Decimal and binary are two different number systems used for representing numbers. The decimal system is the number system we use in everyday life, while the binary system is the foundation of digital computers and electronic devices. Understanding how to convert a decimal number into binary is an important basic skill in mathematics and computer science.
The decimal number system uses ten digits, from 0 to 9. The binary number system uses only two digits: 0 and 1. Although binary numbers may look unfamiliar at first, converting decimal numbers to binary becomes simple once you understand the basic rules and follow the steps carefully.
In this article, you will learn what decimal and binary numbers are, how decimal-to-binary conversion works, the main conversion rules, different methods for converting numbers, and several worked examples for beginners.
What Is the Decimal Number System?
The decimal number system is a base-10 number system. It uses ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
The value of each digit depends on its position. For example, consider the decimal number 583.
Its place values are:
5 × 100 = 500
8 × 10 = 80
3 × 1 = 3
Therefore:
583 = 500 + 80 + 3
The powers of 10 determine the place values in a decimal number.
From right to left, the place values are:
10⁰, 10¹, 10², 10³, …
This is why the decimal system is called base 10.
What Is the Binary Number System?
The binary number system is a base-2 number system. Unlike the decimal system, it uses only two digits:
0 and 1
These two digits are called bits. A bit is the smallest unit of digital information.
The place values in a binary number are based on powers of 2 rather than powers of 10.
From right to left, the binary place values are:
2⁰, 2¹, 2², 2³, 2⁴, …
Their values are:
1, 2, 4, 8, 16, 32, 64, 128, …
For example, the binary number 1011 can be expanded as:
1 × 8 + 0 × 4 + 1 × 2 + 1 × 1
So:
1011 = 8 + 0 + 2 + 1 = 11
Therefore, binary 1011 represents decimal 11.
Why Do Computers Use Binary?
Computers and digital electronic devices work with electrical signals that can have two basic states, such as on and off. These two states can be represented by the binary digits 1 and 0.
For this reason, binary is widely used in:
Computers
Smartphones
Digital electronics
Programming
Computer networks
Data storage
Microprocessors
Digital communication
Humans generally find decimal numbers easier to read, but computers process information using binary representations.
Understanding decimal-to-binary conversion helps explain how ordinary numbers are represented inside digital systems.
What Is Decimal to Binary Conversion?
Decimal to binary conversion means changing a number from the base-10 number system into its equivalent representation in the base-2 number system.
For example:
Decimal 10 = Binary 1010
Both numbers represent the same quantity, but they use different number systems.
The decimal number 10 can be written using powers of 2:
10 = 8 + 2
Since:
8 = 2³
2 = 2¹
the corresponding binary digits are:
1010
The most common method for converting a whole decimal number to binary is the repeated division by 2 method.
Basic Rule for Decimal to Binary Conversion
For a positive whole decimal number, repeatedly divide the number by 2 and record the remainder each time.
The basic rules are:
Divide the decimal number by 2.
Write down the remainder.
Divide the quotient by 2 again.
Continue until the quotient becomes 0.
Read the remainders from bottom to top.
The resulting sequence is the binary number.
Because division by 2 can produce only an even or odd result, every remainder will be either 0 or 1.
An even number gives a remainder of 0.
An odd number gives a remainder of 1.
This is the key idea behind the method.
Step-by-Step Method for Decimal to Binary Conversion
Let us convert decimal 13 into binary.
Start by dividing 13 by 2:
13 ÷ 2 = 6 remainder 1
Now divide the quotient, 6, by 2:
6 ÷ 2 = 3 remainder 0
Continue:
3 ÷ 2 = 1 remainder 1
Finally:
1 ÷ 2 = 0 remainder 1
Now the quotient has reached 0, so the division process stops.
The remainders, from top to bottom, were:
1, 0, 1, 1
But the binary number must be read from the last remainder to the first:
1101
Therefore:
13₁₀ = 1101₂
The small numbers indicate the bases. The subscript 10 means decimal, while the subscript 2 means binary.
Why Are the Remainders Read From Bottom to Top?
This is one of the most important rules to remember.
During repeated division, the first remainder represents the rightmost binary digit, or the 2⁰ place. The next remainder represents the 2¹ place, and so on.
Therefore, the remainders are generated from the least significant bit to the most significant bit.
