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Last updated on October 13, 2025

Decimal Number System

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The decimal number system uses 10 digits (0 to 9) with a base of 10. This number system has been in use since ancient times and is also known as the Arabic number system. Other number systems used in mathematics are binary, octal, and hexadecimal number systems. In this topic, we will be focusing on the decimal number system.

Decimal Number System for US Students
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What is a Decimal Number System?

We use digits from 0 to 9 in the decimal number system. Since the system uses 10 digits, the base is 10. Unless specified, numbers without a base are assumed to be decimal (base 10). In the decimal number system, the place values of a number are read from right to left; the first few place values are ones, tens, hundreds, thousands, and so on. Let’s consider the number 423.

 

Here, 3 is in the ones place (\(3 × 1 = 3\))
  
2 is in the tens place (\(2 \times 10 = 20\))

4 is in the hundreds place (\(4 × 100 = 400 \)

 

Now we can add all of them:  

\(400 + 20 + 3 = 423 \)

 

To find the value of a number, we can multiply each digit by its place value and then add the products together.

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What are the Rules of the Decimal Number System?

The base of a decimal number system is 10, and it includes the digits 0 to 9 to represent the number’s place values. The digit in the tens place is 10 times greater than the digit in the ones place. Here are some rules related to the decimal number system:
 

 

  • 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are the 10 numbers in the decimal number system.
     

 

  • In the decimal number system, when a digit reaches 9, and we add 1, the digit becomes 10. However, since we only have digits from 0 to 9, we write 0 instead of 10. Then, carry over 1 to the next higher place value. So, when we go from 9 to 10, the number on the right becomes 0, and 1 is added to the left side, making it 10. 
     

 

 

Let us take an example to understand the rules better. If the number is \((142)_{10}  \)
\({(142)_{10}} ={ 1 × {10^2} }+ {4× {10^1} }+ {2 × {10^0}}\)

 

If the numbers have a decimal point, then the place value of the numbers after the decimal point continues in decreasing powers of 10.

For instance, if the given number is\({(35.27)_{10}}\)
\({(35.27)_{10}} = {(3 × {10^1})} + (5 × {10^0})+ (2 × {10^{-1}}) + (7 × 10^{-2}) \)

 

Therefore, \({(35.27)_{10}} = 30 + 5 + 0.2 + 0.07 \)

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How to Convert into Decimal Number System?

The binary number system, decimal number system, octal number system, and hexadecimal number system are the four main types of number systems. Base numbers of each number system help in converting one number system to another.

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Binary to Decimal Conversion:

We use only two digits in the binary number system (0 and 1), and its base is 2. So, to convert a binary number to a decimal number, we must multiply every digit of the binary number by a power of 2. The exponent of 2 depends on the position of the binary number. The rightmost digit is multiplied by \(2^0\), the next digit is multiplied by \(2^1\), and so on. After the multiplication process is done, we add up the results to get the converted value. 

 

Multiply each digit of the binary number by 2 raised to the power of its position from right to left, starting at 0. Then add the results together to get the decimal number. 

 

The given binary number is \((1011)_2\).

\({ (1011)_2} = (1 × {2^3}) + (0 × {2^2}) + (1 × {2^1}) + (1 × {2^0})  \)
\( = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)  \)             
\( = 8 + 0 + 2 + 1 \)
\(= 11\)              

 

Thus, \({(1011)_2} = {(11)_{10}}\)

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Octal to Decimal Conversion:

The octal number system has a base of 8. It has 8 digits, from 0 to 7, to represent numbers. To convert an octal number to a decimal number, multiply each digit by the decreasing power of 8 and add the products. Let us convert (167)8 to its decimal form. 

 

\({(167)_8} = {(1 × 8^2)} + ({6 × 8^1)} + {(7 × 8^0)} \)
\(= (1 × 64) + (6 × 8) + (7 × 1) \)
\(= 64 + 48 + 7  \)
\(= 119 \)

 

Thus, \({(167)_8 }= {(119)_{10}} \)
After the conversion, the base power changes from 8 to 10.

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Hexadecimal to Decimal Conversion:

The hexadecimal number system uses 16 symbols: digits from 0 to 9 and letters from A to F. The conversion of a hexadecimal number to a decimal number happens when we multiply each digit of the hexadecimal number by the powers of 16. Once again, the rightmost digit will be multiplied by 160, the next digit by 161, and so on. After multiplying the digits by the powers of 16, we should add the results to obtain the converted value. 

 

\((18)_{16} = (1 × 16^1) + (8 × 16^0)\)

\(= (1 × 16) + (8 × 1) \)
\(= 16 + 8  \)
\(= 24 \)


Thus, \({(18)_{16}} = {(24)_{10}}\)

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How to Convert From Decimal Number System to Other Systems

The conversion of decimal numbers to other number systems is similar to the conversion from a different number system to decimal. The key element in the conversion is the base number of each number system.

