Number Systems and Codes

Chapter

3Number Systems and

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Objectives

• Explain the decimal number system.• Explain integer numbers, fractional

numbers, and real numbers.• Describe the binary number system.• Convert decimal numbers to binary and vice

versa.• Perform binary math operations such as

add, subtract, multiply, and divide.

Objectives

• Explain the hexadecimal number system.• Convert binary numbers to hexadecimal

numbers and vice versa.• Explain the octal number system.• Convert binary numbers to octal numbers

and vice versa.• Explain the binary coded decimal number

system and its use.

Objectives

• Convert decimal numbers to binary coded decimal.

• Explain the Gray code number system and its use.

• Explain ASCII and EBCDIC alphanumeric codes.

Decimal Number System

• Decimal number system:– Has ten (10) digits running from zero (0)

through nine (9). – Decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8,

and 9.– Each digit's value depends on the

position of the digit within the number.

Decimal Number System (Cont.)

• Base, also called the radix:– Multipliers for the digits are numbers

constructed of the number system (10 in the decimal system) raised to the appropriate power.

– 10a

10 = Base or radixa = Power

• Values for the power of ten (10) numbers.

Example:6310 = (6 × 103) + (3 × 102) + (1 × 101) + (0 × 100)= (6 × 1000) + (3 × 100) + (1 × 10) + (0 × 1)= 6000 + 300 + 10 + 0= 6310

Fractional Numbers

• Fractional numbers:– Have weights that are negative powers of

ten (10).

Example: 0.812 = (8 × 10-1) + (1 × 10-2) + (2 × 10-3) = (8 × 0.1) + (1 × 0.01) + (2 × 0.001) = 0.8 + 0.01 + 0.002 = 0.812

Real Numbers

• Real numbers:– Have weights that are both positive and

negative powers of ten (10).– Integer parts.– Fractional parts.

• Value of digits:– Digits.– Digits' positions.

Real Numbers (Cont.)

Example:547.165 = (5 × 102) + (4 × 101) + (7 × 100) + (1 ×

10–1) + (6 × 10–2) + (5 × 10–3)= (5 × 100) + (4 × 10) + (7 × 1) + (1 ×

0.1) + (6 × 0.01) + (5 × 0.001)= 500 + 40 + 7 + 0.1 + 0.06 + 0.005= 547.165

Real Numbers (Cont.)

• Digits to the left of the decimal point represent the integer number.

• Digits to the right of the decimal point represent the fractional number.

Integer Numbers

• Integer numbers:– Have weights that are positive powers of

ten (10).– Can be positive or negative.– Zero is an integer number.

Binary Number System

• Binary number system:– Has a base, also called a radix, of two (2).– Only the digits 1 and 0 are present.

• All binary numbers are created from combinations of these two digits.

Binary Number System (Cont.)

• Least significant bit (LSB):– Binary digit to the far right-hand side of

the number.– Rightmost number has the smallest

value.• Most significant bit (MSB):

– Binary digit to the far left-hand side.

Binary-to-Decimal Conversion

• Converting binary-to-decimal:– Convert each bit to its equivalent decimal

value. – Add all of these decimal values.

Decimal-to-Binary Conversion

• Convert integer part of a decimal number using:– Repeated division.– Successive division.

• Convert fractional part using:– Repeated multiplication.– Successive multiplication.

Decimal-to-Binary Conversion (Cont.)

• Successive division method:– Integer decimal number is divided by two,

repeatedly.– Remainders make up the new binary

number.

Decimal-to-Binary Conversion (Cont.)

• Successive multiplication method:– Convert the fractional part of a decimal

number to binary.• MSB of the binary number:

– Decimal number is multiplied by two and the portion that is carried (if the product is greater than 0).

– Process is continued until the decimal part of the number becomes zero.

Binary Addition

• Decimal number system:– Carry generated when the sum of two

digits is equal to 10.• Binary number system:

– Carry generated when the sum of two binary bits is equal to two.

Binary Addition (Cont.)

• Ways that binary bits are added and how the carry bit is generated:0 + 0 = 00 + 1 = 11 + 1 = 0 with carry of 11 + 1 + 1 = 1 with carry of 1

Binary Subtraction

• In the decimal number system:– When the value of a digit is being

subtracted (the subtrahend) is greater than the value of the digit that it is being subtracted from (the minuend), a borrow from the next higher column (the column to the left) is required.

Binary Subtraction (Cont.)

• Subtraction of binary numbers: – Ways binary bits are subtracted and

when a borrow bit is required:0 – 0 = 0 no borrow1 – 1 = 0 no borrow1 – 0 = 1 no borrow(1) 0 – 1 = 1 with a borrow of 1

Binary Multiplication

• In multiplication, the multiplicand is multiplied by each of the multiplier digits.– Results of each higher multiplier digit are

shifted one place to the left. – When each of the multiplicand’s digits

have been multiplied by the multiplier digits, all of the numbers are added.

Binary Multiplication (Cont.)

