Networking Mathematics Reference

Number Systems

Binary, Decimal, Hexadecimal Conversion

Decimal	Binary	Hexadecimal
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

Powers of 2

\[2^n = \text{value}\]

n	\(2^n\)
0	1
1	2
2	4
3	8
4	16
5	32
6	64
7	128
8	256
9	512
10	1,024

Critical values for networking:

Power	Value	Use
\(2^0\)	1	Single host
\(2^8\)	256	Octet range (0-255)
\(2^{16}\)	65,536	Class B network
\(2^{24}\)	16,777,216	Class A network
\(2^{32}\)	4,294,967,296	Total IPv4 addresses
\(2^{128}\)	\(\approx 3.4 \times 10^{38}\)	Total IPv6 addresses

Power

Value

Use

\(2^0\)

1

Single host

\(2^8\)

256

Octet range (0-255)

\(2^{16}\)

65,536

Class B network

\(2^{24}\)

16,777,216

Class A network

\(2^{32}\)

4,294,967,296

Total IPv4 addresses

\(2^{128}\)

\(\approx 3.4 \times 10^{38}\)

Total IPv6 addresses

IPv4 Subnetting

Subnet Mask Notation

CIDR (Classless Inter-Domain Routing): A method for allocating IP addresses and routing that replaces the older classful network design. CIDR notation specifies an IP address followed by a slash and the number of network bits (e.g., 192.168.1.0/24).

CIDR	Subnet Mask	Binary	Usable Hosts
/8	255.0.0.0	11111111.00000000.00000000.00000000	16,777,214
/16	255.255.0.0	11111111.11111111.00000000.00000000	65,534
/24	255.255.255.0	11111111.11111111.11111111.00000000	254
/25	255.255.255.128	11111111.11111111.11111111.10000000	126
/26	255.255.255.192	11111111.11111111.11111111.11000000	62
/27	255.255.255.224	11111111.11111111.11111111.11100000	30
/28	255.255.255.240	11111111.11111111.11111111.11110000	14
/29	255.255.255.248	11111111.11111111.11111111.11111000	6
/30	255.255.255.252	11111111.11111111.11111111.11111100	2
/31	255.255.255.254	11111111.11111111.11111111.11111110	2 (P2P)
/32	255.255.255.255	11111111.11111111.11111111.11111111	1 (host)

Host Calculation Formula

\[\text{Usable Hosts} = 2^n - 2\]

The subtraction of 2 accounts for the network address and broadcast address.

Where \(n\) = number of host bits (32 - prefix length)

Example: /26 network

\(n = 32 - 26 = 6\)

\(\text{Hosts} = 2^6 - 2 = 64 - 2 = 62\)

Subnet Calculation Formula

\[\text{Number of Subnets} = 2^n\]

Where \(n\) is the number of bits borrowed from the host portion.

Where \(n\) = number of borrowed bits

Example: Subnetting a /24 into /27s

Borrowed bits: \(27 - 24 = 3\)

\(\text{Subnets} = 2^3 = 8\)

Network and Broadcast Address

Network Address: The first address in a subnet, where all host bits are 0. This address identifies the network itself and cannot be assigned to a host. Calculated by performing a bitwise AND between the IP address and subnet mask.
Broadcast Address: The last address in a subnet, where all host bits are 1. Packets sent to this address are delivered to all hosts on the subnet. Calculated by performing a bitwise OR between the network address and the wildcard mask.

Example: 192.168.10.67/26

IP Address:    192.168.10.01000011  (67 in binary)
Subnet Mask:   255.255.255.11000000 (/26)

Network address: AND operation

01000011 AND 11000000 = 01000000 = 64
Network: 192.168.10.64

Broadcast address: OR with inverted mask

01000011 OR 00111111 = 01111111 = 127
Broadcast: 192.168.10.127

Usable range: 192.168.10.65 - 192.168.10.126

VLSM (Variable Length Subnet Masking)

VLSM (Variable Length Subnet Masking): A technique that allows a network to be divided into subnets of different sizes. Unlike fixed-length subnetting, VLSM allocates address space efficiently by matching subnet size to actual requirements, minimizing wasted addresses.

