1. Chapter 3 - Binary Numbering System
CMIT100
Chapter 3 - Binary Numbering System

2. Learning Objectives
Describe Numbering Systems: decimal, binary, and hexadecimal.
Identify how to convert decimal to binary and binary to decimal.
Describe how characters are stored in computer memory.
Illustrate the use of binary in a computer with a focus on IP addresses.

3. Numbering Systems Decimal – Base 10 Binary – Base 2
Hexadecimal – Base 16

4. Base 10 System - Decimal Primary human numbering system Digits 0 – 9
Example: 1, 2, 10, 4,321

5. Base 2 System - Binary Native numbering system for digital computers
Two digits: 0, 1
Each digit is a bit that represents on or off
Eight bits is a byte; also known as an octet
Example of an eight bit binary number:
Which equals?

6. Binary Basics
Binary numbers and arithmetic let you represent any amount you want using just two digits: 0 and 1.
When calculating / converting binary to decimal, we need to understand the exponent numbering system
Defined: The exponent of a number says how many times to use that number in a multiplication.
It is written as a small number to the right and above the base number.
In this example: 82 = 8 × 8 = 64
(The exponent "2" says to use the 8 two times in a multiplication.)
Other names for exponent are index or power.

7. Binary Basics
In IT we need to understand how to convert Bytes (eight 1s and 0s) to decimal, and vice versa.
Understanding Binary, base 2, and exponent math numbering helps us do this
Each digit "1" in a binary number represents a power of two, and each "0" represents zero:
For example:
Binary byte:
To convert think

8. Binary Basics To convert think 27 + 26 + 25 + 24 + 23 + 22 + 21 + 20
The exponents 7 through 0 represent the following decimal numbers
27 = 128
26 = 64
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
20 = 1
Total = 255

9. Binary Basics
When you see a number like “ " you can figure out what it means by adding the powers of 2:
= = 5
= = 10
= = 7

10. Binary Basics
Got it?
Now let’s look at a cheat sheet that can quickly be created.
List the numbers:
Then the binary number:
If the binary number is 0, then the decimal number is zero. If the binary number is 1, then add the decimal number to the total to achieve the decimal number
Adding it up:
0 64 32 4 1
101 Total

11. Binary Basics Converting from decimal to binary:
Using the cheat sheet:
If the decimal number is 200
Is 200 divisible by 128 = yes = 1
200 – 128 = 72
Is 72 divisible by 64 = yes = 1
= 8
Is 8 divisible by 32 = no = 0
Is 8 divisible by 16 = no = 0
Is 8 divisible by 8 = yes = 1
8 – 8 = 0 Is 0 divisible by 4 = no = 0
Is 0 divisible by 2 = no = 0
Is 0 divisible by 1 = no = 0
200 =

12. Why Convert Binary Why do we need to do this?
This is part of understanding the subnetting networks, part of a network administrators job.
Subnetting = creating multiple networks out of one (simple definition)
Applying this conversion also helps with understanding IP v4 IP addressing, used on network devices (computers, printers, switches, etc.)
There is a newer version of IP called IP v6 which uses the hexadecimal numbering system

13. IP v4 TCP / IP v4 properties
(screenshots from Windows Server 2008)
An IP v4 address is shown as a decimal number

14. IP Address: unique 32-bit number
IP v4
IP Address: unique 32-bit number
Divided into four octets separated by periods
0 reserved as placeholder referring to entire group of computers on a network
255 reserved for broadcast transmissions
So you cannot use .0 or .255 as the last octet in your IP
Most common way of expressing IP addresses
Decimal number between 0 and 255 represents each binary octet
Separated by period
Example:

15. IP v4
Each number in dotted decimal address has binary equivalent
Equals

16. Subnets

17. Base 16 - Hexadecimal Digits include: 0 – 9 and A – F Example: 3FA7
Not case sensitive
Example: 3FA7

18. Conversions 1A6F = 0001 1010 0110 1111 = Decimal 6767 F = 15 1111
6 =
A =
1 =

19. Converting between Decimal and Binary and Hexadecimal
HEX B924 =
Binary =
HEX 78E3 =
Binary =
HEX DA75 =
HEX B924 = 47396
Binary =
78E3 = 30947
Binary =

20. Character Representation
American Standard Code for Information Interchange (ASCII)
ASCII is an international standard for alphanumeric characters (letters and numbers), and symbols (- + & etc.) contained within a text file
Essentially, all the things you could type from a standard keyboard
128 Characters
Extended ASCII provides more characters
256 characters
Unicode – A further extension of ASCII
ASCII (American Standard Code for Information Interchange) is the most common format for text files in computers and on the Internet. In an ASCII file, each alphabetic, numeric, or special character is represented with a 7-bit binary number (a string of seven 0s or 1s). 128 possible characters are defined.

21. ASCII
7-bit
128 combinations

22. Extended ASCII
8-bit
256 combinations

23. Unicode Extended ASCII is not enough for international use
One Unicode mapping uses 16 bits per character
65363 Combinations
Unicode is a superset of ASCII
The first 256 characters correspond exactly to the extended ASCII character set

24. Unicode Table (Example)