Decimal Number System: There are various number systems that we usually use. We can classify these systems according to the values of the base of the number system. For example, the number system having the value of the base as \(10\) is called a decimal number system, whereas that with a base of \(2\) is called a binary number system and the number systems having base \(8\) and \(16\) are called octal and hexadecimal number systems respectively.

This article is about the decimal number system, its definition, examples, and conversion of the decimal number system to different types of the number system.

Decimal Number System Introduction

There are four different types of number system.

(i) Binary number system: This number system has a base \(2\). We can represent any number using two digits, \(0\) and \(1\).
(ii) Octal number system: This number system has a base \(8\). We can represent any number using eight digits from \(0\) to \(7\).
(iii) Decimal number system: This number system has a base \(10\). We can represent any number using ten digits from \(0\) to \(9\).
(iv) Hexadecimal number system: This number system has a base \(16\). We can represent any number using ten digits from \(0\) to \(9\) and six characters \(A, B, C, D, E\) and \(F\).

Decimal Number System Definition

The decimal number system comprises ten digits from \(0\) to \(9\) that are \(0, 1, 2, 3, 4, 5, 6, 7, 8\) and \(9\). The base or radix of the decimal number system is \(10\) because the total number of digits in the decimal number system is ten. Therefore, we can express all the other digits with the help of these ten-digit numbers.
The decimal number system is the most common and most accessible number system used in our day-to-day lives. 

Decimal Number System Examples

The number \(786\) is interpreted as 
\( = 7 \times {10^2} + 8 \times {10^1} + 6 \times {10^0}\)
\( = 700 + 80 + 6\)
The number \(2025\) is interpreted as 
\(2025 = 2 \times {10^3} + 0 \times {10^2} + 2 \times {10^1} + 5 \times {10^0} = 2000 + 0 + 20 + 5 = 2025\)
The number \(250.36\) is interpreted as
\(250.36 = 2 \times {10^2} + 5 \times {10^1} + 0 \times {10^0} + 3 \times {10^{ – 1}} + 6 \times {10^{ – 2}}\)
\( = 200 + 50 + 0 + 0.3 + 0.06 = 250.36\)

We will use three rules on the decimal number system to write the further numbers:
1. Write numbers from \(0\) to \(9\).
2. Once we reach the digit \(9\), we will make the rightmost digit as \(0\) and add \(1\) to the left digit to make it \(10\).
3. Then, on the rightmost digit, we write until \(9\), and when we reach \(19\), we use \(0\) on the rightmost digit and add \(1\) to the left, so we get \(20\).
4. Similarly, when we reach \(99\), we use \(0\) s in both of these digits places and add \(1\) to the left, giving us \(100\).

How to Read Decimal Numbers?

In single digits from \(0\) to \(9\) the numbers are read as it is. But in the case of two-digit numbers, the right digit means the same, but the left digit means \(10\) times, i.e., in number \(26, 6\) is \(6\) itself, but \(2\) represents \(20\). Altogether it forms \(26\).

If we take a \(3\)-digit number, the rightmost digit means the same, and the middle one is \(10\) times the digit, leftmost digit \(100\) times the digit. 
The number \(546\) means \((5 \times 100) + (4 \times 10) + 6\).

Decimal Number System Conversion

Conversion From Other Number Systems to Decimal

How to convert Binary to Decimal?
Let us consider a binary number with \(n\) digits:
\({d_{n – 1}} \ldots {d_3}{d_2}{d_1}{d_0}\)
The conversion of binary to decimal number is obtained by the sum of the product of binary digits \(\left( {{d_r}} \right)\) and their power of \(2\left( {{2^n}} \right):\)
Decimal number \( = {d_0} \times {2^0} + {d_1} \times {2^1} + {d_2} \times {2^2} + {d_3} \times {2^3} + \ldots \)

Example: Convert \({(10110)_2}\) into a decimal number. 
The binary number given is \(10110\)
Positional weights of each digit from left are \(4, 3, 2, 1, 0\), respectively.
The positional weights for each of the digits are written below each digit. 

