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293 LearnersLast updated on December 4, 2025
Decimal to Binary Conversion
In mathematics and computer science, numbers can be represented in various numeral systems, such as decimal, binary, octal, or hexadecimal. The decimal system is the most commonly used and is based on the number 10, employing digits from 0 to 9. The binary system, however, is based on the number 2, using only the digits 0 and 1. Decimal to binary conversion is essential in computing and digital electronics, as computers use binary language to process data. In this topic, we will learn how to convert decimal numbers to binary.
What is Decimal?
A decimal system is a base-10 numeral system, which is the standard system for denoting integers and non-integers. It is the most widely used system in daily life for expressing numbers.
The decimal system utilizes ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Each digit's position in a number represents a power of 10, making it straightforward to perform arithmetic operations.
What is Binary?
The binary numeral system, or base-2 system, represents numeric values using two symbols: 0 and 1.
It is the foundation of all binary code, which is used in computing and digital systems.
Each digit in a binary number represents a power of 2, and the binary system is essential for computer operations and digital circuit design.
What Is Decimal to Binary Conversion?
Decimal to binary conversion is the process of changing a number from the decimal system (base 10) into the binary system (base 2).
The decimal system is the number system we use every day, made up of digits from 0 to 9. In contrast, the binary system is used in computers and digital devices and includes only two digits: 0 and 1.
Computers store and process all information in binary. Converting decimal numbers to binary helps students understand how digital systems represent data, perform calculations, and execute commands.
This conversion builds a strong foundation for computer science, coding, logic circuits, and understanding how digital technology works.
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Decimal to Binary Formula
To convert a decimal number to binary, follow these steps:
- Repeatedly divide the decimal number by 2.
- Write down the remainder after each division (this will be either 0 or 1).
- Continue dividing the quotient by 2 until the quotient becomes 0.
- Read the remainders from bottom to top to get the binary number.
Formula:
Binary = Read the remainders of (Decimal ÷ 2) from bottom to top
\( N_{i+1} = \left[ \frac{N_i}{2} \right] \)
This method converts the integer part of a decimal number to binary by repeatedly dividing by 2 until the quotient reaches 0.
How to Convert Decimal to Binary?
Converting a decimal number to binary involves dividing the number by 2 and keeping track of the remainders.
Step-by-Step Process to Convert Decimal to Binary
Step 1: Write down the decimal number.
Step 2: Divide the number by 2.
Step 3: Record the remainder (0 or 1).
Step 4: Repeat the process with the quotient until it is 0.
Step 5: The binary number is obtained by reading the remainders in reverse order.
Decimal to Binary Conversion Table
When dealing with numbers, we often use decimal and binary systems.
Here is a conversion table to help understand how decimal numbers are represented in binary.
Common Mistakes and How to Avoid Them in Decimal to Binary Conversion
When converting decimal numbers to binary, beginners often make mistakes. Here are some common errors to help understand the conversion process better.
Mistake 1
Misunderstanding the division process
Learners sometimes confuse the division process, forgetting to track the remainders properly, leading to incorrect binary results.
Mistake 2
Incorrect order of remainders
Students might list the remainders in the order they are calculated, instead of reversing them to form the binary number. For example, instead of writing the binary for 10 as 1010, they might incorrectly write it as 0101.
Mistake 3
Ignoring the quotient
Some learners overlook the quotient in each division step, stopping the process prematurely, which results in an incomplete binary representation.
Mistake 4
Using subtraction instead of division
Beginners may mistakenly use subtraction instead of division to convert decimal numbers, leading to incorrect binary conversions.
Mistake 5
Misinterpreting powers of 2
Students often confuse the value of binary digits by misinterpreting the powers of 2, affecting the accuracy of binary representation.
Decimal to Binary Conversion Examples
Problem 1
Convert 45 to binary
45 in decimal is 101101 in binary.
Explanation
To convert 45:
45 ÷ 2 = 22 remainder 1
22 ÷ 2 = 11 remainder 0
11 ÷ 2 = 5 remainder 1
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Binary: 101101
Problem 2
Convert 78 to binary.
Solution: 78 in decimal is 1001110 in binary.
Explanation
Convert 78:
78 ÷ 2 = 39 remainder 0
39 ÷ 2 = 19 remainder 1
19 ÷ 2 = 9 remainder 1
9 ÷ 2 = 4 remainder 1
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Binary: 1001110
Problem 3
What is 15 in binary?,A sports merchandise shop in New York City is preparing digital codes for customers who buy official NFL jerseys. One customer purchases a limited-edition jersey costing $24, and the shopโs system requires converting the decimal number 24 into binary to generate a discount voucher. What is 24 in binary? Solution:
15 in decimal is 1111 in binary.
11000
Explanation
Convert 15:
15 ÷ 2 = 7 remainder 1
7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Binary: 1111
Since the system uses binary for generating voucher IDs, we convert 24 from decimal to binary.
We repeatedly divide by 2 and record remainders:
24 ÷ 2 = 12 → remainder 0
12 ÷ 2 = 6 → remainder 0
6 ÷ 2 = 3 → remainder 0
3 ÷ 2 = 1 → remainder 1
1 ÷ 2 = 0 → remainder 1
Reading remainders bottom-up gives 11000.
This code is later printed on the customer's NFL receipt.
Problem 4
A Walgreens pharmacy in Chicago prints dosage instruction barcodes using binary numbers. A cold medicine formula includes an ingredient ID number of 37, which must be converted to binary before printing. What is 37 in binary?
100101
Explanation
Walgreens converts decimal numbers like 37 into binary for internal tracking.
Divide repeatedly by 2:
37 ÷ 2 = 18 → remainder 1
18 ÷ 2 = 9 → remainder 0
9 ÷ 2 = 4 → remainder 1
4 ÷ 2 = 2 → remainder 0
2 ÷ 2 = 1 → remainder 0
1 ÷ 2 = 0 → remainder 1
Read from bottom to top → 100101.
This binary ID goes into their nationwide medicine-tracking database.
Problem 5
A sports merchandise shop in New York City is preparing digital codes for customers who buy official NFL jerseys. One customer purchases a limited-edition jersey costing $24, and the shopโs system requires converting the decimal number 24 into binary to generate a discount voucher. What is 24 in binary?
11000
Explanation
Since the system uses binary for generating voucher IDs, we convert 24 from decimal to binary.
We repeatedly divide by 2 and record remainders:
24 ÷ 2 = 12 → remainder 0
12 ÷ 2 = 6 → remainder 0
6 ÷ 2 = 3 → remainder 0
3 ÷ 2 = 1 → remainder 1
1 ÷ 2 = 0 → remainder 1
Reading remainders bottom-up gives 11000.
This code is later printed on the customer's NFL receipt.
FAQs on Decimal to Binary
1.What is the binary of 7?
2.What is 50 in binary?
3.Is 128 a power of 2?
4.How do I convert 64 to binary?
5.Who invented binary code?
6.Does NASA use binary code?
Important Glossaries for Decimal to Binary Conversion
Conversion: The process of changing one number from one numeral system to another, such as from decimal to binary. Binary Number: A number expressed in the base-2 numeral system, using only the digits 0 and 1. Decimal Number: A number expressed in the base-10 numeral system, using digits from 0 to 9. Remainder: The amount left over after division, crucial for determining binary digits. Power of 2: Numbers like 1, 2, 4, 8, etc., which are the result of raising 2 to an integer power.
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
