Computers and Math Binary, Hex, and Decimal Oh My Copyright © Texas Education Agency, 2015. All...

Computers and MathBinary, Hex, and Decimal Oh My

BACK TO THE FOUNDATIONA brief review of decimal notation

Numbers Review

a. Natural numbers

b. Integers

c. Positional numbering system

d. Number base

Exponents

Any number to the 0 power = 1

Example: 20 = 1

Any number to the 1st power = that number

Example: 21 = 2

Decimals

7 2 57

one hundreds

2 tens

5 ones

Used with permission: Robyn Segrest May 12, 2015.

7 * 102 = 7 * 100 = + 700

2 * 101 = 4 * 10 = + 20

5 * 100 = 5 * 1 = + 5

CONVERTING FROM BINARY TO DECIMAL

11112 = 1510

Binary

• Base 2

• Uses 0 and 1

Binary Table

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

This is where binary digits (0s and 1s) go

Place holder value

Decimal equivalent

Convert 000011102

from binary to decimal

000011102 = 1410

128 * 0 = 0

64 * 0 = 0

32 * 0 = 0

16 * 0 = 0

8 * 1 = 8

4 * 1 = 4

2 * 1 = 2

1 * 0 = 0 +

0 0 0 0 1 1 1 0

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 0

YOUR TURNBinary to Decimal

CONVERTING FROM DECIMAL TO BINARY

1410 = 11102

Binary Conversion

Can be done two different ways– Long division– Subtraction

Long Division

Convert 3510 to binary

Binary bits

01000112 = 3510

Subtraction

3710 = 001000112

37 - 32 = 3

3 – 2 = 1

1 – 1 = 0

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

Convert 3710 from decimal to binary

YOUR TURNDecimal to Binary

HEXADECIMAL BASE 16

F16 = 1510

Hexadecimal

Characters– 0 – 9– A – F

Can convert directly from hex to decimal but not from decimal to hex

Hexadecimal (Hex) Binary Decimal

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

A 1010 10

B 1011 11

C 1100 12

D 1101 13

E 1110 14

F 1111 15

Hexadecimal Table

165 164 163 162 161 160

1048576 65536 4096 256 16 1

This is where hexadecimal characters (0 - F) would go

Place holder value

Decimal equivalent

Convert FF16 from to decimal

FF16 = 25510

15 * 16 = 240

15 * 1 = +15

163 162 161 160

65536 4096 16 1

YOUR TURN

Hexadecimal to Binary

DECIMAL TO HEX CONVERSION

Convert 5410 to Hexadecimal

54- 32 = 2222 – 16 = 66 – 4 = 22 – 2 = 0 5410 = 0011 01102

0011 01103 6

5410 = 3616

Convert from decimal to binary

Convert from binary to hex

Convert 12810 to Hexadecimal

128 - 128 = 012810 = 1000 00002

1000 00008 0

12810 = 8016Binary to hex

Decimal to binary

YOUR TURN

Decimal to Hexadecimal

IP ADDRESSING

Every node on a network must have a unique IP address: Internet Protocol version 4

• Unique 32-bit number– Divided into four octets divided by decimal points

» EX: 192.168.0.3

• Separated into classes

Commonly Used IPv4 Classes

Used with permission: Robyn Segrest March 29, 2015.

Class First Octet Shared Octets

Number of Networks

Maximum Addressable

A 1 – 126 1 126 16,777,214

B 128 – 191 2 > 16,000 65,534

C 192 – 223 3 > 2,000,000 254

Highest decimal number an octet may be is 255– Each octet is equal to eight binary bits

• 111111112 = 25510

Running out of IPv4 addresses due to 32-bit limitation

Combined with a ‘subnet mask’ to increase number of addressable nodes on a network

Subnet Mask

Used with permission: Robyn Segrest March 29, 2015.

Class First Octet Default Subnet Mask

A 1 – 126 255.0.0.0

B 128 – 191 255.255.0.0

C 192 – 223 255.255.255.0

Composed of 128 bits– Eight 16-bit fields

• Represented by hexadecimal numbers– Divided into groups of four hexadecimal characters

» EX: FEE3:00FF:003D:0000:0000:0000:3015:AABC– Multiple fields with zero values can be abbreviated

» EX: 00EE – EE» 0000 - 0

Assigning IP Addresses

Can be done manually or by DHCP (Dynamic Host Configuration Protocol)

– Static IP address• Manually assigned• Does not change• Human error in duplicating addresses can cause connection

issues

– Dynamic IP addressing• Assigned automatically by a DHCP server• Most common and simplest method

Boolean Operators

Also called Logical Operators or just bool

Logical operators that return true or false

Three basic Boolean operators

Order of Boolean operations

Truth Tables

ANDInput 1 Input 2 Output

0 0 00 1 01 0 01 1 1

ORInput 1 Input 2 Output

0 0 00 1 11 0 11 1 1

!ANDInput 1 Input 2 Output

0 0 10 1 11 0 11 1 0

!ORInput 1 Input 2 Output

0 0 10 1 01 0 01 1 0

ANDING

Every IP address has a default subnet mask– Class A – 255.0.0.0– Class B – 255.255.0.0– Class C – 255.255.255.0

In order to locate a specific node on the network a computer must AND the IP address with the appropriate subnet mask in binary

Based on our truth tables we should know the following– 1 AND 1 = 1– 1 AND 0 = 0– 0 AND 1 = 0– 0 AND 0 = 0

Finding the Network Address

EXAMPLE:• IP address: 192.168.0.10• Subnet mask: 255.255.255.0

First we convert both to binary:

IP address: 11000000.10101000.00000000.00001010 (192.168.0.10)

Subnet Mask: 11111111.11111111.11111111.00000000 (255.255.255.0)

AND: 11000000.10101000.00000000.00000000 (192.168.0.0)

The result of ANDing is the network address.

YOUR TURN

Converting IP Addresses

Computers and Math Binary, Hex, and Decimal Oh My Copyright © Texas Education Agency, 2015. All...

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