# Decimal to Binary – Definition With Examples

8 minutes read

Created: December 28, 2023

Last updated: January 15, 2024

Welcome to another learning adventure at Brighterly, your hub for fun and engaging mathematics! Here at Brighterly, we firmly believe that numbers have stories to tell and mysteries to unravel. That’s why today, we are taking a closer look at a fascinating pair of numeric systems – Decimal and Binary numbers. As we journey together through this exciting topic, we will explore the very building blocks of numbers as we know them and delve into the exciting world of digital computation, a realm where binary reigns supreme. So, let’s flip the switch and illuminate the world of Decimal and Binary numbers!

## What are Decimal and Binary Numbers?

When we talk about numbers, we often picture the familiar digits from 0-9. This is what we call the decimal number system, or base 10 system, which is widely used in everyday life. But there’s another fascinating system, called the binary number system, that uses only 0s and 1s. It’s the foundation of all things digital – from computers and smartphones to futuristic robots and AI.

## Definition of Decimal Numbers

The decimal number system is what we commonly use. It is a base 10 number system, meaning it uses 10 digits from 0 to 9. Every position in a decimal number represents a power of 10. For example, in the number 345, the digit ‘5’ is in the ones place (10^0), ‘4’ is in the tens place (10^1), and ‘3’ is in the hundreds place (10^2).

## Definition of Binary Numbers

On the other hand, binary numbers form a base 2 system, which means they use just two digits – 0 and 1. Binary numbers follow a similar pattern as decimal numbers where each position represents a power of 2. For instance, the binary number 101 represents 1*(2^2) + 0*(2^1) + 1*(2^0) = 4 + 0 + 1 = 5 in the decimal system.

## Properties of Decimal and Binary Numbers

Let’s now explore some interesting properties of these two number systems.

### Properties of Decimal Numbers

- Unique Representation: Each number in the decimal system has a unique representation and cannot be expressed in any other way using the same digits.
- Addition and Subtraction: Operations like addition, subtraction, multiplication, and division follow specific rules, much like the ones you’ve learned in school.
- Decimal Point: The decimal point is used to represent fractions in the decimal system.

### Properties of Binary Numbers

- Simplicity: Binary numbers are simple to work with because they only use two digits: 0 and 1.
- Operations: Like decimal numbers, binary numbers also follow specific rules for arithmetic operations. However, these rules are generally simpler.
- Bit and Byte: In the realm of computers, a binary digit is called a ‘bit’. Eight of these ‘bits’ form a ‘byte’.

## Difference Between Decimal and Binary Numbers

The main difference between decimal and binary numbers lies in the base they use – decimal uses base 10, while binary uses base 2. This difference might seem trivial, but it greatly impacts the way numbers are represented, calculated, and utilized, especially in computing and digital electronics.

## Conversion from Decimal to Binary

Understanding the conversion process from decimal to binary is vital, particularly for those interested in computer science, electronics, and mathematics. Let’s break down this conversion process step by step.

### Steps in Converting Decimal Numbers to Binary Numbers

- Start with your decimal number and divide it by 2.
- Write down the remainder. This will either be a 0 (if the number is even) or a 1 (if the number is odd).
- Continue this process, dividing the quotient from the previous step by 2 and recording the remainder, until your quotient is 0.
- Your binary number is the sequence of remainders, read from the bottom up.

### Writing Binary Equivalents of Decimal Numbers

Following our step-by-step guide, let’s convert the decimal number 13 into binary.

- 13 divided by 2 equals 6 remainder 1
- 6 divided by 2 equals 3 remainder 0
- 3 divided by 2 equals 1 remainder 1
- 1 divided by 2 equals 0 remainder 1

Reading the remainders from the bottom up gives us 1101. So, 13 in decimal is 1101 in binary!

## Practice Problems on Decimal to Binary Conversion

Let’s put theory into practice by converting some decimal numbers into binary.

### Example 1: Converting Decimal 7 to Binary

Let’s start with the number 7:

- 7 divided by 2 equals 3 remainder 1
- 3 divided by 2 equals 1 remainder 1
- 1 divided by 2 equals 0 remainder 1

Reading the remainders from the bottom up, we get 111. So, the binary equivalent of the decimal number 7 is 111.

### Example 2: Converting Decimal 18 to Binary

Now let’s convert the number 18:

- 18 divided by 2 equals 9 remainder 0
- 9 divided by 2 equals 4 remainder 1
- 4 divided by 2 equals 2 remainder 0
- 2 divided by 2 equals 1 remainder 0
- 1 divided by 2 equals 0 remainder 1

Reading the remainders from the bottom up, we get 10010. Therefore, the binary equivalent of the decimal number 18 is 10010.

### Example 3: Converting Decimal 25 to Binary

Let’s try converting 25:

- 25 divided by 2 equals 12 remainder 1
- 12 divided by 2 equals 6 remainder 0
- 6 divided by 2 equals 3 remainder 0
- 3 divided by 2 equals 1 remainder 1
- 1 divided by 2 equals 0 remainder 1

Reading the remainders from the bottom up, we get 11001. Therefore, the binary equivalent of the decimal number 25 is 11001.

### Example 4: Converting Decimal 32 to Binary

Finally, let’s convert 32:

- 32 divided by 2 equals 16 remainder 0
- 16 divided by 2 equals 8 remainder 0
- 8 divided by 2 equals 4 remainder 0
- 4 divided by 2 equals 2 remainder 0
- 2 divided by 2 equals 1 remainder 0
- 1 divided by 2 equals 0 remainder 1

Reading the remainders from the bottom up, we get 100000. Therefore, the binary equivalent of the decimal number 32 is 100000.

These examples should help you to better understand the process of decimal to binary conversion. Practice more to get familiar with the process. You can check your answers using this Decimal to Binary Converter.

## Frequently Asked Questions on Decimal to Binary Conversion

### Why do we need to convert decimal numbers into binary?

Binary numbers form the basis of all computing systems. By understanding how to convert decimal numbers into binary, we can grasp how computers store and manipulate data, making it a key concept for budding programmers and computer scientists.

### Is binary only used in computers?

Although binary is primarily associated with computers and digital electronics, it also finds applications in certain areas of mathematics and physics, like set theory and quantum mechanics.

### How do I get better at decimal to binary conversion?

Practice, practice, and practice! Start with smaller numbers and gradually work your way up to bigger ones. Using the steps provided in this guide, you will soon find the process of converting decimals to binary numbers becomes intuitive.

## Conclusion

Journeying through the landscapes of Decimal and Binary numbers with us at Brighterly, we hope you’ve gleaned a sense of awe and understanding about these two essential numeric systems. They are more than mere numbers – they are the cornerstone of our digital age. By understanding the nuances of decimal and binary, we not only unravel the secrets of our everyday computations but also peek into the inner workings of the digital world that surrounds us.

The fascinating conversion between decimal and binary opens up pathways into the world of computer science, offering a solid foundation for any young learner interested in coding, digital electronics, or artificial intelligence. Keep practicing the conversion steps we’ve provided, and soon, the translation from decimal to binary will become second nature.

Remember, at Brighterly, we’re committed to making your mathematical journey engaging, exciting, and enlightening. Stay tuned for more in-depth explorations into the vast world of mathematics. Happy learning!

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