Difference Between Decimal and Binary Number System
Last Updated :
25 Jul, 2024
Decimal Number is defined by the number which has a whole number and the fractional part separated by a decimal point. While Binary Number is defined by a number which is expressed as in the base-2 numeral system. It is always written in the form of 0 and 1. In math, a system that is used to represent numbers in different forms is called number system.
Decimal and Binary Number System
A mathematical value that is used to count, measure, and perform various arithmetic calculations is called a number. Based on properties, numbers are classified into different types, such as natural numbers, whole numbers, fractions, rational and irrational numbers, and so on.
In a similar way, we have different kinds of number systems based on different properties, such as a binary number system, an octal number system, a decimal number system, and a hexadecimal number system. We can convert a number from any number system into any of the other three number systems.
We have curated the difference between Decimal and Binary Number in detail below.
Difference between Decimal and Binary Number
Below is the difference between Decimal and Binary number system.
| Number System | Decimal Number System | Binary Number System |
|---|
Definition | The number which has a whole number and the fractional part separated by a decimal point is called Decimal Number system. | A number system that expresses a number in terms of 0 and 1 digits is a binary number system. |
| Base | 10 | 2 |
| Digits Used | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | 0, 1 |
| Usage | Everyday life and general arithmetic | Computers and digital systems |
| Number Representation | More compact for the same values due to base 10 | Longer for large values due to base 2 |
| Calculation Method | Standard arithmetic operations (taught in general education) | Binary arithmetic operations (simpler but different from decimal) |
Decimal Number System
The number system that represents a number in terms of 0 to 9 digits is a decimal number system. A decimal number system has ten digits, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The base of a number in this system is 10. In the decimal number system, a number is expressed in terms of powers of 10, i.e., the positions of successive digits to the left of the decimal point represent units, tens, hundreds, thousands, and so on.
Some examples of numbers in the decimal number system are (23)10,(123)10, (5547)10, (6531)10, and so on. In everyday life, we most frequently represent numbers using the decimal number system.
For example, (123)10 in powers of 10 is expressed as 1 × 102 + 2 × 101 + 3 × 100. 1 is in the hundreds place, 2 is in the tens place and 3 is in the units place.
Binary Number System
A number system that expresses a number in terms of 0 and 1 digits is a binary number system. A binary system has only two digits, i.e., 0 and 1. The base of a number in this system is 2. In a binary number system, a number is expressed in terms of powers of 2. Â For example, a decimal number 26 is expressed as (11010)2 in a binary system. The binary digits 0 and 1 are used in all computer coding and languages like C, C++, Java, etc., to write a program and encode any digital data.
A binary number (1011)2 is expressed in powers of 2 as (1 × 23) + (0 × 22) + (1 × 21) + (1 × 20).
 Decimal number |  Binary number |
|---|
1 | 01 |
2 | 10 |
3 | 11 |
4 | 100 |
5 | 101 |
6 | 110 |
7 | 111 |
8 | 1000 |
9 | 1001 |
10 | 1010 |
Conversion of Binary and Decimal Number System
Binary numbers can be converted to Decimal numbers and Decimal numbers can be converted back to Binary numbers by using the following rules:
Decimal to Binary Conversion
To convert a decimal number into a binary number, divide the given number by 2 continuously till we get the quotient as 1. Note that we have to write the numbers from downwards to upwards.
Example: Convert (31)10 into a binary number.
Solution:
Therefore, (31)10 is expressed as (11111)2
Binary to Decimal Conversion
The number system that expresses a number in terms of 0 and 1 digits is a binary number system, whereas a decimal number system represents a number in terms of 0 to 9 digits. Now, to convert a binary number into a decimal number, multiply each digit of the binary number by the power of 2.
If B = an-1...a3a2a1a0 is a binary number that has n digits, then the respective decimal number will be
D = (an-1 × 2n-1) +...+(a3 × 23) + (a2 × 22) + (a1 × 21) + (a0 × 20).
Example: Convert (11001)2 to a decimal number.
Solution:
The given binary number is (11001)2.
(11001)2 = (1 × 24) + (1 × 23) + (0 × 22) + (0 × 21) + (1 × 20)
= 16 + 8 + 0 + 0 + 1 = (25)10
Therefore, the binary number (11001)2 is expressed as (25)10.
Solved Example on Conversion of Binary and Decimal Number
Example 1: Convert (65)10 into a binary number.
Solution:
Therefore, (65)10 is expressed as (1000001)2
Example 2: Convert (10101)2 to a Decimal number.
Solution:
The given binary number is (10101)2.
(10101)2 = (1 × 24) + (0 × 23) + (1 × 22) + (0 × 21) + (1 × 20)
= 16 + 0 + 4 + 0 + 1 = (21)10
Therefore, the binary number (10101)2 is expressed as (21)10.
Example 3: Convert (111001)2 to a Decimal number.
Solution:
The given binary number is (111001)2.
(10101)2 = (1 × 25) + (1 × 24) + (1 × 23) + (0 × 22) + (0 × 21) + (1 × 20)
= 32 + 16 + 8 + 0 + 0 + 1 = (57)10
Therefore, the binary number (111001)2 is expressed as (57)10.
Example 4: Convert (46)10 into a binary number.
Solution:
Therefore, (46)10 is expressed as (101110)2.
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