As you see decimal, binary, octal, and hexadecimal number schemes are positional value number systems. To Conversion from Decimal Number System to any Other Number System Step by Step. we fair essential to add the formation of each digit with its positional value. Here we are going to learn another conversion among these number schemes.

**Decimal Number to Binary Number** System

Decimal numbers can be different to binary by recurrent division of the number by 2 while footage the remainder. Let’s look at an example to understand how this occurs.

**The Decimal Numbering System**

In the decimal, base-10 (den), or denary numbering system, each integer number column has standards of units, tens, hundreds, thousands, etc as we change along with the number from right to left. Mathematically these values are written as 100, 101, 102, 103, etc. Then respectively position to the left of the decimal point indicates an increased positive power of 10. Likewise, for fractional numbers, the weight of the number converts more negative as we move from left to right, 10-1, 10-2, 10-3, etc.

So we can see that the “decimal numbering system” has a base of 10 or *modulo-10* (sometimes called MOD-10) with the position of each digit in the decimal system representative the scale or weight of that digit as q is equal to “10” (0 through 9). For example, 20 (twenty) is the same as saying 2 x 10^{1} and therefore 400 (four hundred) is the same as saying 4 x 10^{2}.

The value of any decimal number will be equal to the sum of its digits increased by their respective weights. For example: N = 6163_{10} (Six Thousand One Hundred and Sixty-Three) in a decimal set-up is equal to:

6000 + 100 + 60 + 3 = 6163

or it can be written reflecting the weight of each digit as:

( 6×1000 ) + ( 1×100 ) + ( 6×10 ) + ( 3×1 ) = 6163

or it can be written in polynomial form as:

( 6×10^{3} ) + ( 1×10^{2} ) + ( 6×10^{1} ) + ( 3×10^{0} ) = 6163

Where in this decimal numbering system example, the left most digit is the most important digit, or MSD, and the right most digit is the least significant digit or LSD. In other arguments, the digit 6 is the MSD since its left most position carries the most mass, and the number 3 is the LSD as its right most position carries the least weight.

**The Binary Numbering System**

The **Binary Numbering System** is the most important numbering system in all digital and computer-based systems and binary numbers trail the same set of rules as the decimal numbering system. But distinct from the decimal system which practices powers of ten, the binary numbering system works on powers of two giving a binary to the decimal alteration from base-2 to base-10.

Digital logic and computer systems use fair two values or states to represent a condition, a logic level “1” or a logic level “0”, and each “0” and “1” is considered to be a single digit in a Base-of-2 (bi) or “binary numbering system”.

In the binary numbering system, a binary number such as 101100101 is expressed with a string of “1’s” and “0’s” with each digit along the string from right to left captivating a value twice that of the previous digit. But as it is a binary digit it can only have a value of either “1” or “0” therefore, q is equal to “2” (0 or 1) with its location representative its mass within the string.

As the decimal number is a biased number, changing from decimal to binary (base 10 to base 2) will also produce a weighted binary number with the right-hand most bit being the **Least Significant Bit** or **LSB**, and the left-hand most bit being the **Most Significant Bit** or **MSB**

The remains are to be deliver from bottom to top to get the binary equal.

43_{10} = 101011_{2}

**Decimal Number to Octal Number System**

- Decimal numbers can be different from octal by repeated division of the number by 8 while recording the remainder. Let’s look at an example to understand how this happens.

**Decimal Number:** All the numbers to the base ten are called decimal numbers. These are the usually used numbers, which are 0-9. It has an equal integer part and a decimal part. It is parted by a decimal point (.). Numbers on the left of the decimal are integers and numbers on the right of the decimal are the decimal part. Sample: (236.89)_{10}, (54.2)_{10}, etc.

**Octal number: **These are the numbers with base 8. If x is a number formerly the octal number is meant as x_{8}. It contains digits from 0 to 7. Sample: (212)_{8}, (121)_{8}, etc.

**Decimal to Octal Table**

Decimal | Equivalent Octal Number | Decimal | Equivalent Octal Number |

0 | 0 | 9 | 11 |

1 | 1 | 10 | 12 |

2 | 2 | 11 | 13 |

3 | 3 | 12 | 14 |

4 | 4 | 13 | 15 |

5 | 5 | 14 | 16 |

6 | 6 | 15 | 17 |

7 | 7 | 16 | 20 |

8 | 10 | 17 | 21 |

Reading the remains from bottom to top,

473_{10} = 731_{8}

**Decimal Number to Hexadecimal** Number System

Decimal numbers can be transformed to octal by repeated division of the number by 16 while recording the remainder. Let’s look at an example to understand how this occurs.

**Decimal to Hexadecimal Table**

To convert the decimal number system to hex, students have to remember the table given below, to solve the problems in a quick way.

Decimal Number | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

Equivalent Hexadecimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |

**Decimal to Hexadecimal Conversion with Steps**

To convert a decimal number into hex, follow the below-given, steps;

- First, divide the decimal number by 16, as the number as an integer.
- Keep aside the remainder left.
- Over, divide the amount by 16 and repeat till you get the quotient value equal to zero.
- Now take the values of the remainder’s left in the reverse order to get the hexadecimal numbers.

**Note:** Remember, from 0 to 9, the numbers will be calculated as the same in the decimal system. But from 10 to 15, they are expressed in alphabetical order such as A, B, C, D, E, F and so on.

Let us take an example to understand the stages assumed above for decimal to hex conversion.

**Example:** Convert (960)_{10} into hexadecimal.

**Solution: **Following the step,

- First, divide 960 by 16.

960 ÷ 16 = 60 and remainder = 0

- Again, divide quotient 60 by 16.

60 ÷ 16 = 3 and remainder 12.

- Again dividing 3 by 16, will leave proportion=0 and rest = 3.
- Now taking the remainder in reverse order and substituting the equivalent hexadecimal value for them, we get,

3→3, 12→C and 0→0

Consequently, (960)_{10} = (3C0)_{16}

This example must have made you comprehend the conversion technique of decimal to hex. Let us resolve a few more examples to get a moral repetition over it.

Reading the remains from bottom to top we get,

423_{10} = 1A7_{16}