Basic Logic 2.1

Basic Numbering Systems

Technician Series

©Paul GodinUpdated Dec 2014

prgodin @ gmail.com

Basic Logic 2.2

Numbering Systems

Binary numbers represent a value, and it is important to express these into values that we understand.

Binary numbers need to be converted to decimal and hexadecimal values to make them easier to describe and understand.

Basic Logic 2.3

Converting Binary to Decimal

Each position represents a “weight”

25+24+21= 32+16+2 = 50 in decimal

1100102

20

21222324

25

Each position has a weight that is a factor of 2. Add all positions that have a binary “1” (ignore “0”)

Basic Logic 2.4

Short-Cuts

◊ You may be able to calculate some values in your head if you remember the weight of each position:

_ _ _ _ _ _ _ _ _ _

20

1

27

128

22

423

824

1625

3226

64

21

228

256

29

512

Basic Logic 2.5

Short-Cuts Examples

Value remembered

0010 = 2

1000 = 8

10000 = 16

Value determined through remembered value.

0011 is one more than 0010 0011 = 2 +1 = 3

1011 is three more than 1000 1011 = 8 +3 = 11

01111 is one less than 10000 01111 = 16 - 1 = 15

0011

1011

1111

Number

Basic Logic 2.6

Exercise 1

0101 = _________

1001 = _________

10101 = ________

110 = ___________

11011 = _________

1011011 = ________

Convert the following Binary numbers to Decimal:

Basic Logic 2.7

Converting Decimal to BinaryPowers of 2 Method

Powers of 2: The decimal value is successively reduced by the largest power of 2 that will fit.

57 – 32 = 25 32 is position #6

_ _ _ _ _ _

25 – 16 = 9 16 is position #5

9 – 8 = 1 8 is position #4

1 – 1 = 0 1 is position #1

Example: Convert 57 to binary

Blank positions become 0

1 1 1 10 0

Animated

Basic Logic 2.8

Converting Decimal to BinaryDivision by 2 Method

Division by 2: The decimal value is successively divided by 2 and the remainder is recorded.

57 ÷ 2 = 28, remainder 1

_ _ _ _ _ _

28 ÷ 2 = 14, remainder 0

Example: Convert 57 to binary

1 1 1 10 0Animated

14 ÷ 2 = 7, remainder 07 ÷ 2 = 3, remainder 13 ÷ 2 = 1, remainder 1

1 ÷ 2 = 0, remainder 1

Basic Logic 2.9

Exercise 2

12 = _________

17 = _________

124 = ________

8 = ___________

33 = _________

155 = ________

Convert the following Decimal numbers to Binary:

Basic Logic 2.10

Converting Hexadecimal to Decimal

Each position represents a “weight”

11x163 + 0 + 3x161 + 2x160 =

B03216

160

161162

163

Each position has a weight that is a factor of 16. Multiply the value (converted to decimal) by its weight.

(11 x 4096) + (3x16) + (2x1) =

45,056 + 48 + 2 = 45,106

Basic Logic 2.11

Converting Decimal to HexadecimalDivision by 16 Method

Division by 16: The decimal value is successively divided by 16 and the remainder is recorded.

45,106 ÷ 16 = 2819, remainder 2

_ _ _ _

2819 ÷ 16 = 176, remainder 3

Example: Convert 45,10610 to Hexadecimal

B 20 3Animated

176 ÷ 16 = 11, remainder 011 ÷ 16 = 0, remainder 11

Basic Logic 2.12

Exercise 3

1210 = _________16

1710 = _________16

12410 = ________16

1816 = ___________10

3B16 = _________10

ABC16 = ________10

Convert the following numbers:

Basic Logic 2.13

Converting Hexadecimal to Binary

The Hexadecimal numbering system is used because of the ease converting between it and a 4-bit binary value. Each digit is converted to Decimal then converted to Binary.

Example: Convert B032h to Binary

B03216

_ _ _ _ _ _ _ _

1110 010 310 210

01 10_ _ _ _0 10 000 001_ _ _ _0 112

Basic Logic 2.14

Converting Binary to Hexadecimal

To convert from Binary to Hexadecimal, convert each grouping of 4 bits to Decimal, then to a Hexadecimal digit.

Example: Convert 1011 0000 0011 0010 to Hexadecimal

B03216

1110 010 310 210

01 100 10 000 0010 112

Basic Logic 2.15

Counting in Binary

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

Basic Logic 2.16

Binary Counting

When counting up, 1 is added to the current state to determine the next state.

In Binary, when adding “1” to a “1”, the result is “10” (zero and carry the one).

Example:

1011

111

100

Basic Logic 2.17

Adding in Binary

◊ Example:

1101+0101

1101+0101

0

11101

+010110

11101

+0101010

11

1101+010110010

11

Basic Logic 2.18

Adding in Binary

101+011

1 + 1 = 10

0

1 Carry 1

1 + 1 = 100

1

Carry 11 + 1 = 10

0

1

1

Carry 1

Animated

Basic Logic 2.19

Binary Codes

◊ Binary numbers are used to represent values such as alphanumeric characters.

◊ One encoding system is ASCII, a standard that defines the representation of an alphanumeric character using binary.

◊ Example: The letter A in Binary is 100 0001

Note ASCII code is 7 bit but there are several 8 bit codes available.

Basic Logic 2.20

Exercise

Determine what the following says (7-bit ASCII):

010010010010000001010111011010010110111000100001

Google for it.

Basic Logic 2.21

Grey Code

When counting in Binary there are instances where two or more bits change at the same time. Examples: 001 to 010, 011 to 100.

In mechanical systems, such as those using an encoding wheel, it is desirable to have just one bit change at a time to allow for more uniform transitions with rotation.

In Digital Electronic communications, these multi-bit changes may interfere with one another.

Encoder Disk(Wikipedia - PD)

Basic Logic 2.22

Grey Code

000001011010110111101100

00011110

0000000100110010011001110101010011001101111111101010101110011000

2-b

it G

rey C

od

e

4-b

it G

rey C

od

e

3-b

it G

rey C

od

e

Basic Logic 2.23

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©Paul R. Godinprgodin°@ gmail.com