• How binary bits are multiplied:0 × 0 = 00 × 1 = 01 × 0 = 01 × 1 = 1

Binary Division

• Division:– How many times a number called the

divisor can be subtracted from another number called the dividend.

– Similar to decimal division.

Binary Division (Cont.)

• Subtraction routine:– Continues until the remainder is less than

the dividend.• Quotient:

– Number of times the divisor can be subtracted from the dividend.

Hexadecimal Number System

• Hexadecimal number system:– Used in medium to larger programmable

logic controllers (PLCs).– Often just called hex.– Used to display the PLC memory map

and to carry out the math instructions.– Created by grouping four binary bits.– Hexadecimal numbers use base of 16

and require fifteen digits.

Hexadecimal Number System (Cont.)

– Uses the numbers zero (0) through nine (9) for the first ten digits.

– Ten (10) through fifteen (15) are represented by letters A through F. • Digits representing 10, 11, 12, 13, 14, and 15

are displayed as A, B, C, D, E, and F respectively.

– Capital letter H or the small letter h is placed at the end of a hexadecimal number.

Hexadecimal Binary Decimal0 0000 01 0001 12 0010 23 0011 34 0100 45 0101 56 0110 67 0111 78 1000 89 1001 9A 1010 10B 1011 11C 1100 12D 1101 13E 1110 14F 1111 15

• To form hex numbers from binary numbers, group binary number bits in groups of four.

• Swap out each group of four binary bits with its hex equivalent.

Example:111100010111 becomes 1111 0001 0111 which is

transformed to F 1 7 or F13h.

• Reverse the process to convert to binary.

• Take each hex digit and swap it with the binary 4-bit equivalent.

Example:The number 37Ah becomes 0011 0111 1010 or

001101111010.

Octal Number System

• Octal number system:– Has a base of eight.– Therefore, only the digits zero (0) through

seven (7) are present.– Used to display the memory maps and

carry out the math instructions in some MicroLogix controllers.

Octal Number System (Cont.)

Octal Binary Decimal 0 000 0

1 001 1 2 010 2

3 011 3 4 100 4 5 101 5 6 110 6 7 111 7

• Form an octal number from a binary number by grouping binary number bits in groups of three.

• Swap out each group of three binary bits with its octal equivalent octal digit.

Example:• The number 11001111 becomes 011 001 111

which is transformed to 3 1 7 or 317O.

• Reverse the process to convert to binary.– Take each octal digit and swap it with the

binary 3-bit equivalent.

Example:• The number 246 becomes 010 100 110 or

10100110.

• The capital letter O is placed at the end of an octal number.

Binary Coded Decimal (BCD) System• Each 4 bits of binary numbers are

grouped to form a decimal number.• Four bit groups representing 10, 11,

12, 13, 14, and 15 are not valid numbers in the BCD number system.

• Only digits zero (0) through nine (9) are valid.

Binary Coded Decimal (BCD) System (Cont.)

Binary Coded Decimal (BCD) Decimal 0000 0 0001 1

0010 2 0011 3 0100 4 0101 5 0110 6

0111 7 1000 8

1001 9

Binary Coded Decimal (BCD) System (Cont.)

• To convert a number from any number system to BCD, first convert the number to decimal.

• Then, represent each decimal digit with a 4-bit binary number code.

Gray Code

• Servo control:– Closed-loop industrial control system that

is used to control the speed and position of an electric motor shaft.

• Optical encoders:– Generate codes that represent the

angular positions of the motor shafts.– Usually in a format called Gray code.

Gray Code (Cont.)

• Gray code system:– Similar to binary. – Converted to binary prior to arithmetic

operation. – Used for detecting the position of a motor

shaft. – Only one digit changes state (turns on or

off) as a number increments or decrements.

Gray Code (Cont.)

• Gray code system (continued):– Speed of the operation of a control

system increases.• Only one digit changes with every rotation of

the optical encoder disc attached to the motor shaft.

– Difficulty in doing arithmetic operations.

Gray Code (Cont.)Gray code Binary Decimal

0000 0000 0 0001 0001 1 0011 0010 2 0010 0011 30110 0100 40111 0101 50101 0110 60100 0111 71100 1000 81101 1001 9 1111 1010 10 1110 1011 11 1010 1100 12 1011 1101 13 1001 1110 141000 1111 15

Alphanumeric Code

• Alphanumeric codes represent:– Letters– Numbers– Special symbols

American Standard Code for Information Interchange (ASCII)• 7-bit code• Used to represent alphabets, numbers,

characters, and control codes in computer systems.

• One hundred and twenty eight (27 = 128) different characters can be represented.

• Data is often sent in bytes (groups of eight bits).

• Eighth bit in one byte of transmitted data used for a parity bit.

Extended Binary Coded Decimal Interchange Code (EBCDIC)

• 8-bit code. • Alphanumeric code commonly encountered in

computer mainframe equipment.• Two hundred and fifty six (28 = 256) different

characters can be represented.– Uppercase and lowercase.– Numbers (1 thru 9).– Punctuation, control codes, and other characters.

• Commonly called Extended ASCII.

Alphanumeric Codes

Control Codes

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