Binary Number	\(1\)	\(0\)	\(1\)	\(1\)	\(0\)
Power of \(2\)	\({2^4}\)	\({2^3}\)	\({2^2}\)	\({2^1}\)	\({2^0}\)

Hence the decimal equivalent number is given as: 
\(1 \times {2^4} + 0 \times {2^3} + 1 \times {2^2} + 1 \times {2^1} + 0 \times {2^0}\)
\( = 16 + 0 + 4 + 2 + 0\)
\( = {(22)_{10}}\)

How to Convert Octal to Decimal?

To convert octal to decimal number system, a number with base \(8\) is converted into a number with base \(10\) by multiplying each digit of octal number by decreasing power of \(8\).
Example: Convert \({(3462)_8}\) into a decimal number. 
The positional weights for each of the digits are written below each digit. 
The given digit is \(3 4 6 2\).
Positional weights of each digit from left are \(3 2 1 0\).

Hence, the decimal equivalent number is given as:
\(3 \times {8^3} + 4 \times {8^2} + 6 \times {8^1} + 2 \times {8^0}\)
\( = 1536 + 256 + 48 + 2\)
\( = {(1842)_{10}}\)

How to Convert Hexadecimal to Decimal?

In hexadecimal to decimal conversion, a number with the base \(16\) is converted into a number with base \(10\) by multiplying each digit of hexadecimal number by decreasing the power of \(16\).
Example: Convert \({15_{16}}\) in the decimal numeral system.
Multiply each digit with the decreasing power of \(16\).
\({15_{16}} = 1 \times {16^1} + 5 \times {16^0}\)
\( = 16 + 5\)
\( = {21_{10}}\)

Conversion of Decimal to Other System

How to Convert Decimal to Binary?
Example: Convert \({13_{10}}\) to binary.
Step 1: Divide the given number by \(2\).
Divide \(13\) by \(2\)
Step 2: Take the quotient for the next iteration. And the remainder for the binary digit.
\(\frac{{13}}{2}\) and remainder \(1\) 
Step 3:  Divide the obtained quotient again by \(2\).
\(\frac{6}{2}\) and remainder is \(0\) 
Step 4: Repeat the steps until we get a quotient equal to \(0\).
\(\frac{3}{2}\) and remainder is \(1\) 
\(\frac{1}{2}\) and remainder is \(1\)
So, we collect the remainders in order last to first to get \({1011_2}\).
\({13_{10}} = {1011_2}\)

How to Convert Decimal to Octal?
Decimal to Octal conversion is the same as a decimal to number should be divided by \(8\).
Example: Convert \({60_{10}}\) into octal number system 
Divide \(60\) by \(8\).
\(\frac{{60}}{8}\) and remainder is \(4\) (Most Significant Bit or MSB)
\(\frac{7}{8}\) remainder is \(7\) (Least Significant Bit or LSB)
we count the remainder from LSB to MSB. So, we collect the remainders we get 
\({60_{10}} = {74_8}\)

How to Convert Decimal to Hexadecimal?
The conversion from decimal to hexadecimal is the same as a decimal to binary just instead of \(2\), the number should be divided by \(16\).
Example: Convert \({110_{10}}\) to a hexadecimal number system,
Divide the given number by \(16\) 
\(\frac{{110}}{{16}} = 6\) remainder is \(14\) 
\(\frac{6}{{16}} = 0\) remainder is \(6\)
(replace \(10, 11, 12, 13, 14, 15\) by \(A, B, C, D, E, F\) respectively) 
Hence \(14\) is replace by \(E\).
So, \({110_{10}} = 6E\)

Hexadecimal Number System

A hexadecimal number system has sixteen \((16)\) alphanumeric values,  \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A,B,C,D,E\) and \(F\). This number system has a base of \(16\) because it has \(16\) alphanumeric values. Here A represents the number \(10, B\) represents the number \(11, C\) represents the number \(12, D\) represents the number \(14, E\) represents the number \(15\), and \(F\) represents the number \(16\).

