tenth part").

8. Binary – Hex Conversion:
To convert binary to hexadecimal you need to
break it down into nibbles (blocks of 4 bits).
1 0 0 0
1st Nibble
1 1 0 1
2nd Nibble
(The binary number 10001101 in denary is: 141)
Binary to Denary Calculation Table
Denary Values 128 64 32 16 8 4 2 1
Binary Values 1 0 0 0 1 1 0 1
Explanation: 128 + 8 + 4 + 1 = 141

9. Binary – Hex Conversion (2):
We now convert each nibble into Denary;
Binary to Denary Calculation Table – First Nibble
Denary Values 128 64 32 16 8 4 2 1
Binary Values 0 0 0 0 1 0 0 0
First Nibble Value: 8
Binary to Denary Calculation Table – Second Nibble
Denary Values 128 64 32 16 8 4 2 1
Binary Values 0 0 0 0 1 1 0 1
Second Nibble
Value:
13

10. Binary – Hex Conversion (3):
We now convert 8 and 13 into Hexadecimal;
Remember that 13 = D in hexadecimal:
141 would be represented as 8D.
Hex Values: 0 1 2 3 4 5 6 7
Denary Values: 0 1 2 3 4 5 6 7
Hex Values: 8 9 A B C D E F
Denary Values: 8 9 10 11 12 13 14 15
Hex Conversion: 8D

11. Your Turn:
Binary-Hex conversions
1. Convert 11111111 to hex
2. Convert 11011011 to hex
3. Convert 10010011 to hex
4. Convert 11000011 to hex
5. Convert 00110110 to hex

12. Your Turn:
Denary-Hex conversions
1. Convert 4010 to hex
2. Convert 6410 to hex
3. Convert 14010 to hex
Note:
Simply convert your Denary values to
Binary, then Binary to Hex

13. Your Turn:
Binary-Hex conversions –
Answers:
1. Convert 11111111 to hex –
2. Convert 11011011 to hex -
3. Convert 10010011 to hex –
4. Convert 11000011 to hex –
5. Convert 00110110 to hex -
Note: Hex is also written; 3616
0Fh
DBh
93h
C3h
36h

14. Your Turn:
Denary-Hex conversions
Answers:
6. Convert 40 to hex -
7. Convert 64 to hex -
8. Convert 140 to hex -
28h
40h
8Ch

15. Hex - Binary Conversion:
To convert hexadecimal to binary you just reverse
the process.
Convert each part of the hexadecimal number into
nibbles of binary numbers.
For Example:
Calculation Table
Hex Value 8 D
Denary Values 8 4 2 1 8 4 2 1
Binary Values 1 0 0 0 1 1 0 1

16. Hex – Binary
Conversion Table:
Denary Binary Hex Denary Binary Hex
0 0000 0 8 1000 8
1 0001 1 9 1001 9
2 0010 2 10 1010 A
3 0011 3 11 1011 B
4 0100 4 12 1100 C
5 0101 5 13 1101 D
6 0110 6 14 1110 E
7 0111 7 15 1111 F

17. Your Turn:
Hex-Binary conversions
1. Convert 4D to binary
2. Convert 2F to binary
3. Convert 72 to binary
4. Convert 90 to binary
5. Convert 3B to binary
Note: Hex is also written; 3616

18. Your Turn:
Hex-Binary conversions
Answers:
1. Convert 4D to binary -
2. Convert 2F to binary -
3. Convert 72 to binary -
4. Convert 90 to binary -
5. Convert 3B to binary -
01001101
00101111
01001000
01011010
00111011

19. Your Turn:
Hex - Denary conversions
1. Convert 4D to denary -
2. Convert 2F to denary -
3. Convert 91 to denary -
4. Convert AA to denary -
5. Convert F1 to denary -

20. Your Turn:
Hex - Denary conversions
Answers:
1. Convert 4D to denary -
2. Convert 2F to denary -
3. Convert 91 to denary -
4. Convert AA to denary -
5. Convert F1 to denary -
77
47
145
170
241

21. Hex in Action:

22. Why Do We Need IPv6?
• The driver for the uptake of 128-bit IPv6 will be the
shortage of 32-bit IPv4 addresses on the internet. IPv6
is also more secure. For example, it can overcome the
lack of security and prioritization of IPv4 datagrams.
• In the mid-term we are beset with compatibility problems
because IPv4-only clients cannot communicate with
IPv6-only routers. Thus for most business scenarios
migrating to an IPv6-only network is not the answer just
yet.
• Until IPv4 is switched off, networks will need to cater for
both protocol stacks, and develop strategies to work
seamlessly with both types of IP node.

23. Five Useful IPv6 Concepts
• Stateful IP addresses are given out by a DHCP server. Usually DHCP in addition
to the IPv6 hex number, the clients get the address of the default gateway and
probably a DNS server or two.
• Stateless IPv6 addresses are assigned by the host itself, rather like APIPA in
IPv4. This is what happens if there is no DHCP or manual address assignment.
• Link-local IPv6 addresses only allow connections with neighbours on that subnet
or 'link'. You can identify Link-local addresses because they begin with FE80, also
(FC and FD) naturally, Link-local addresses are not forwarded by routers.
• Site-local means the IPv6 is routable, but not to the internet, thus hosts with Site-
Local IPv6 addresses can use private (not ICANN) IP addresses, AND can
connect to any other Site-local address within the organisation. Such site-local
addresses all start with FEC0.
• Neighbour Discovery (ND) This concepts means that machines determine
information about their nearest router. The idea is also that if an IPv6 stack can
obtain information about other nodes, then you won't get the problem of duplicate
IP addresses.