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Decimal to Binary Conversion:

Decimal numbers can be converted to binary by dividing the number repeatedly by 2 until the quotient becomes 0. In every step, the remainder is noted (either 1 or 0). In the end, the remainders are written from bottom to top to get the binary equivalent of the decimal number.

 

Dividend Remainder
\({138\div 2} = 69\) 0
\({69 \div 2} = 34\) 1
\(34\div2 = 17\) 0
\(17\div2 = 8\) 1
\(8\div2 = 4\) 0
\(4\div2 = 2\) 0
\(2\div2 = 1\) 0
\(1\div2 = 0\) 1

 

Now write the remainder from bottom to top. Thus, the binary number of \({{(138)}_{₁₀} = {10001010}_₂} \).

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Decimal to Octal Conversion:

Decimal numbers can be converted to octal by dividing the number by 8 repeatedly until the quotient is less than 8. In every step, the remainder is noted. After the division process, the remainders are written from bottom to top to obtain the converted value. In the division process, the first remainder is known as the least significant digit (LSD), and the last remainder is called the most significant digit (MSD). Let’s see how to convert \({(65)_{10}}\) to an octal number.

 

Dividing by 8 Quotient Remainder
\(65\div8\) 8 1
\(8\div8\) 1 0
\(1\div8\) 0 1

 

Write the numbers obtained as remainders from bottom to top. 

Thus,\( {(65)_{10}} = {(101)_{8}}\)

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Decimal to Hexadecimal Conversion:

Each decimal number will be divided by the base number of hexadecimal (16) until the quotient becomes 0. For example, convert the decimal number \((150)_{10} \) to hexadecimal. 

 

Dividing by 16 Quotient Remainder
\(150\div16\) 9 6
\(6\div16\) 0 6

 

Now write the remainder from the bottom to the top to get the hexadecimal number. Thus,  \({(150)_{10}} = {(96)_{16}}\)

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Tips and Tricks to Master the Decimal Number System

Mastering the decimal system helps students to understand the place value, perform accurate calculations, and apply numbers confidently in real-life situations. In this section, we will learn a few tips and tricks to master the decimal number system. 

 

  • Understand Place Value: Always remember that each digit’s place represents a power of 10. Leftward digits increase in value, and rightward digits (after the decimal) decrease in value.
     
  • Memorize Base Values: Know the base of each number system: Binary = 2, Octal = 8, Decimal = 10, Hexadecimal = 16.
     
  • Use Expanded Form: Break numbers into their expanded form (e.g., 345.27 = 300 + 40 + 5 + 0.2 + 0.07) to understand the contribution of each digit.
     
  • Practice Conversions: Convert numbers between decimal, binary, octal, and hexadecimal to strengthen your grasp of bases and place values.
     
  • Visualize with Charts: Use place value charts to clearly see the value of each digit in large or decimal numbers.
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Common Mistakes and How to Avoid Them in Decimal Number System

It is easy to make mistakes while dealing with the decimal number system. Even the slightest of errors can change the final result completely. Therefore, it is important to avoid commonly made mistakes, and some of them are mentioned below:  

Mistake 1

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Confusion Between Place Values of Digits

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Students should learn the place values and the proper arrangement of them in a decimal number system. If they forget the place values, then the result will be incorrect. Keep in mind that the digit in the tens place is 10 times greater than the digit in the ones place.

 

For example, we can expand the number \((142)_{10} \)as:

\((142)_{10} = 1 × 10^2 + 4 × 10^1 + 2 × 10^0\)

Mistake 2

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Misunderstanding Numbers with Decimal Point

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Students should be taught how to properly expand numbers containing a decimal point. In the decimal number system, the place values of digits after the decimal point follow decreasing powers of 10.

 

For instance, if the given number is \((25.2)_{10}\)
\({(25.2)_{10}} = 2 × 10^1 + 5 × 10^0 + 2 × 10^ -1 +  7 × 10^-2   \)

 

Therefore, \((25.27)_{10} = 20 + 5 + 0.2 + 0.07\)

Mistake 3

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Forgetting the Base Number of the Decimal Number System 

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Students should understand that the decimal number system is based on powers of 10. It uses 10 digits, from 0 to 9, to represent the place value of numbers. As we move to the left, each place value increases by a power of 10. Forgetting the base (10) can lead to incorrect place values and wrong results.

 

For example, the decimal number is written as  \((436)_{10}\), and the base number is written with the number.

Mistake 4

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Forgetting the Base Numbers of Other Number Systems

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When converting a binary, octal, or hexadecimal number into a decimal number, each digit must be multiplied by the respective base number to get the decimal number. Therefore, students should learn the base number of each number system. The base value of the binary number system is 2, the octal number system has 8 as the base value, and the base value of the hexadecimal number system is 16.

