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Understanding Number Systems: Decimal, Binary, and Hexadecimal

Photo by Alexander Sinn on Unsplash


In everyday life, we use numbers all the time, whether for counting, telling time, or handling money. The number system we’re most familiar with is the decimal system, but computers use other systems, such as binary and hexadecimal.

Let’s break down these number systems to understand how they work.

What is a Number System?

A number system is a way of representing numbers using a set of symbols and rules. The most common number systems are:

  • Decimal (Base 10)
  • Binary (Base 2)
  • Hexadecimal (Base 16)

Each system has a different “base” that tells us how many unique digits (symbols) are used to represent numbers.

Decimal Number System (Base 10)

This is the system we use daily. It has 10 digits, ranging from 0 to 9.

Example:

The number 529 in decimal means:

  • 5 × 1⁰² + 2 × 1⁰¹ + 9 × 1⁰⁰ = 500 + 20 + 9 = 529

Each position represents a power of 10, starting from the rightmost digit.

Why Base 10?

Decimal is base 10 because it has 10 digits (0–9). It likely originated because humans have 10 fingers, making it natural to count in groups of 10.

Binary Number System (Base 2)

Computers use the binary system because they operate using two states: on and off, which are represented by 1 and 0. Binary has only these two digits: 0 and 1.

Example:

The binary number 1011 is calculated as:

  • 1 × ²³ + 0 × ²² + 1 × ²¹ + 1 × ²⁰ = 8 + 0 + 2 + 1 = 11 (decimal)

How Computers Use Binary

Each binary digit (bit) represents a power of 2. Computers use this system because it is efficient for processing electronic signals, which can easily be in an “on” (1) or “off” (0) state.

Hexadecimal Number System (Base 16)

The hexadecimal system is base 16, meaning it uses 16 symbols. These are:

  • 0–9 (for values 0 to 9)
  • A-F (for values 10 to 15)

Example:

The hexadecimal number 2F3 is calculated as:

  • 2 × 1⁶² + F (15) × 1⁶¹ + 3 × 1⁶⁰ = 512 + 240 + 3 = 755 (decimal)

Why Use Hexadecimal?

Hexadecimal is often used in computing because it’s a compact way to represent binary numbers. One hexadecimal digit can represent four binary digits (bits), making it easier to read and manage large binary numbers.

Converting Between Systems

It’s easy to convert between decimal, binary, and hexadecimal once you understand the basics. Let’s look at how:

Decimal to Binary

Take the decimal number and keep dividing by 2, writing down the remainder each time, until the quotient is 0. Then, read the remainders in reverse.

Example: Convert 13 to binary.

  • 13 ÷ 2 = 6 (remainder 1)
  • 6 ÷ 2 = 3 (remainder 0)
  • 3 ÷ 2 = 1 (remainder 1)
  • 1 ÷ 2 = 0 (remainder 1)

Read the remainders backward: 1101 (binary)

Binary to Hexadecimal

Group the binary digits in sets of four (starting from the right), then convert each group to its hexadecimal equivalent.

Example: Convert 11011101 to hexadecimal.

  • Group: 1101 1101
  • 1101 = D (13 in decimal), 1101 = D

So, 11011101 (binary) is DD (hexadecimal).

Hexadecimal to Decimal

Multiply each digit by its place value in powers of 16, then add them up.

Example: Convert 1A3 to decimal.

  • 1 × 1⁶² + A (10) × 1⁶¹ + 3 × 1⁶⁰ = 256 + 160 + 3 = 419 (decimal)

Why Learn Different Number Systems?

Understanding these number systems is essential for working with computers and programming. Binary is crucial because it’s the language of computers. Hexadecimal helps simplify binary, especially in coding, debugging, and understanding memory addresses.

Summary

  • Decimal (Base 10) is the system we use every day.
  • Binary (Base 2) is how computers process data using 1s and 0s.
  • Hexadecimal (Base 16) is used in computing to represent binary numbers more compactly.

By learning how these number systems work and how to convert between them, you’ll have a strong foundation for understanding how computers store and process data.

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