Number Systems

Decimal Numbers

Decimal stands for 10

Decimal is the base 10 Number System

It has 10 unique symbols

0 1 2 3 4 5 6 7 8 9

The largest symbol is the base – 1

10 – 1 = 9

9 is the largest symbol in Decimal numbers

Decimal Numbers

1000 100 10 units

4 3 5 6

4000

300

50

+ 6

435610

Decimal NumbersWeight origin 103 102 101 100

Weights 1000 100 10 1

Number MSD 6 3 6 4 LSD

6000

300

60

+ 4

636410

Binary Numbers

Binary stands for 2Binary is the base 2 number systemIt has two unique symbols

0 and 1The largest symbol in Binary is the base – 1

2 – 1 = 11 is the largest symbol in the Binary Number system

Binary Numbers

Weight origin 23 22 21 20

Weights 8 4 2 1

Number 1 0 1 12

8

0

2

+ 1

1110

Counting in BinaryWeight 8 4 2 1 decimal

0 0 0 0 0

0 0 0 1 1

0 0 1 0 2

0 0 1 1 3

0 1 0 0 4

0 1 0 1 5

0 1 1 0 6

0 1 1 1 7

1 0 0 0 8

1 0 0 1 9

1 0 1 0 10

1 0 1 1 11

1 1 0 0 12

1 1 0 1 13

1 1 1 0 14

1 1 1 1 15

Converting from Decimal to BinaryThe decimal number 55 = ____ In binary

55 ÷ 2 = 27.5* 1 LSB

27 ÷ 2 = 13.5 1

13 ÷ 2 = 6.5 1

6 ÷ 2 = 3.0 0

3 ÷ 2 = 1.5 1 1 ÷ 2 = 0.5 1 MSB

110111 = 5510

* If the remainder is .5 place a 1 to the right. If 0 place 0 to right

Octal Number System

Octal stands for Eight

Octal is the base 8 Number System

It has 8 unique symbols

0 1 2 3 4 5 6 7

The largest symbol is the base – 1

8 – 1 = 77 is the largest symbol in the Octal number system

Octal Number System

Weight origin 83 82 81 80 Weights 512 64 8 1

Number 2 1 6 78

1024 64 48 + 7

114310

Hexadecimal Number System

Hex = 6 and Decimal = 10 6+10 = 16

Hex is the base 16 Number System

It has 16 symbols

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

It must have 16 UNIQUE Symbols

0 1 2 3 4 5 6 7 8 9 A B C D E F

The largest symbol is 16 – 1 = 15 or F

Hexadecimal Number System

Weight origin 163 162 161 160

Weights 4096 256 16 1

Number 1 2 A F 16

4096

512

160

+ 15

478310

Octal Conversions

The largest symbol in Octal is 7

It requires three binary bit to represent the number 7 1112

To convert a binary number to Octal, start with the LSB and mark off groups of 3.

101011010010 LSB

101-011-010-010 Write Octal symbol below

5 3 2 28

Octal Conversions

To convert an Octal number to binary:

Write the three bit binary equivalent below

1 5 6 48

001 101 110 100 = 11011101002

Leading zeros can be left off.

Convert Binary to Hex

The largest symbol in Hex is 15 or F

It requires four binary bits to represent the number 15 or the symbol F 11112

To convert a binary number to Hex, start with the LSB and mark off groups of 4.

1011001110101110LSB

1011-0011-1010-1110 Write Hex symbols below

B 3 A E 16

Convert Hex to Binary

To convert Hex to Binary, write the 4 bit Binary equivalent below each Hex digit.

A 6 3 F 16

1010 0110 0011 1111 = 10100110001111112