24. IPv6 Changes in Windows 8
• Any operating systems running a dual stack (IPv4 and IPv6) is going to face
connectivity problems. Naturally, if there is connectivity for both IPv4 and IPv6
then Windows 8 (or 7) will favour the IPv6 path. What irritated Windows 7 users is
where the OS cannot detect an IPv6 path and there is a delay while it figures out
how long to wait trying the non-existent IPv6 path.
• In Windows 8, Microsoft has developed a better algorithm than Windows 7, it
checks the state of the IPv6 path at initial configuration. If no IPv6 connectivity
exists it will be marked as unreachable, and the IPv4 will seek the traditional IPv4
route.
• There are also changes on the Windows Server 2012, in particular NAT64/DNS64
is now built-in. This caters for networks running IPv6 internally, but using IPv4 for
the internet. Incidentally, PowerShell v3 on the Server 2012 provides better
cmdlets to manage IPv6 configuration options.

25. IPv6 Maths - See the Big Picture
• Experts tell us that IPv4 would generate 4,294,967,296 possible IP addresses. In
practice it turned out there were only about 17 million useful addresses.
• With the 128bit IPv6 addresses, the same experts say there should be
340,300,000,000,000,000,000,000,000,000,000,000,000,000 IP addresses.
• However, there may be as few as: 18,000,000,000,000 useful IPv6
addresses. This shortfall is partly due to reserved and unassigned bits in the
128bit address.
• The other reason for this reduced number of usable IP addresses is a design
feature whereby 64-bits are taken up with the Interface ID (Mac Number).
• Even with this surprisingly low estimate, it still means that everybody on the planet
could be given 3,000 IP addresses. One day, we could see one IPv6 for the
computer, one for the phone, car, fridge, cooker and every other appliance - then
some.

26. IPv6 Address
Making Sense of the Actual Hex Numbers
• IPv6 uses hexadecimal, which is base 16 this is why you now see IP addresses
containing not only numbers, but also the letters ABCDEF, for example:
2001:0618:71B3:08C3:1319:8C2D:0271:6017
• As you can see, 128-bit numbers are split into 8 groups of 16bit.
• IPv4 addresses are base 10, another difference is that each IPv6 group is
separated by a colon rather than a dot. It is readily apparent that this base 16
scheme helps to increase the available IP addresses. Surprisingly, the hex letters
are not case sensitive.
• Obviously, private networks won't need any where near the full range of IPv6
numbers; as a result many of the address values will be zero. In this
circumstance look for compression of the zeros, instead of
FD01:0000:0000:0000:0000:0000:0000:0005, you will see FD01::5. Note the
double colon :: indicating compression of the intervening zeros. Thus the term
'compression' in IPv6 refers to the notation and not to the protocol packets
themselves. Remember that you can only use the double colon once in each IP
address.

27. IPv6 and MAC Address
• The biggest reason that there will be fewer IPv6 addresses than the theoretical
maximum is that each 64-bit number contains the MAC address of the host.
• While incorporating the hardware address reduces the available IPv6 nodes, it
makes this protocol more efficient, secure and useful than IPv4.
• Note how the DUID* above contains the Physical Address (MAC Address).
• *DUID = DHCP Unique Identifier.
IAID = Application Unique Identifier

29. 0h 1h 2h 3h 4h 5h 6h 7h 8h 9h Ah Bh Ch Dh Eh Fh
0h 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240
1h 1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241
2h 2 18 34 50 66 82 98 114 130 146 162 178 194 210 226 242
3h 3 19 35 51 67 83 99 115 131 147 163 179 195 211 227 243
4h 4 20 36 52 68 84 100 116 132 148 164 180 196 212 228 244
5h 5 21 37 53 69 85 101 117 133 149 165 181 197 213 229 245
6h 6 22 38 54 70 86 102 118 134 150 166 182 198 214 230 246
7h 7 23 39 55 71 87 103 119 135 151 167 183 199 215 231 247
8h 8 24 40 56 72 88 104 120 136 152 168 184 200 216 232 248
9h 9 25 41 57 73 89 105 121 137 153 169 185 201 217 233 249
Ah 10 26 42 58 74 90 106 122 138 154 170 186 202 218 234 250
Bh 11 27 43 59 75 91 107 123 139 155 171 187 203 219 235 251
Ch 12 28 44 60 76 92 108 124 140 156 172 188 204 220 236 252
Dh 13 29 45 61 77 93 109 125 141 157 173 189 205 221 237 253
Eh 14 30 46 62 78 94 110 126 142 158 174 190 206 222 238 254
Fh 15 31 47 63 79 95 111 127 143 159 175 191 207 223 239 255
Toconvertbetweendecimaland
hexisquitesimplewiththischart:

30. Number Systems – Task:
Number Systems Working Together
Using a structured approach, complete a range of conversions between
the number systems discussed previously. The conversions needed
are:
• Decimal to Binary, Octal, Hexadecimal
• Binary to Decimal, Octal, Hexadecimal
• Octal to Decimal, Binary, Hexadecimal
• Hexadecimal to Decimal, Binary, Octal