Last updated on July 9th, 2025
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.
We use digits from 0 to 9 in the decimal number system. Since there are 10 numbers, the base number is 10. Unless stated otherwise, any number expressed without a base belongs to this number system, and its base is 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 × 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:
Let us take an example to understand the rules better. If the number is (142)10
(142)10 = 1 × 102 + 4 × 101 + 2 × 100
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 × 101) + (5 × 100 )+ (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 20, the next digit is multiplied by 21, and so on. After the multiplication process is done, we add up the results to get the converted value.
Here, we must multiply each binary number by the decreasing power of 2. Then add the results together to get the decimal number.
The given binary number is (1011)2.
(1011)2 = (1 × 23) + (0 × 22) + (1 × 21) + (1 × 20)
= (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 × 82) + (6 × 81) + (7 × 80)
= (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 × 161) + (8 × 160)
= (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.
Dividend | Remainder |
138/2 = 69 | 0 |
69/2 = 34 | 1 |
34/2 = 17 | 0 |
17/2 = 8 | 1 |
8/2 = 4 | 0 |
4/2 = 2 | 0 |
2/2 = 1 | 0 |
1/2 = 0 | 1 |
Now write the remainder from bottom to top. Thus, the binary number of (138)10 is 100010102.
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 |
65/8 | 8 | 1 |
8/8 | 1 | 0 |
1/8 | 0 | 1 |
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.
Dividing by 16 | Quotient | Remainder |
150/16 | 9 | 6 |
6/16 | 0 | 6 |
Now write the remainder from the bottom to the top to get the hexadecimal number. Thus, (150)10 = (96)16.
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:
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:
Convert the binary number (1010)₂ to a decimal number.
(10)10
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 × 23) + (0 × 22) + (1 × 21) + (0 × 20)
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
Convert (67)₈ to a decimal number.
(55)10
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 × 81) + (7 × 80)
= (6 × 8) + (7 × 1)
Now multiply the values:
= 48 + 7
= 55
Thus, (67)8 = (55)10
Convert (2B)₁₆ to a decimal number.
(43)10
The given hexadecimal number is (2B)16.
(2B)16 = (2 × 161) + (B × 160)
In the hexadecimal number system, B is equal to 11.
= (2 × 161) + (11 × 160)
= (2 × 16) + (11 × 1)
Now we can multiply the values:
= 32 + 11
= 43
Therefore, (2B)16 = (43)10
Convert (12)_10 to a binary number.
(1100)2
The given number is (12)10. To convert a decimal number to binary, we will divide each digit by 2 and list the numbers obtained as remainders.
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
Convert (18)_10 to an octal number.
(22)8
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
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.
: She loves to read number jokes and games.