# Decimal Number System

In the number system, each number is represented by its base. If the base is 2 it is a binary number, if the base is 8 it is an octal number, if the base is 10, then it is called decimal number system and if the base is 16, it is part of the hexadecimal number system. The conversion of decimal numbers to any other number system is an easy method. But to convert other base number systems into decimal numbers requires practice. In this article, let us learn more on the decimal number system and the conversion from a decimal number system to other systems here in detail.

**Table of Contents:**

- Definition
- Conversion from Other Bases to Decimal Number System
- Decimal Number System to Other Bases

## What is the Decimal Number System?

In the **decimal number system**, the numbers are represented with base 10. The way of denoting the decimal numbers with base 10 is also termed as decimal notation. This number system is widely used in computer applications. It is also called the base-10 number system which consists of 10 digits, such as, 0,1,2,3,4,5,6,7,8,9. Each digit in the decimal system has a position and every digit is ten times more significant than the previous digit. Suppose, 25 is a decimal number, then 2 is ten times more than 5. Some examples of decimal numbers are:-

(12)_{10}, (345)_{10}, (119)_{10}, (200)_{10}, (313.9)_{10}

A number system which uses digits from 0 to 9 to represent a number with base 10 is the decimal system number. The number is expressed in base-10 where each value is denoted by 0 or first nine positive integers. Each value in this number system has the place value of power 10. It means the digit at the tens place is ten times greater than the digit at the unit place. Let us see some more examples:

(92)_{10} = 9×10^{1}+2×10^{0}

(200)_{10} = 2×10^{2}+0x10^{1}+0x10^{0}

The decimal numbers which have digits present on the right side of the decimal (.) denote each digit with decreasing power of 10. Some examples are:

(30.2)_{10}= 30×10^{1}+0x10^{0}+2×10^{-1}

(212.367)_{10} = 2×10^{2}+1×10^{1}+2×10^{0}+3×10^{-1}+6×10^{-2}+7×10^{-3}

**Also, read:**

## Conversion From Other Bases to Decimal Number System

Let us see here how in number system conversion, we can convert any other base system such as binary, octal and hexadecimal to the equivalent decimal number.

### Binary to Decimal

In this conversion, a number with base 2 is converted into number with base 10. Each binary digit here is multiplied by decreasing power of 2. Let us see one example:

Example: Convert (11011)
_{2} to decimal number.Solution: Given (11011) We need to multiply each binary digit with the decreasing power of 2. That is; 1×2 =16+8+0+2+1 =27 Therefore, (11011) |

### Octal to Decimal

In this conversion, a number with base 8 is converted into number with base 10. Each digit of octal number here is multiplied by decreasing power of 8. Let us see one example:

Example: Convert 121
_{8} into the equivalent decimal number.Solution: Given (121) Here, we have to multiply each octal digit with the decreasing power of 8, such as; 1×8 =64+16+1 =81 |

### Hexadecimal to Decimal

In this conversion, a number with base 16 is converted into number with base 10. Each digit of hex number here is multiplied by decreasing power of 16. Let us understand with the help of an example:

Example: Convert 12
_{16} into a decimal number.Solution: Given 12 Multiply each digit with decreasing power of 16 to obtain an equivalent decimal number. 1×16 =16+2 =18 |

## Decimal Number System to Other Bases

Earlier we learned about converting other base number systems into a decimal number, Here we will learn how to convert a decimal number into different base numbers. Let us see one by one.

### Decimal to Binary

To convert a decimal number into an equivalent binary number we have to divide the original number system by 2 until the quotient is 0, when no more division is possible. The remainder so obtained is counted for the required number in the order of LSB (Least significant bit) to MSB (most significant bit). Let us go through the example.

Example: Convert 26
_{10} into a binary number.Solution: Given 26 Divide 26 by 2 26/2 = 13 Remainder →0 (MSB) 13/2 = 6 Remainder →1 6/2 = 3 Remainder →0 3/2 = 1 Remainder →1 ½ = 0 Remainder →1 (LSB) Hence, the equivalent binary number is (11010) |

### Decimal to Octal

Here the decimal number is required to be divided by 8 until the quotient is 0. Then, in the same way, we count the remainder from LSB to MSB to get the equivalent octal number.

Example: Convert 65.
_{10} into an octal numberSolution: Given 65 Divide by 8 65/8 = 8 Remainder →1 (MSB) 8/8 = 1 Remainder →0 ⅛ = 0 Remainder →1 (LSB) Hence, the equivalent octal number is (101) |

### Decimal to Hexadecimal

The given decimal number here is divided by 16 to get the equivalent hex. The division of the number continues until we get the quotient 0.

Example: Convert 127_{10} to a hexadecimal number.
Solution: Given 127 Divide by 16 127/16 = 7 Remainder →15 7/16 = 0 Remainder → 7 In the hexadecimal number system, alphabet F is considered as 15. Hence, 127 |

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