Decimal Number System
A Decimal number system is the number system that we use on a daily basis based on the 10 digits. In mathematics, a number system is considered the notation of numbers by using digits or symbols. The number system consists of four main types namely binary number system, decimal number system, octal number, and hexadecimal number system. The decimal number system is also known as the HinduArabic or Arabic number system since in ancient civilizations it was difficult to multiply and divide large numbers by using hands. Let us learn more about the decimal number system.
1.  Definition of Decimal Number System 
2.  Conversion From Others to Decimal Number System 
3.  Conversion From Decimal Number System to Others 
4.  FAQs on Decimal Number System 
Definition of Decimal Number System
Decimal number system is the number system we use every day and uses digits from 0  9 i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9. The base number of the decimal number system is 10 as the total number available in this number system is 10. If any number is represented without a base, it means that its base is 10. For example: \(73_{10}, 132_{10}, 5267_{10}\) are some examples of numbers in the decimal number system.
Rules on Decimal Number System
While writing a decimal number system, we always express it in base 10 where each value is denoted by 0 or the first nine positive integers. Each value has a place value of the power of 10 which means the digit at the tens place is 10 times greater than the digit at the one's place. Here are a few points or rules to remember while writing in the decimal number system.
 In the decimal number system, the numbers are from 0  9.
 Once 9 is reached, we make the rightmost number 0 and add 1 to the left which becomes 10.
 As and when we reach a digit with 9, we always add 1 so that it number increased to the next.
Each number in the decimal number system has a place value of the power of 10. Let us look at an example for a better understanding, \((134)_{10}\) = 1 × 10^{2} + 3 × 10^{1} + 4 × 10^{0}, \((78)_{10}\) = 7 × 10^{1} + 8 × 10^{0}. A number with a decimal point in the decimal number system is expressed in the decreasing power of 10 after the decimal point. For example, \((24.5)_{10}\) = 2 × 10^{1} + 4 × 10^{0} + 5 × 10^{1}.
Conversion From Others to Decimal Number System
The number system consists of four types namely, binary number system, octal number system, decimal number system, and hexadecimal number system. Each of these has its own base numbers that help in converting one number system to the other. Let us look at how to convert binary to decimal, octal to decimal, and hexadecimal to decimal.
Binary to Decimal Conversion
The binary number can be converted to a decimal number by expressing each digit as a product of the given number 1 or 0 to the respective power of 2. The base number of the binary number system is 2 and after conversion, the base number becomes 10. If a binary number has n digits, B = \(a_{n1}...a_{3}a_{2}a_{1}a_{0}\), the decimal number for it is given as, D = (\((a)_{0}\) × 2^{0}) + (\((a)_{1}\) × 2^{1}) + (\((a)_{2}\) × 2^{2}) + ...
For example: Convert binary number \((10111)_{2}\) to its decimal form.
Given binary number as \((10111)_{2}\).
We need to multiply each binary digit with the decreasing power of 2 and add the products.
= (1 × 2^{4}) + (0 × 2^{3}) + (1 × 2^{2}) + (1 × 2^{1}) + (1 × 2^{0})
= 1 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 1 × 1
= 16 + 0 + 4 + 2 + 1
= 23
Therefore, \((10111)_{2}\) = \((23)_{10}\).
Octal to Decimal Conversion
The conversion of octal numbers to decimals numbers is done using the octal base number which is 8. The number is expanded with the base of 8 where each number is multiplied with the decreasing power of 8 and further added to obtain the decimal number. The decimal number system has a base of 10 after the conversion.
For example: Convert octal number \((278)_{8}\) to its decimal form.
\((278)_{8}\)_{ }= 2 x 8^{2} + 7 x 8^{1} + 8 x 8^{0}
= 2 x 64 + 7 x 8 + 8 x 1
= 128 + 15 + 8
= 151
Therefore, \((278)_{8}\) = \((151)_{10}\).
Hexadecimal to Decimal Conversion
The base number of hexadecimal is 16 and to convert hexadecimal to decimal we use the number 16. The number is expanded with the base of 16 where each number is multiplied with the decreasing power of 16 and further added to obtain the decimal number. The decimal number system has a base of 10 after the conversion.
For example: Convert hexadecimal number \((14)_{16}\) to its decimal form.
\((14)_{16}\) = 1 × 16^{1} + 4 × 16^{0}
= 1 × 16 + 4 × 1
= 16 + 4
= 20
Therefore, \((14)_{16}\) = \((20)_{10}\).
Conversion From Decimal Number System to Others
Converting a decimal number to another number system is similar to converting any number system to a decimal number system. Each of the base numbers is required for conversion. Let us look at the conversions.
Decimal to Binary Conversion
To convert a decimal number to binary we need to divide the given number by 2 until the quotient is equal to 0. We keep the remainders aside during the division process. Once the quotient is equal to zero, we write the remainder along with the last numbers starting from the bottom to the top to obtain the binary number.
For example: Convert decimal number \((128)_{10}\) to binary.
Dividend  Remainder 