That is why they must be read in reverse order.
For example, when converting 13:
| Division | Quotient | Remainder |
|---|---|---|
| 13 ÷ 2 | 6 | 1 |
| 6 ÷ 2 | 3 | 0 |
| 3 ÷ 2 | 1 | 1 |
| 1 ÷ 2 | 0 | 1 |
Reading the remainders from bottom to top gives:
1101
So:
13 = 1101₂
Example 1: Convert Decimal 8 to Binary
Let us convert 8 using repeated division by 2.
8 ÷ 2 = 4 remainder 0
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read the remainders from bottom to top:
1000
Therefore:
8₁₀ = 1000₂
This makes sense because:
1000₂ = 1 × 8 = 8
Example 2: Convert Decimal 20 to Binary
Divide 20 repeatedly by 2:
20 ÷ 2 = 10 remainder 0
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Read the remainders from bottom to top:
10100
Therefore:
20₁₀ = 10100₂
We can check the answer using binary place values:
10100₂ = 16 + 4 = 20
So the conversion is correct.
Example 3: Convert Decimal 25 to Binary
Now convert 25.
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read the remainders upward:
11001
Therefore:
25₁₀ = 11001₂
Check the result:
11001₂ = 16 + 8 + 1 = 25
The answer is correct.
Example 4: Convert Decimal 50 to Binary
Let us convert 50.
50 ÷ 2 = 25 remainder 0
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading the remainders from bottom to top gives:
110010
Therefore:
50₁₀ = 110010₂
Check:
32 + 16 + 2 = 50
So the conversion is correct.
Example 5: Convert Decimal 100 to Binary
Now consider a larger number: 100.
100 ÷ 2 = 50 remainder 0
50 ÷ 2 = 25 remainder 0
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read the remainders from bottom to top:
1100100
Therefore:
100₁₀ = 1100100₂
Check:
64 + 32 + 4 = 100
So the conversion is correct.
An Alternative Method: Using Powers of 2
Another useful method is to convert a decimal number by identifying the powers of 2 that add up to the number.
Suppose we want to convert 37 into binary.
The relevant powers of 2 are:
32, 16, 8, 4, 2, 1
Start with the largest value that does not exceed 37.
37 − 32 = 5
The value 16 is too large, so its binary digit is 0.
The value 8 is also too large, so its digit is 0.
The value 4 fits:
5 − 4 = 1
The value 2 is too large, so its digit is 0.
The value 1 fits:
1 − 1 = 0
Now write the digits corresponding to:
32, 16, 8, 4, 2, 1
They are:
1 0 0 1 0 1
Therefore:
37₁₀ = 100101₂
This method is especially useful when you want to understand what each binary digit means.
Decimal to Binary Conversion Rules to Remember
For beginners, the following rules are the most important:
Rule 1: Binary Uses Only 0 and 1
A binary number cannot contain digits such as 2, 3, 5, or 9.
Valid examples include:
1010
11001
100101
Rule 2: Divide by 2
For whole-number decimal-to-binary conversion, repeatedly divide the number by 2.
Rule 3: Record Every Remainder
Every division produces a remainder of either 0 or 1.
Never skip a remainder.
Rule 4: Continue Until the Quotient Is 0
Stop only when the final quotient becomes 0.
Rule 5: Read the Remainders Backward
This is perhaps the most commonly forgotten rule.
Read the last remainder first and the first remainder last.
Rule 6: Check the Answer
Convert the binary result back into decimal using powers of 2. This provides a simple way to verify your answer.
Common Mistakes Beginners Make
Decimal-to-binary conversion is simple, but a few mistakes are common.
Reading the Remainders in the Wrong Direction
If you read the remainders from top to bottom, you will usually get the wrong answer.
Always read them from bottom to top.
Forgetting the Final Remainder
When the quotient becomes 1, divide it by 2 one final time. This produces a remainder of 1 and a quotient of 0.
For example:
1 ÷ 2 = 0 remainder 1
That final 1 is important.
Using Decimal Place Values
Binary place values are based on powers of 2, not powers of 10.
Do not use:
1, 10, 100, 1000
for binary place values.
Instead use:
1, 2, 4, 8, 16, 32, 64, …
Putting Digits Other Than 0 and 1 in a Binary Number
A binary number can contain only 0 and 1.
