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.

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

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.

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

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.

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

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.

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.

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

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.

Write the numbers obtained as remainders from bottom to top. 

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

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. 

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

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.

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.