128/2 = 64  0 
64/2 = 32  0 
32/2 = 16  0 
16/2 = 8  0 
8/2 = 4  0 
4/2 = 2  0 
2/2 = 1  0 
1/2 = 0  1 
Write the remainder from bottom to top i.e. in the reverse chronological order. This will give the binary equivalent of 128. Therefore, the binary equivalent of decimal number \((20)_{10}\) is \((10000000)_{2}\).
Decimal to Octal Conversion
To convert decimal to octal, the decimal number is divided by 8 to the reminder that is obtained from the previous digit. The first remainder is the least significant digit(LSD) and the last remainder is the most significant digit(MSD). Once the quotient is less than 8, we obtain the octal number by writing the remainder in reverse order. Let us understand the conversion with the help of an example.
For example: Convert the decimal number \((45)_{10}\) to an octal number.
Divide 45 by 8 until the remainder is less than 8.
Dividing by 8  Quotient  Remainder 
45/8  5  5 
5/8  0  5 
Writing the octal number from bottom to top. \((45)_{10}\) = \((55)_{8}\).
Decimal to Hexadecimal Conversion
The decimal to hexadecimal conversion is done similarly to the other two number systems. The base number of hexadecimal is 16 so the number needs to be divided by 16 until the quotient is zero. A hexadecimal number system uses digits only from 0  9 and 10  15 uses alphabets like A, B, C, D, E, F respectively. Let us look at an example.
Convert hexadecimal number \((120)_{10}\) to decimal.
Divide 120 by 16 until the quotient is zero.
Dividing by 16  Quotient  Remainder 
120/16  7  8 
7/16  0  7 
To obtain the hexadecimal number we write the numbers from the bottom to the top. Therefore, \((120)_{10}\) = \((78)_{16}\).
Related Topics
Here are a few related topics to the decimal number system, take a look.
Examples on Decimal Number System

Example 1: Prove that binary number \((1000100)_{2}\) can be converted to the decimal number \((68)_{10}\).
Solution: The binary number \((1000100)_{2}\) = (0 × 2^{0}) + (0 × 2^{1}) + (1 × 2^{2}) + (0 × 2^{3}) + (0 × 2^{4}) + (0 × 2^{5}) + (1 × 2^{6}) = 64 + 4 = \((68)_{10}\)
Therefore, the binary number \((1000100)_{2}\)_{ }can be converted to the binary number \((68)_{10}\).

Example 2: Convert octal number 163.22 to a decimal number.
Solution:
\((163.22)_{8}\) = 1 × 8^{2} + 6 × 8^{1 }+ 3 × 8^{0} + 2 × 8^{1} + 2 × 8^{2}
\((163.22)_{8}\) = 1 × 64 + 6 × 8 + 3 × 1 + 2 × 1/8 + 2 × 1/8^{2}
\((163.22)_{8}\) = 64 + 48 + 3 + 0.25 + 0.03125
\((163.22)_{8}\) = 115.28125
Therefore, \((163.22)_{8}\) = \((115.28125)_{10}\).

Example 3: Convert decimal number \((5321)_{10}\) to hexadecimal.
Solution: Divide 5321 by 16 until the quotient is zero.
Dividing by 16 Quotient Remainder 5321/16 332 9 332/16 20 12 20/16 1 4 1/16 0 1 When we write the number from the bottom we get 14129. Hexadecimal numbers are written as alphabets from 10  15. Therefore, \((5321)_{10}\) = \((14C9)_{16}\).
FAQs on Decimal Number System
What is Decimal Number System?
Decimal number system is also known as the HinduArabic number system is used on a daily basis. The numbers used in the decimal number system are from 0  9. Once the number reaches 9, we add a number to make it double digits. The base number of the decimal number system is 10 and it helps in converting a number from one number system to the other.
Where is Decimal Number System Used?
Decimal number system plays an important role in the development of science and technology since the base number is 10. Other number systems like binary, octal, and hexadecimal number systems also are used in microprocessor programming.
What are the Four Types of Number System?
In mathematics, there are four types of number systems, they are:
 Binary number system  The base number is 2
 Octal number system  The base number is 8
 Decimal number system  The base number is 10
 Hexadecimal number system  The base number is 16
What is the Octal Equivalent of Decimal Number 100?
Divide the number 100 by 8 until the quotient is zero.
100/8 = 12, remainder is 4
12/8 = 1, remainder is 4
1/8 = 0, remainder is 1
Write the numbers from the bottom to the top. Therefore, \((100)_{10}\ = \((144)_{8}\.
What is Decimal Number System in Computers?
Decimal number system used the base number as 10 using digits from 0 to 9. A numerical quantity can be represented by using these 10 digits of the decimal number system. This type of number system is also known as the positional value system as the value of the digits depends on their position.
What is Decimal Number System Also Called As?
Decimal number system is also known as the base 10 positional numeral system as this type of number system has the base number as 10 and uses digits from 0 to 9. The decimal number system is the standard system for denoting integer and noninteger numbers.
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