For example, 1021 is not a valid binary number.
Not Checking the Result
A quick check can catch many mistakes. Add the powers of 2 represented by the 1s in your binary answer and compare the result with the original decimal number.
Decimal to Binary Conversion Table
The following table can help beginners recognize common conversions quickly.
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 11 | 1011 |
| 12 | 1100 |
| 13 | 1101 |
| 14 | 1110 |
| 15 | 1111 |
| 16 | 10000 |
This table shows an important pattern. Whenever a number reaches a power of 2, the binary representation becomes a 1 followed by zeros.
For example:
2 = 10
4 = 100
8 = 1000
16 = 10000
32 = 100000
How to Verify a Decimal-to-Binary Conversion
Verification is an excellent habit when learning number-system conversions.
Suppose you converted decimal 45 and obtained:
101101
Now assign the binary place values:
32, 16, 8, 4, 2, 1
The binary digits are:
1 0 1 1 0 1
Multiply each digit by its corresponding place value:
1 × 32 = 32
0 × 16 = 0
1 × 8 = 8
1 × 4 = 4
0 × 2 = 0
1 × 1 = 1
Add them:
32 + 8 + 4 + 1 = 45
Therefore:
101101₂ = 45₁₀
The conversion is correct.
Decimal Numbers Greater Than 255
The same conversion process works for much larger decimal numbers.
For example, whether the number is 10, 100, 500, 1,000, or 10,000, you can repeatedly divide by 2 until the quotient becomes zero.
The number of binary digits increases as the decimal number becomes larger.
For example:
255₁₀ = 11111111₂
This contains eight binary digits.
A decimal number from 0 to 255 can be represented using eight binary bits when leading zeros are allowed.
For example:
5 = 00000101
The leading zeros do not change the value. They are often used when a fixed number of bits is required.
What Are Leading Zeros in Binary?
A leading zero is a zero placed before the first 1 in a binary number.
For example:
101 = 5
can also be written as:
00000101 = 5
Both represent the same value.
Leading zeros are useful when binary numbers need to have a fixed length, such as 8 bits.
However, when performing a basic decimal-to-binary conversion, you normally do not need to add leading zeros unless the question specifically asks for a fixed number of bits.
Decimal 0 in Binary
Zero is a special but simple case.
Decimal zero is represented as:
0₂
There is no need to perform repeated division for this case.
Therefore:
0₁₀ = 0₂
A Simple Strategy for Beginners
If you are learning decimal-to-binary conversion for the first time, use this five-step strategy:
Step 1: Write the decimal number.
Step 2: Divide it by 2.
Step 3: Record the remainder.
Step 4: Continue dividing the quotient by 2 until it becomes 0.
Step 5: Read all remainders from bottom to top.
For example, to convert 18:
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 upward:
10010
Therefore:
18₁₀ = 10010₂
With a little practice, this process becomes quick and almost automatic.
Conclusion
Decimal-to-binary conversion is the process of representing a decimal number using only the binary digits 0 and 1. The most common method for converting whole decimal numbers is repeated division by 2. At every step, record the remainder and continue dividing the quotient until it reaches zero. The remainders must then be read from bottom to top to obtain the binary number.
The most important ideas to remember are simple: decimal is base 10, binary is base 2, binary uses only 0 and 1, and binary place values are powers of 2. Once you understand these rules and practice a few examples, converting decimal numbers to binary becomes much easier. This basic skill also provides a useful foundation for learning computer science, programming, digital electronics, and other areas where binary numbers are essential.
FAQs
1. What is decimal to binary conversion?
Decimal to binary conversion is the process of changing a number from the decimal number system into its equivalent binary representation. The decimal system uses ten digits, from 0 to 9, while the binary system uses only 0 and 1. For example, decimal 10 is represented as binary 1010. Both numbers have the same value but use different number systems. Decimal-to-binary conversion is important because computers and digital devices use binary to represent and process information. Beginners can convert whole decimal numbers using repeated division by 2 and then reading the remainders from bottom to top.
