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Decimal to Binary Conversion – Steps, Process & Solved Examples
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Decimal to Binary conversion is a topic in the Number System that is often asked in exams. It involves changing a number from the decimal system (which uses numbers from 0 to 9) to the binary system (which uses only 0 and 1). To convert a decimal number into binary, we divide the number by 2 again and again until the result becomes 0. At each step, we note the remainder. Finally, we write all the remainders in reverse order — from the last one to the first — to get the binary number.
In this method, each division gives us a part of the binary code. This process helps us understand how computers and digital systems store and use numbers. In this article, you will learn what decimal and binary numbers are, how to convert between them using simple steps, and also go through solved examples and frequently asked questions for better understanding.
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Read about Binary Numbers here.
Decimal to Binary Conversion
Each place value in the decimal (base ten) numeral system has ten possible values (0,1,2,3,4,5,6,7,8, or 9) in the decimal (base ten) numeral system. The binary (base two) number system, on the other hand, has two possible values for each place value, which are represented as 0 or 1. Because electronic computers use the binary system as their internal language, serious computer programmers should know how to convert from decimal to binary. To understand how to convert decimal numbers to binary numbers, let us first understand what are decimal and binary numbers.
Decimal Number System Definition
The decimal number system is also called the base-10 number system because it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is the most common number system we use in daily life—for counting, measuring, money, and many other things. In this system, each digit has a place value based on powers of 10.
For example, in the number 452, the digit 2 is in the units place (10⁰), 5 is in the tens place (10¹), and 4 is in the hundreds place (10²). As we move to the left, the place value of each digit becomes 10 times bigger. This pattern continues with thousands (10³), ten-thousands (10⁴), and so on.
The decimal system is very easy to understand and use, which is why it’s the standard number system for most of the world.
Binary Number System Definition
The binary numbers system or the base 2 system constitutes only two digits that are 0 and 1. Computers manipulate and store all of their data, including numbers, words, films, pictures, and music, using the binary number system. Because the number “2” does not exist in this system, 1 + 1 = 10. George Boole, a British mathematician, produced seminal work in 1854 describing an algebraic system of logic based on the Binary System, which became known as Boolean algebra. His logical calculus would play a key role in the development of digital electrical circuitry.
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How to Convert 3 to Binary in Simple Steps
We want to change the number 3 from decimal (base 10) to binary (base2).
Step 1:
Divide 3 by 2.
3 ÷ 2 = 1 with a remainder of 1
Step 2:
Now divide the quotient (1) by 2.
1 ÷ 2 = 0 with a remainder of 1
Step 3:
Now write the remainders from bottom to top (reverse order).
So, we get: 11
Therefore, 3 in binary is 11.
How to Convert Decimal to Binary Numbers?
Converting a decimal number to binary is popularly done by dividing the digit by 2 and writing out the remainder aside.
By repeatedly dividing a number by two and recording the result, decimal values can be transformed into binary.
Steps to be followed
Conversion of Integral Decimal Numbers
- Divide the number by 2.
- Get the integer quotient for the next iteration.
- Get the remainder for the binary digit.
- Repeat the above steps until the quotient is equal to 0.
Next is the conversion of the fraction part.
Learn about Properties of Rational Numbers here.
Conversion of Fractional Decimal Numbers
Fractional numbers can be transformed to binary form by successive multiplication by 2. That is in every step, the digit before the decimal point is transferred to a binary record and the same process is replicated among the remaining fraction.
The last step is decided when the fraction part is zero or it is terminated when the desired precision is achieved.
Take a look at an example to see how this works.
The remainders are to be read from bottom to top to obtain the binary equivalent.
4310=1010112. Also, read about Relations and Functions here
How to Convert 2 to Binary
The number 2 in decimal (normal number system) is written as 10 in binary. Binary numbers use only 0 and 1.
Steps to Convert 2 to Binary:
Step 1:
Divide 2 by 2:
- 2 ÷ 2 = 1 with a remainder of 0
Step 2:
Now divide the result (1) by 2:
- 1 ÷ 2 = 0 with a remainder of 1
Step 3:
Write the remainders in reverse order (from bottom to top):
- So we get 10
Therefore, 2 in binary is 10
How to Convert 3 to Binary
The number 3 in decimal becomes 11 in binary.
Steps to Convert 3 to Binary:
Step 1:
Divide 3 by 2:
- 3 ÷ 2 = 1 with a remainder of 1
Step 2:
Now divide the result (1) by 2:
- 1 ÷ 2 = 0 with a remainder of 1
Step 3:
Write the remainders from bottom to top:
- So we get 11
Therefore, 3 in binary is 11
Decimal to Binary Conversion Table
Decimal
Binary
Decimal
Binary
0
0000
11
1011
1
0001
12
1100
2
0010
13
1101
3
0011
14
1110
4
0100
15
1111
5
0101
16
10000
6
0110
17
10001
7
0111
18
10010
8
1000
19
10011
9
1001
20
10100
10
1010
Decimal |
Binary |
Decimal |
Binary |
0 |
0000 |
11 |
1011 |
1 |
0001 |
12 |
1100 |
2 |
0010 |
13 |
1101 |
3 |
0011 |
14 |
1110 |
4 |
0100 |
15 |
1111 |
5 |
0101 |
16 |
10000 |
6 |
0110 |
17 |
10001 |
7 |
0111 |
18 |
10010 |
8 |
1000 |
19 |
10011 |
9 |
1001 |
20 |
10100 |
10 |
1010 |
Decimal to Binary Conversion Examples
Let’s see some examples of Decimal to Binary Conversion that come in exams.
