# Numbering Systems and Codes 222

The class “Numbering Systems and Codes” explains the numbering systems used with PLCs, as well as the process for converting between different numbering systems. PLCs use numbering systems to process data and perform calculations. These systems include the decimal system, binary system, octal system, and hexadecimal system. PLC operators also use codes based on numbering systems, such as binary coded decimal, when entering information into an input/output module on a PLC.

After taking this course, users will be familiar with the characteristics and conversion processes for numbering systems used with a PLC. This helps operators understand the internal operations of a PLC, which may simplify troubleshooting, reduce downtime, and improve productivity.

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## Class Details

Class Name:
Numbering Systems and Codes 222
Difficulty:
Intermediate
Number of Lessons:
21

## Class Outline

• Numbering Systems and PLCs
• Numbering Systems
• The Decimal System
• Counting in the Decimal System
• The Binary System
• Numbering System Review
• Converting Binary to Decimal
• Converting Decimal to Binary
• Negative Binary Numbers
• Binary System Calculations Review
• The Octal System
• Octal and Decimal Conversions
• Octal and Binary Conversions
• Octal and Hexadecimal Systems Review
• Codes
• PLC Interfaces and Codes
• Final Review

## Objectives

• Describe numbering systems and PLCs.
• Describe the main characteristics of numbering systems.
• Describe the decimal system.
• Describe the binary system.
• Explain how to convert binary numbers to decimal numbers.
• Explain how to convert decimal numbers to binary numbers.
• Explain how to add binary numbers.
• Describe how to represent negative numbers in binary.
• Describe the octal system.
• Explain how to convert between octal numbers and decimal numbers.
• Explain how to convert between octal numbers and binary numbers.
• Explain how to convert between hexadecimal numbers and decimal numbers.
• Explain how to convert between hexadecimal numbers and binary numbers.
• Describe the use of codes in PLCs.
• Describe how codes are used with PLC interfaces.