2. How do you convert a decimal number to binary?
To convert a whole decimal number to binary, repeatedly divide the number by 2. After each division, record the remainder, which will always be either 0 or 1. Continue dividing the quotient by 2 until the quotient becomes 0. Finally, read all the recorded remainders from bottom to top. For example, converting 13 gives remainders 1, 0, 1, and 1. Reading them in reverse order gives 1101. Therefore, decimal 13 is equal to binary 1101. This repeated-division method is one of the easiest and most reliable methods for beginners.
3. Why do we divide by 2 when converting decimal to binary?
We divide by 2 because binary is a base-2 number system. In a binary number, each position represents a power of 2, such as 1, 2, 4, 8, 16, and so on. Division by 2 separates a decimal number into these powers of 2. The remainder tells us whether a particular binary position contains 0 or 1. An even number produces a remainder of 0, while an odd number produces a remainder of 1. Repeatedly dividing by 2 therefore reveals the binary digits of the original decimal number.
4. Why are the remainders read from bottom to top?
The remainders are read from bottom to top because the first remainder represents the rightmost binary digit, which has the smallest place value. Each following division reveals the next binary position moving toward the left. Therefore, the remainders are produced in reverse order compared with how we normally write a binary number. For example, converting 13 produces remainders 1, 0, 1, and 1. The first 1 belongs to the 2⁰ position, so it must appear at the right side. Reading the remainders upward gives 1101, which is the correct binary representation of 13.
5. What is the binary equivalent of decimal 10?
The binary equivalent of decimal 10 is 1010. This can be found by repeatedly dividing 10 by 2. First, 10 divided by 2 gives 5 with remainder 0. Next, 5 divided by 2 gives 2 with remainder 1. Then, 2 divided by 2 gives 1 with remainder 0. Finally, 1 divided by 2 gives 0 with remainder 1. Reading the remainders from bottom to top gives 1010. We can also verify it using powers of 2: 1010₂ equals 8 + 2, which gives 10.
6. What are the binary place values?
Binary place values are based on powers of 2. Starting from the rightmost position, the place values are 2⁰, 2¹, 2², 2³, 2⁴, and so on. Their numerical values are 1, 2, 4, 8, 16, 32, 64, 128, and so forth. For example, in the binary number 10101, the place values are 16, 8, 4, 2, and 1. Only positions containing a 1 contribute to the total. Therefore, 10101₂ represents 16 + 4 + 1, which equals decimal 21.
7. Can every decimal whole number be converted into binary?
Yes. Every non-negative whole decimal number can be represented in binary. The binary system can represent zero and any positive whole number using combinations of 0 and 1. The number of binary digits required depends on how large the decimal number is. For example, decimal 5 is 101 in binary, while decimal 100 is 1100100. Larger numbers simply require more binary positions. The repeated-division-by-2 method works for all whole decimal numbers, regardless of their size. However, converting decimal fractions, such as 0.5 or 2.75, requires a different method.
8. What is the easiest way to check a binary conversion?
The easiest way to check a decimal-to-binary conversion is to convert the binary result back into decimal using powers of 2. For example, suppose you obtained 101101 for decimal 45. The binary place values are 32, 16, 8, 4, 2, and 1. The digits containing 1 correspond to 32, 8, 4, and 1. Adding them gives 32 + 8 + 4 + 1 = 45. Because the result matches the original decimal number, the conversion is correct. This verification method is especially useful when learning binary numbers.
9. What are common mistakes in decimal to binary conversion?
Common mistakes include reading the remainders in the wrong direction, forgetting the final remainder, stopping the division too early, and using incorrect place values. Beginners sometimes read the remainders from top to bottom instead of bottom to top, producing an incorrect binary number. Another common mistake is forgetting that binary uses only 0 and 1. It is also important to remember that binary place values are powers of 2, not powers of 10. A simple way to avoid mistakes is to write every division clearly, record every remainder, read the remainders upward, and then check the final answer.
10. Why is learning decimal to binary conversion important?
Learning decimal-to-binary conversion provides a basic understanding of how numbers are represented in computers and digital systems. Computers use binary because electronic circuits can work with two basic states, commonly represented as 0 and 1. Understanding binary helps beginners build a foundation for computer science, programming, digital electronics, data representation, and computer engineering. Decimal-to-binary conversion is also a useful introduction to other number systems, such as hexadecimal and octal. Once you understand how decimal numbers can be represented using powers of 2, many concepts in computing become easier to understand.