Solved Example 1: Convert the decimal number 25 into a binary number.
Solution:
Follow the steps below to convert 25 into a binary system.
- Divide the number 25 by 2.
- Get the integer quotient for the next iteration.
- Get the remainder for the binary digit.
- Repeat the above steps until the quotient is equal to 0.
Solved Example 2: Convert the decimal number 0.35 into a binary number.
Solution:
Multiply by 2 and record the carry in the integral position for fractional decimal figures. When the carries are read down, the equivalent binary fraction is produced, as seen in the sample below.
Thus the fractional binary number is .01011, i.e., 0.01011.
Solved Example 3: Convert the decimal number 18 into a binary number
Solution:
We use the division-by-2 method:
Division |
Quotient |
Remainder |
18 ÷ 2 |
9 |
0 |
9 ÷ 2 |
4 |
1 |
4 ÷ 2 |
2 |
0 |
2 ÷ 2 |
1 |
0 |
1 ÷ 2 |
0 |
1 |
Now, write the remainders from bottom to top: 10010
Therefore, the binary form of 18 is: 10010
Hope this article on the Decimal to Binary was informative. Get some practice of the same on our free Testbook App. Download Now!
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FAQs For Decimal To Binary
How many Digits are Used in the Decimal Number System?
The number system having the value of the base as 1010 is called a decimal number system. From 0 to 9, the decimal number system has ten digits: 0,1,2,3,4,5,6,7,8, and 9. The decimal number system's base, or radix, is 10 because the decimal number system's total number of digits equals ten. As a result, we may use these ten-digit integers to express all of the other digits. In our daily lives, we employ the decimal number system since it is the most frequent and accessible.
What is decimal number system with example?
The decimal number system is also known as the base 10 numbers system because it uses ten digits from 0 to 9. The decimal number system is the one we use frequently in our day-to-day life. In the decimal number system, the position progressive is towards the left of the decimal point represented by units, tens, hundreds, thousands, and so on. Every position outlines a specific power of the base (10). For example, the decimal number 1245 consists of the digit 5 in the unit’s position, 4 in the tens position, 2 in the hundreds position, and 1 in the thousands position.
What are the 4 types of number system?
The four common types of Number systems are:Decimal Number System: Has base 10. The decimal number system is also known as the base 10 numbers system because it uses ten digits from 0 to 9. The decimal number system is the one we use frequently in our day-to-day life.Binary Number System: Has base 2. The binary numbers system or the base 2 system constitutes only two digits that are 0 and 1. Computers manipulate and store all of their data, including numbers, words, films, pictures, and music, using the binary number system.Octal Number System: Has base 8. The octal numbers system is represented with the base of 8, that is it and uses digits from 0 to 7 i.e., 0, 1, 2, 3, 4, 5, 6, and 7 to represent numbers. The term octal is used to describe numbers with an eight-digit base.Hexadecimal Number System: Has base 16. A hexadecimal system is represented with base 16. This implies in the hexadecimal system there are 16 hex numbers. The hexadecimal (sometimes known as base 16 or just hex) numeric system is a positional numeral system used in mathematics and computing.
What are Binary Numbers?
The binary numbers system or the base 2 system constitutes only two digits that are 0 and 1. Computers manipulate and store all of their data, including numbers, words, films, pictures, and music, using the binary number system. Because the number "2" does not exist in this system, 1 + 1 = 10. George Boole, a British mathematician, produced seminal work in 1854 describing an algebraic system of logic based on the Binary System, which became known as Boolean algebra. His logical calculus would play a key role in the development of digital electrical circuitry.
What are Octal Numbers?
The octal numbers system is represented with the base of 8, that is it uses digits from 0 to 7 i.e., 0, 1, 2, 3, 4, 5, 6, and 7 to represent numbers. The term octal is used to describe numbers with an eight-digit base. Octal numbers have a wide range of applications and significance, including use in computers and digital numbering systems. When queried by ground radar, airplane transponders broadcast a "squawk" code, which is expressed as a four-octal-digit number. On the radar screen, this code is used to identify different aircraft.
What are Hexadecimal Numbers?
A hexadecimal system is represented with base 16. This implies in the hexadecimal system there are 16 hex numbers. The hexadecimal (sometimes known as base 16 or just hex) numeric system is a positional numeral system used in mathematics and computing. Hexadecimal is utilized in the Base16 transfer encoding, which divides each byte of plaintext into two 4-bit values and two hexadecimal digits. Hexadecimal numbers are commonly used by software developers and system designers because they provide a human-friendly representation of binary-coded data. Each hexadecimal digit, commonly known as a nibble, represents four bits (binary digits) (or nybble).
What is the binary equivalent of 25?
To convert 25 to binary: 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 So, 25 in binary is 11001.