Number Systems Notes

Module Title: Number Systems
Module Objective: Calculate numbers between decimal, binary, and hexadecimal systems.
Topic Title and Objectives:
- Binary Number System: Calculate numbers between decimal and binary systems.
- Hexadecimal Number System: Calculate numbers between decimal and hexadecimal systems.

Binary numbering system consists of 1s and 0s, called bits.
Decimal numbering system consists of digits 0 through 9.
Hosts, servers, and network equipment use binary addressing to identify each other.
Each address is made up of a string of 32 bits, divided into four sections called octets.
Each octet contains 8 bits (or 1 byte) separated by a dot.
For ease of use by people, this dotted notation is converted to dotted decimal.

Positional notation means that a digit represents different values depending on the “position” the digit occupies in the sequence of numbers.
Decimal Positional Notation:
- Radix: 10
- Positions: 3, 2, 1, 0
- Calculation: $(10^3)$ , $(10^2)$ , $(10^1)$ , $(10^0)$
- Position Value: 1000, 100, 10, 1 (Thousands, Hundreds, Tens, Ones)
- Example: Decimal Number (1234)
  - Digits: 1, 2, 3, 4
  - Calculation: 1 x 1000, 2 x 100, 3 x 10, 4 x 1
  - Sum: 1000 + 200 + 30 + 4
  - Result: 1,234
Binary Positional Notation:
- Radix: 2
- Positions: 7, 6, 5, 4, 3, 2, 1, 0
- Calculation: $(2^7)$ , $(2^6)$ , $(2^5)$ , $(2^4)$ , $(2^3)$ , $(2^2)$ , $(2^1)$ , $(2^0)$
- Position Value: 128, 64, 32, 16, 8, 4, 2, 1
- Example: Binary Number (11000000)
  - Digits: 1, 1, 0, 0, 0, 0, 0, 0
  - Calculation: 1x128, 1x64, 0x32, 0x16, 0x8, 0x4, 0x2, 0x1
  - Sum: 128 + 64 + 0 + 0 + 0 + 0 + 0 + 0
  - Result: 192

Convert 11000000.10101000.00001011.00001010 to decimal.
- (11000000) = 192
- (10101000) = 168
- (00001011) = 11
- (00001010) = 10
Result: 192.168.11.10

The binary positional value table is useful in converting a dotted decimal IPv4 address to binary.
Start in the 128 position (the most significant bit).
Is the decimal number of the octet (n) equal to or greater than 128?
- If no, record a binary 0 in the 128 positional value and move to the 64 positional value.
- If yes, record a binary 1 in the 128 positional value, subtract 128 from the decimal number, and move to the 64 positional value.
Repeat these steps through the 1 positional value.
Example: Convert decimal 168 to binary
- Is 168 > 128? - Yes, enter 1 in 128 position and subtract 128 (168-128=40)
- Is 40 > 64? - No, enter 0 in 64 position and move on
- Is 40 > 32? - Yes, enter 1 in 32 position and subtract 32 (40-32=8)
- Is 8 > 16? - No, enter 0 in 16 position and move on
- Is 8 > 8? - Equal. Enter 1 in 8 position and subtract 8 (8-8=0)
- No values left. Enter 0 in remaining binary positions
- Decimal 168 is written as 10101000 in binary

Routers and computers only understand binary, while humans work in decimal.
It is important to gain a thorough understanding of these two numbering systems and how they are used in networking.

To understand IPv6 addresses, you must be able to convert hexadecimal to decimal and vice versa.
Hexadecimal is a base sixteen numbering system, using the digits 0 through 9 and letters A to F.
It is easier to express a value as a single hexadecimal digit than as four binary bits.
Hexadecimal is used to represent IPv6 addresses and MAC addresses.

IPv6 addresses are 128 bits in length. Every 4 bits is represented by a single hexadecimal digit.
That makes the IPv6 address a total of 32 hexadecimal values.
Each four hexadecimal character group is referred to as a hextet.

Follow the steps listed to convert decimal numbers to hexadecimal values:
1. Convert the decimal number to 8-bit binary strings.
2. Divide the binary strings in groups of four starting from the rightmost position.
3. Convert each four binary numbers into their equivalent hexadecimal digit.
Example: 168 converted into hex using the three-step process.
- 168 in binary is 10101000.
- 10101000 in two groups of four binary digits is 1010 and 1000.
- 1010 is hex A and 1000 is hex 8, so 168 is A8 in hexadecimal.

Follow the steps listed to convert hexadecimal numbers to decimal values:
1. Convert the hexadecimal number to 4-bit binary strings.
2. Create 8-bit binary grouping starting from the rightmost position.
3. Convert each 8-bit binary grouping into their equivalent decimal digit.
Example: D2 converted into decimal using the three-step process:
- D2 in 4-bit binary strings is 1101 and 0010.
- 1101 and 0010 is 11010010 in an 8-bit grouping.
- 11010010 in binary is equivalent to 210 in decimal, so D2 is 210 is decimal

Binary is a base two numbering system that consists of the numbers 0 and 1, called bits.
Decimal is a base ten numbering system that consists of the numbers 0 through 9.
Binary is what hosts, servers, and networking equipment uses to identify each other.
Hexadecimal is a base sixteen numbering system that consists of the numbers 0 through 9 and the letters A to F.
Hexadecimal is used to represent IPv6 addresses and MAC addresses.
IPv6 addresses are 128 bits long, and every 4 bits is represented by a hexadecimal digit for a total of 32 hexadecimal digits.
To convert hexadecimal to decimal, you must first convert the hexadecimal to binary, then convert the binary to decimal.
To convert decimal to hexadecimal, you must first convert the decimal to binary and then the binary to hexadecimal.