Solved Examples on Decimal Number System

Q.1. Convert \(42A{D_{16}}\) into a decimal number.
Ans: The positional weights for each of the digits can be written below each digit. 
The hexadecimal number given is \(42AD\).  
Positional weights of each digit from left are \(3 2 1 0\) .
Hence, the decimal equivalent number is given as
\(4 \times {16^3} + 2 \times {16^2} + 10 \times {16^1} + 13 \times {16^0}\)
\( = 16384 + 512 + 160 + 13\) 
\( = {(17069)_{10}}\)

Q.2. Convert \({362.35_8}\) into a decimal number.
Ans: The positional weights for each of the digits can be written below each digit. 
The octal number given is \(3 6 2 . 3 5\). 
Positional weights of each digit from left are \(2 1 0-1-2\). 
Hence the decimal equivalent number is given as:
\( = 3 \times {8^2} + 6 \times {8^1} + 2 \times {8^0} + 3 \times {8^{ – 1}} + 5 \times {8^{ – 2}}\)
\( = 192 + 48 + 2 + 0.375 + 0.078125\)
\( = {(242.453125)_{10}}\)

Q.3. Convert \({26_{10}}\) into a binary number.
Ans: Given \({26_{10}}\) is a decimal number.
Divide \(26\) by \(2\)
\(\frac{{26}}{2} = 13\) Remainder \(→0\) (MSB)
\(\frac{{13}}{2} = 6\) Remainder \(→1\)
\(\frac{6}{2} = 3\) Remainder \(→0\)
\(\frac{3}{2} = 1\) Remainder \(→1\)
\(\frac{1}{2} = 0\) Remainder \(→1\) (LSB)
Hence, the equivalent binary number is \((11010)2\)

Q.4. Convert \({(426)_{10}}\) into an octal number.
Ans:

Division	Quotient	Generated remainder
\(\frac{{426}}{8}\)	\(53\)	\(2\)
\(\frac{{53}}{8}\)	\(6\)	\(5\)
\(\frac{6}{8}\)	\(0\)	\(6\)

Hence the converted octal number is \({(652)_8}\)

Q.5. Convert \({(11001)_2}\) into a decimal number. 
Ans: The positional weights for each of the digits can be written in italics below each digit. 
The binary number given is \(1 1 0 0 1 \)
Positional weights of each digit from left are \(4 3 2 1 0\). 
Hence the decimal equivalent number is given as:
\(1 \times {2^4} + 1 \times {2^3} + 0 \times {2^2} + 0 \times {2^1} + 1 \times {2^0}\)
\( = 16 + 8 + 0 + 0 + 1\)
\( = {(25)_{10}}\) 

Summary

This article discussed the number system, four types of number systems viz binary, decimal, octal, and hexadecimal number system. Also, we have discussed how to read the decimal number system and conversions of the decimal number system.

FAQs on Decimal Number System

Q.1. How do you write a decimal number system?
Ans: We will use three rules on the decimal number system to write the further numbers
1. Write numbers from \(0\) to \(9\).
2. Once we reach the digit \(9\), we will make the rightmost digit as \(0\) and add \(1\) to the left digit to make it \(10\).
3. Then, on the rightmost digit, we write until \(9\), and when we reach \(19\), we use \(0\) on the rightmost digit and add \(1\) to the left, so we get \(20\).
4. Similarly, when we reach \(99\), we use \(0\)s in both of these digits places and add \(1\) to the left, giving us \(100\).

Q.2. What are the \(4\) types of number system?
Ans: The four types of number systems are binary, decimal, octal and hexadecimal.

Q.3. What is a decimal number system?
Ans: The decimal number system comprises ten digits from \(0\) to \(9\) that are \(0, 1, 2, 3, 4, 5, 6, 7, 8\) and \(9\). The base or radix of the decimal number system is \(10\) because the total number of digits in the decimal number system is ten. Therefore, we can express all the other digits with the help of these ten-digit numbers.

Q.4. Do numbers look the same in all languages?
Ans: No, numbers don’t look the same in all the languages. For example, in roman numerals, the numbers are written as \(I, II, III, IV\) etc. In normal we write numbers as \(1,2,3,4\) so on. In the binary number system, we write numbers using \(0\) and \(1\) only. In the decimal number system, we write numbers using the digits from \(0\) to \(9\).

Q.5. What is the number system example?
Ans: The number \(525\) is interpreted as
\( = 5 \times {10^2} + 2 \times {10^1} + 5 \times {10^0}\)
\( = 500 + 20 + 5\)