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Understanding Number Bases
An Essential Skill in Analytical Mathematics
What Are Number Bases?
In mathematics and electronics, a number base defines how many digits a system uses to represent numbers. While most people are familiar with the decimal number system (Base 10), students in Electrical and Electronic Technology programs need to understand binary (Base 2) and hexadecimal (Base 16) as well.
These number systems are essential in:
Reading and designing digital circuits
Programming PLCs and microcontrollers
Working with logic gates, memory addresses, and control signals
Understanding how to convert between number bases builds confidence in interpreting data, troubleshooting hardware, and communicating with digital devices.
How Do We Write Number Bases?
To avoid confusion between number systems, we use a subscript to show which base a number belongs to.
When writing a binary number, we write a small subscript 2 after the number to indicate it’s base 2.
Example: 1011₂ means binary 1011 (not one thousand eleven).A decimal number may include a subscript 10, like 47₁₀, though we often leave it out since decimal is the default.
A hexadecimal number is written with a subscript 16, like 2F₁₆.
These subscripts help clarify which number system is being used, especially when converting between binary, decimal, and hexadecimal.
Counting Powers in a Number System
In any number system, we count the powers of the base from right to left. Each digit in a number has a place value, and that place value is determined by raising the base to a power.
The rightmost digit is always the base to the power of 0.
Each position to the left increases the exponent by 1.
No matter the base, place values grow by powers of the base, starting with 0 on the right and increasing as you move left.
Base 10: Decimal Number System
What is the Decimal System?
The decimal number system (Base 10) is the standard number system used in everyday life. It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Each digit has a place value that represents a power of 10.
Decimal Example: 472
472 = (4×10² )+(7×10¹ )+(2×10⁰ ) = 400 + 70 + 2
This system is intuitive because it’s what we use for money, measurements, and counting.
Base 2: Binary Number System
What is the Binary System?
The binary number system (Base 2) is used by digital electronics and computers. It uses only two digits: 0 and 1.
Every operation in a digital circuit — from storing memory to turning on an LED — is based on binary logic: ON (1) or OFF (0).
Binary Example: 1011₂
1011₂ = (1×2³ )+(0×2²)+(1×2¹)+(1×2⁰)
Binary is the language of embedded systems, programmable devices, and digital communication.
Base 16: Hexadecimal Number System
What is the Hexadecimal System?
The hexadecimal system (Base 16) is a compact way to represent binary numbers. It uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and A–F (where A = 10, B = 11, ..., F = 15)
Hex numbers are commonly used in:
Color codes (like #FF0000 for red)
Memory addresses
Instruction sets in microcontrollers
Hexadecimal Example: 2F₁₆
2F₁₆ =(2×16¹)+(15×16⁰)
Hex is a bridge between binary and human readability in digital electronics.
How to Convert Between Number Systems
When working with number systems like binary, decimal, and hexadecimal, it's important to know how to convert from one base to another. Each system uses a different base, but the process of converting between them follows logical steps based on place value and division or multiplication.
Being able to convert numbers accurately is a valuable skill in electronic and electrical technology, especially when working with digital circuits, programming devices, or interpreting data. Below are step-by-step methods to help you convert between these number systems with confidence.
How to Convert Decimal to Binary (Step-by-Step Guide)
The binary number system (Base 2) is used in a wide range of industries including electronics, computer science, IT, cybersecurity, robotics, and automation. While the decimal system (Base 10) is used in everyday life, most modern machines and digital systems operate internally using binary values.
Being able to convert decimal to binary is essential for:
Working with digital circuits, logic gates, and PLCs
Understanding memory addresses and bit-level data
Programming microcontrollers, sensors, or embedded systems
Learning how to manually convert decimal values to binary helps you build a solid foundation in digital logic and machine-level data processing — skills used in many technical and high-demand careers.
Below is a step-by-step method for converting any decimal number to binary using repeated division.
Step-by-Step Method: Repeated Division by 2
To convert a decimal number (Base 10) to binary (Base 2), follow these steps:
Step 1: Divide the decimal number by 2. Write down the quotient and the remainder. The remainder will always be 0 or 1 in binary.
Step 2: Divide the new quotient by 2 again. Continue dividing each new quotient by 2, writing down the remainder each time, until the quotient becomes 0.
Step 3: Read the remainders from bottom to top. Once you finish, read all the remainders in reverse order — from the last remainder up to the first. This is your binary number.
Example: Convert 19 (Decimal) to Binary
Tip:
Binary numbers grow right to left, so the rightmost digit (called the least significant bit or LSB) is the smallest value.
Always double-check by converting your binary answer back to decimal (using powers of 2) to verify your result.
Why Learn to Convert Binary to Decimal?
The binary number system (Base 2) is used across a wide range of industries, not just in electronic technology. It’s the foundation of how modern systems store, process, and transmit data.
You’ll encounter binary in fields such as:
Electronics and Electrical Engineering
Computer Science and Information Technology
Cybersecurity and Data Encryption
Automation and Robotics
Telecommunications and Networking
Biomedical Equipment Technology
Automotive Systems and Embedded Software
In all of these areas, digital devices operate using binary logic — ON/OFF, 1s and 0s — to carry out instructions, store memory, or trigger operations. Being able to convert binary to decimal helps you interpret and troubleshoot real-world systems that use binary internally but display outputs in decimal.
Learning this conversion bridges the gap between how humans communicate and how machines operate.
How to Convert Binary to Decimal (Step-by-Step Guide)
To convert a binary number to its decimal equivalent, follow this three-step method:
Step 1: Write down the binary number. Start by identifying each digit’s position from right to left. The rightmost digit is always in position 0, and each position increases by 1 as you move to the left.
Step 2: Assign powers of 2 to each digit. Each digit in a binary number is multiplied by 2 raised to the power of its position.
Step 3: Add all the results together. Once all digits are multiplied by their powers of 2, add the results to get the decimal equivalent.
Example: Convert 1011₂ to Decimal
Step 1: Label each digit with a power of 2 (starting from the right)
Tip:
Always label positions from right to left, starting at 0.
If you're ever unsure, double-check by working backward: convert the decimal result back into binary using division by 2.
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