VLSM Design Process

List networks by size (largest first)
Assign smallest subnet that fits each requirement
Allocate sequentially to avoid overlap

Example: Subnetting 10.0.0.0/24

Requirements: - Network A: 100 hosts - Network B: 50 hosts - Network C: 25 hosts - Network D: 2 hosts (P2P link)

Allocation:

Network	Hosts Needed	Prefix	Subnet	Range
A	100	/25 (126 hosts)	10.0.0.0/25	.1-.126
B	50	/26 (62 hosts)	10.0.0.128/26	.129-.190
C	25	/27 (30 hosts)	10.0.0.192/27	.193-.222
D	2	/30 (2 hosts)	10.0.0.224/30	.225-.226

IPv6 Addressing

Address Structure

IPv6 Address Structure: A 128-bit address written as eight groups of four hexadecimal digits separated by colons (e.g., 2001:0db8:85a3:0000:0000:8a2e:0370:7334). The address space is divided into a global routing prefix (typically 48 bits), subnet ID (16 bits), and interface ID (64 bits).

IPv6: 128 bits total
├── Global Routing Prefix: 48 bits (ISP assigned)
├── Subnet ID: 16 bits (65,536 subnets)
└── Interface ID: 64 bits (host portion)

IPv6 Notation Rules

Leading zeros in each group can be omitted
One sequence of consecutive all-zero groups can be replaced with ::

Example

Full: 2001:0db8:0000:0000:0000:0000:0000:0001

Compressed: 2001:db8::1

IPv6 Prefix Lengths

Prefix	Use	Addresses
/32	ISP allocation	\(2^{96}\)
/48	Site allocation	\(2^{80}\)
/64	Single subnet	\(2^{64}\)
/128	Single host	1

Prefix

Use

Addresses

/32

ISP allocation

\(2^{96}\)

/48

Site allocation

\(2^{80}\)

/64

Single subnet

\(2^{64}\)

/128

Single host

1

IPv6 Address Types

Type	Prefix	Example
Global Unicast	2000::/3	2001:db8::/32
Link-Local	fe80::/10	fe80::1
Unique Local	fc00::/7	fd00::/8
Multicast	ff00::/8	ff02::1 (all nodes)
Loopback	::1/128	::1

Type

Prefix

Example

Global Unicast

2000::/3

2001:db8::/32

Link-Local

fe80::/10

fe80::1

Unique Local

fc00::/7

fd00::/8

Multicast

ff00::/8

ff02::1 (all nodes)

Loopback

::1/128

::1

Bitwise Operations

AND Operation (Subnet Calculation)

AND Truth Table

A	B	A AND B
0	0	0
0	1	0
1	0	0
1	1	1

Result is 1 only when both inputs are 1.

Used to find network address:

IP:     11000000.10101000.00001010.01000011 (192.168.10.67)
Mask:   11111111.11111111.11111111.11000000 (/26)
AND:    11000000.10101000.00001010.01000000 (192.168.10.64)

OR Operation (Broadcast Calculation)

OR Truth Table

A	B	A OR B
0	0	0
0	1	1
1	0	1
1	1	1

Result is 1 when either input is 1.

Used with inverted mask to find broadcast:

Network: 11000000.10101000.00001010.01000000 (192.168.10.64)
Wildcard: 00000000.00000000.00000000.00111111 (0.0.0.63)
OR:       11000000.10101000.00001010.01111111 (192.168.10.127)

Wildcard Mask

\[\text{Wildcard Mask} = 255.255.255.255 - \text{Subnet Mask}\]

Wildcard masks are the inverse of subnet masks, used in ACLs and OSPF configurations.

Examples

Subnet mask: 255.255.255.192

Wildcard: \(255 - 192 = 63\) → 0.0.0.63

Used in ACLs: permit ip 192.168.10.64 0.0.0.63 any

Quick Reference Tables

Subnet Cheat Sheet

CIDR	Mask	Hosts	Subnets from /24	Block Size	Magic Number
/24	.0	254	1	256	256
/25	.128	126	2	128	128
/26	.192	62	4	64	64
/27	.224	30	8	32	32
/28	.240	14	16	16	16
/29	.248	6	32	8	8
/30	.252	2	64	4	4
/31	.254	2	128	2	2
/32	.255	1	256	1	1

Private Address Ranges (RFC 1918)

Class	Range	CIDR	Hosts
A	10.0.0.0 - 10.255.255.255	10.0.0.0/8	16,777,214
B	172.16.0.0 - 172.31.255.255	172.16.0.0/12	1,048,574
C	192.168.0.0 - 192.168.255.255	192.168.0.0/16	65,534

Special Addresses

Address	CIDR	Purpose
0.0.0.0	/8	This network
127.0.0.0	/8	Loopback
169.254.0.0	/16	Link-local (APIPA)
224.0.0.0	/4	Multicast
255.255.255.255	/32	Broadcast

Address

CIDR

Purpose

0.0.0.0

/8

This network

127.0.0.0

/8

Loopback

169.254.0.0

/16

Link-local (APIPA)

224.0.0.0

/4

Multicast

255.255.255.255

/32

Broadcast