Introduction to Number Systems: Decimal, Binary, Octal and Hexadecimal Explained

Why Do We Even Need More Than One Number System?

Suppose someone handed you a light switch and asked you to count how many times you flicked it. You'd naturally use decimal, 1, 2, 3, and so on. Now suppose that switch is actually a transistor inside a PLC, switching millions of times a second. Decimal stops being useful, and a completely different way of counting takes over.

We count in tens because, quite literally, we have ten fingers. That's decimal, base 10, and it's so natural to us that we rarely stop to think of it as a choice at all. But a PLC, a microcontroller, or any digital circuit doesn't have fingers. It has transistors, and a transistor only really understands two states clearly: current flowing, or current not flowing. On or off. That's it. So computers and PLCs settled on a number system built around exactly two digits, binary, base 2.

Here's the catch. Binary numbers get long, fast. A decimal number like 200 becomes 11001000 in binary, eight digits just to represent three. Now imagine reading pages of PLC memory addresses like that. It would be exhausting and genuinely error prone. So engineers came up with two shortcuts, octal (base 8) and hexadecimal (base 16), specifically because they compress binary into something a human can actually read comfortably, while still translating back to binary perfectly, digit for digit.

Decimal: The System You Already Know By Heart

Decimal is a positional system. That just means where a digit sits matters as much as what the digit actually is. Take the number 453. The 4 isn't just "four", it's four hundreds. The 5 is five tens. The 3 is three ones. Written properly, that's (4 × 10²) + (5 × 10¹) + (3 × 10⁰). Every number system we're about to look at, binary, octal, hexadecimal, works on this exact same positional idea. Only the base changes.

Binary: What a PLC Actually Understands

Binary uses only two digits, 0 and 1. Each one is called a bit, short for binary digit. Group four bits together and you get a nibble. Group eight bits, or two nibbles, and you get a byte. Sixteen bits, or two bytes, make up a word, which happens to be exactly the register size most PLCs use internally. Thirty two bits make a double word.

For example, a single 16-bit PLC word filled entirely with 1s represents the decimal number 65,535. That's the practical ceiling of what one basic PLC register can hold, and it's exactly why programmers care about data type sizes when storing large counts or timer values.

Watch: Number Systems Introduction, Decimal, Binary, Octal and Hexadecimal

This video walks through the same conversions covered above with additional worked examples.

Octal: Binary's Older, Simpler Shorthand

Octal groups binary digits into sets of three, since 2³ equals 8, matching octal's eight digits perfectly. This made octal genuinely convenient in early computing, when many systems naturally worked in groups of 12 or 24 bits. Some PLC families still carry this legacy today, for instance AutomationDirect's DirectLOGIC series addresses its memory in octal, a detail that surprises engineers used to newer hex-based systems.

Hexadecimal: The System Most Modern PLCs Actually Speak

Hex groups binary into sets of four, since 2⁴ equals 16. That's a perfect match to a nibble, which is exactly why hex became the go-to shorthand in modern computing and automation. Brands like Allen-Bradley and Mitsubishi lean heavily on hexadecimal for memory addressing and data display, precisely because one hex digit maps cleanly onto exactly four binary digits, no remainder, no awkward grouping.

BCD: A Fifth System Worth Knowing, Even If It's Not a "Real" Base

Have you ever looked at a seven-segment display on an old PLC panel and wondered how it shows plain decimal digits when the processor underneath only speaks binary? That's Binary Coded Decimal, or BCD, doing its job. BCD isn't really a new base at all. It takes each individual decimal digit, one at a time, and encodes just that single digit using 4 bits.

Suppose we want to represent decimal 165 in BCD. We don't convert the whole number to binary at once. Instead, each digit gets its own 4-bit group: 1 becomes 0001, 6 becomes 0110, and 5 becomes 0101. Strung together, that's 000101100101. Compare that to true binary, where 165 is simply 10100101, noticeably shorter. BCD trades that compactness for something else, human readability, since each group maps directly back to a familiar decimal digit without any extra conversion step.

There's a genuine gotcha here worth remembering. BCD only allows the bit patterns for 0 through 9 in each 4-bit group. The patterns for 10 through 15, the same ones hexadecimal happily uses as A through F, are simply invalid in BCD. Feed an invalid BCD pattern into certain PLC instructions, and some processors will flag a fault rather than guess at what you meant.

Try It Yourself: Live Number System Converter

Enter a number in any base below, and watch it convert to all four systems instantly.

Quick Reference: All Four Systems Side by Side

Have you ever noticed how the same value looks completely different depending on which system you write it in? That's exactly the point, and exactly why a PLC address label like "A5" only makes sense once you know it's being read in hex, not decimal.

Which Number System Does Your PLC Actually Use?

Where This Actually Matters on the Job

Quick FAQs: Number Systems

External References

• GeeksforGeeks: Base Conversions for Number Systems
• AutomationDirect: Numbering Systems Explained
• AutomationDirect: DL405 User Manual, Numbering Systems Appendix