## Glossary

Vocabulary Term Definition
2's complement A method of representing negative binary numbers by placing a 1 signed bit in front of the MSD, inverting the subsequent digits, and adding a 1 to the LSD. The 2's complement method is the most common method of representing negative numbers in the binary system.
addresses The unique identification of the location of the data for an input device, output device, or other saved data in the CPU's memory. Addresses may be displayed using numbers in the decimal system.
alphanumeric Consisting of letters and numbers. Alphanumeric codes are entered into PLC programming devices.
American Standard Code for Information Interchange ASCII. An alphanumeric code that uses 7 bits in various combinations to represent up to 128 different letters, numbers, and symbols. The American Standard Code for Information Interchange is used to transfer data to and from computers.
ASCII American Standard Code for Information Interchange. An alphanumeric code that uses 7 bits in various combinations. ASCII is used to transfer data to and from computers.
automate Using mechanical systems that operate with limited human interaction. PLCs can be used to automate a manufacturing process.
base The amount of unique numbers or symbols used to represent numerical quantities in a numbering system. The base, or radix, indicates the range of a numbering system.
base 10 system A method of mathematical notation that uses combinations of the numbers 0–9 to represent all numerical quantities. In the base 10 system, or decimal system, each successive place value column from right to left is 10 times greater than the previous place value column.
base 16 system A method of mathematical notation that uses combinations of the numbers 0–9 and letters A–F to represent all numerical values. In the base 16 system, or hexadecimal system, each successive place value column from right to left is 16 times greater than the previous place value column.
base 2 system A method of mathematical notation that uses combinations of the numbers 0 and 1 to represent all numerical quantities. In the base 2 system, or binary system, each successive place value column from right to left is 2 times greater than the previous place value column.
base 8 system A method of mathematical notation that uses combinations of the numbers 0–7 to represent all numerical quantities. In the base 8 system, or octal system, each successive place value column from right to left is 8 times greater than the previous place value column.
BCD Binary coded decimal. A code system that uses series of four bits to represent the decimal numbers 0–9. BCD was developed as an easy method of converting decimal numbers to a binary format.
binary coded decimal BCD. A code system that uses series of four bits to represent the decimal numbers 0–9. Binary coded decimal was developed as an easy method of converting decimal numbers to a binary format.
binary system A method of mathematical notation that uses combinations of the numbers 0 and 1 to represent all numerical quantities. In the binary system, or base 2 system, each successive place value column from right to left is 2 times greater than the previous place value column.
bits A single binary digit. A bit is represented by a 1 or a 0 in the base 2, or binary, system.
byte A group of eight binary bits. A byte is represented by a single digit in the base 8, or octal, system.
central processing unit CPU. The microprocessor device inside a computer that controls system activities. The central processing unit executes a PLC program and performs all logic activities for the system.
codes Any system that uses numbers or symbols to represent information other than simple numerical quantities. Codes allow operators to enter information into a PLC.
complement To reverse binary digits, with all 1s becoming 0s and all 0s becoming 1s. Complementing, or inverting, is a way to represent negative binary numbers.
CPUs Central processing unit. The microprocessor device inside a computer that controls system activities. A CPU executes a PLC program and performs all logic activities for the system.
data A collection of numbers, information, and values about a process or product. Data is transmitted between computers and machines.
decimal system A method of mathematical notation that uses combinations of the numbers 0–9 to represent all numerical quantities. In the decimal system, or base 10 system, each successive place value column from right to left is 10 times greater than the previous place value column.
discrete input module A PLC component that receives high-voltage input signals from discrete input devices. A discrete input module receives two-state electrical signals.
electrical signals Voltage, current, or frequency that communicates information between two devices. Electrical signals are either analog or discrete.
hexadecimal system A method of mathematical notation that uses combinations of the numbers 0–9 and letters A–F to represent all numerical quantities. In the hexadecimal system, or base 16 system, each successive place value column from right to left is 16 times greater than the previous place value column.
hexadecimal system A method of mathematical notation that uses combinations of the numbers 0–9 and letters A–F to represent all numerical values. In the hexadecimal system, or base 16 system, each successive place value column from right to left is 16 times greater than the previous place value column.
HMI Human machine interface. A peripheral device that connects to a PLC and allows an operator to monitor a program and interact with the PLC. An HMI may have a graphical or textual display.
human machine interface HMI. A peripheral device that connects to a PLC and allows an operator to monitor a program and interact with the PLC. A human machine interface may have a graphical or textual display.
input/output devices I/O devices. A component that connects to an input/output module on a PLC and performs an action. Each input/output device, or field device, correlates to a bit in a PLC register.
input/output modules I/O modules. The portion of a PLC that contains connection points for input/output devices and circuity to convert electrical current. An input/output module may be input only, output only, or a combination.
interface A hardware configuration that interconnects devices for the purposes of communication. Interfaces for PLCs include thumbwheels and human machine interfaces.
invert To reverse binary digits, with all 1s becoming 0s and all 0s becoming 1s. Inverting, or complementing, is a way to represent negative binary numbers.
least significant digit LSD. The digit with the lowest place value in a number. The least significant digit is the rightmost column of the number.
LSD Least significant digit. The digit with the lowest place value in a number. The LSD is the rightmost column of the number.
memory The portion of a computer where data is stored. Memory storage can be built into the CPU of a PLC or in the form of a removable device.
most significant digit MSD. The digit with the highest place value in a number. The most significant digit is the leftmost column of the number.
MSD Most significant digit. The digit with the highest place value in a number. The MSD is the leftmost column of the number.
negative numbers A unit representing a value less than zero. Negative numbers are shown in the decimal system by a minus sign, but the binary system may use a 1 signed bit in front of the most significant digit to show a negative number.
network A group of hardware devices connected through a communication link. Networks allow multiple devices to communicate and share information.
nibble A group of four binary bits. A nibble can be represented by one digit in the base 16, or hexadecimal, system.
numbering system A method of mathematical notation that uses a range of numbers or other symbols arranged in a series of place value columns to represent specific numerical quantities. The most common numbering systems are decimal, binary, octal, and hexadecimal.
octal system A method of mathematical notation that uses combinations of the numbers 0–7 to represent all numerical quantities. In the octal system, or base 8 system, each successive place value column from right to left is 8 times greater than the previous place value column.
place value A numerical value assigned to a specific column in a number. The place value is determined by the numbering system's base.
PLC Programmable logic controller. A processor-driven device that uses logic-based software to provide electrical control to machines and processes. PLCs replace many physical relays in a process.
power A number that indicates how many times a number is multiplied by itself. A power can be shown as a smaller number placed above and to the right of a number.
programmable logic controllers PLC. A processor-driven device that uses logic-based software to provide electrical control to machines and processes. Programmable logic controllers replace many physical relays in a process.
programming device An instrument attached to a PLC that is used to enter programs into a CPU's memory. Programming devices include personal computers and handheld devices.
quotient The result of a division operation. The quotient of the division problem 24 divided by 2 is 12.
radix The amount of unique numbers or symbols used to represent numerical quantities in a numbering system. The radix, or base, indicates the range of a numbering system.
range The span of available numbers or symbols in a numbering system. Ranges are described from smallest value to highest value.
registers A data file in a PLC that stores bit files. Registers are available in different bit sizes, but 8-bit and 16-bit are common.
remainders r. The quantity left over when one number is divided by another. Remainders are used to maintain whole numbers.
signed bit The leftmost digit in a binary number that indicates whether the number is positive or negative. Signed bits are also used in the 2's complement method.
superscript A small number, letter, or symbol that appears above and to the right of a larger number, letter, or symbol. Superscript numbers are used to indicate mathematical powers.
thumbwheel An interface device that allows an operator to enter numerical information into a PLC input module. A thumbwheel inputs decimal numbers as binary coded decimal numbers.
Unicode An international encoding standard that uses 16 bits in various combinations to represent up to 65,536 different characters. Unicode is designed to be used with different languages and platforms.
units A single number, character, or symbol that is used to represent a numerical quantity in a numbering system. Units in numbering systems may be numbers or letters.
word A group of 16 binary digits. A word can represent decimal values ranging from 0–65,535.