Mistake 5

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Incorrect Conversion into Decimal Number System

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 Whenever we convert a particular number system to decimal, we should remember to multiply the number’s digits by the decreasing power of its base. After the multiplication process, we should also remember to add up the products to get the correct decimal form. The conversion of a hexadecimal number to a decimal number is shown below:  

 

\((18)_{16} = (1 × 16^1) + (8 × 16^0)\)

          \( = (1 × 16) + (8 × 1) \)

           \(= 16 + 8  \)

          \( = 24 \)

Thus, \((18)_{16} = (24)_{10}\)

 
\(\)

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Real-life Applications of Decimal Number System

The decimal number system plays a crucial role in determining the place value of each digit. The place value of each digit in a number is based on powers of 10. Here are some real-world applications of the decimal number system: 

 

  • The decimal system helps students determine the place value of digits and perform arithmetic operations and build a string foundation in mathematics

 

  • The decimal number system helps us determine the value of the money we have. For example, if we have $288, we can easily understand its value, such as we have two hundred eighty-eight dollars. 

 

  • Mobile numbers, PIN codes, and numerical passwords are based on the decimal number system. It helps us to recognize and memorize, and use these numbers efficiently.
     
  • From weighing items to measuring lengths, volumes, and temperatures, the decimal system ensures accuracy and standardization in everyday measurements.
     
  • The decimal system is used to record hours, minutes, and seconds, enabling consistent and precise time calculations.
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Solved Examples of Decimal Number System

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Problem 1

Convert the binary number (1010)₂ to a decimal number.

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\((10)_{10}\)

Explanation

Here, we must multiply each digit of the binary number by the decreasing power of 2, starting from the rightmost digit. The given binary number is  \({(1010)_2} \)

\({(1010)_2}  =  (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0)  \) 
Now we can calculate the individual terms: 
\(           = (1 × 8) + (0 × 4) + (1 × 2) + (0 × 1)  \)
Then, add the results together to get the decimal number. 

\( = 8 + 0 + 2 + 0\)

 

\(= 10\)

Thus, \(​​(1010)_{2} = (10)_{10}\)

 

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Problem 2

Convert (67)₈ to a decimal number.

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\((55)_{10}\)

Explanation

We multiply each digit of the octal number (67)8 by the decreasing power of 8, beginning at the rightmost digit. After that, we add the products to convert an octal number to a decimal number.

 

The given number is \((67)_8\).

\((67)_8 = (6 × 8^1) + (7 × 8^0) \)

\(= (6 × 8) + (7 × 1)  \)

 

Now multiply the values:

       \(= 48 + 7  \)
       \(= 55 \)
Thus, \((67)_8 = (55)_{10}\)

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Problem 3

Convert (2B)₁₆ to a decimal number.

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\((43)_{10}\)

Explanation

The given hexadecimal number is \((2B)_{16}\).

\( (2B)_{16} = (2 × 16^1) + (B × 16^0)\)

 

In the hexadecimal number system, B is equal to 11. 

 


           \(= (2 × 16^1) + (11 × 16^0)\)
          \( = (2 × 16) + (11 × 1)\)

 

 

Now we can multiply the values:
           \(= 32 + 11\)
           \(= 43  \)

 

Therefore, \((2B)_{16} = (43)_{10}\)

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Problem 4

Convert (12)_10 to a binary number.

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\((1100)_2\)

Explanation

The given number is \((12)_{10}\). To convert a decimal number to binary, we will divide the decimal number by 2 repeatedly, noting the remainder each time.

 

Divide 12 by 2:
      \(12 ÷ 2 = 6\)    Remainder = 0

 

Divide 6 by 2:
     \(6 ÷ 2 = 3\)   Remainder = 0

 

Divide 3 by 2:
     \(3 ÷ 2 = 1\)    Remainder = 1 

 

Divide 1 by 2:
     1 ÷ 2 = 0    Remainder = 1 

 

Now, we can write the remainder from bottom to top:
   \(    (12)_{10} = (1100)_{2}\)

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Problem 5

Convert (18)_10 to an octal number.

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\((22)_8 \)

Explanation

Here, the given number is \(18_{10}\). To convert a decimal number to an octal number, we divide the decimal number by 8, which is the base of the octal number system. We then note the quotients and remainders, continuing the division until the quotient becomes 0. 

 

Divide 18 by 8:
    \(18 ÷ 8 =  2\) (Quotient)  Remainder = 2

 

Divide the quotient 2 by 8:
     \(2 ÷ 8 = 0\) (Quotient)    Remainder = 2

 

Write the remainder from bottom to top: 
     \((18)_{10} = (22)_8 \)

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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