**Number System Conversions:**

The number system, which we have discussed in the previous article named “let’s operate the number system”. In which we talk about the relationship of (Decimal, Binary, Octal, and Hexadecimal). We can convert a number of a particular system to its equivalent number in other systems.

**Conversion from Decimal system to other systems:**

The method of converting from decimal to another system is by successive division by the base of the objective system. The remainders of these divisions are arranged in reverse order.

## Decimal to Binary Conversions:

To convert from decimal to binary, divide the decimal number by 2, which is the base of the binary system.

Here we have decimal 57 and going to convert into its binary equivalent.

The binary number is obtained by arranging the remainders from bottom to top. Thus, the equivalent binary number is : (111001)

## Decimal to Octal Conversion:

To convert a decimal number to its octal equivalent, divide the decimal number successively by 8 in a similar manner as was done with decimal to binary conversions. The remainders are read in reverse order to obtain the required octal number.

Here we have decimal digits 1769 and going to convert into its octal to equivalent:

Thus, we got octal number which is (3351).

## Decimal to Hexadecimal conversions:

Decimal to Hexadecimal conversions can be carried out using the same technique as shown above binary and octal conversion with the division being done by 16.

Here we have decimal digits 6958 and going to convert into its hexadecimal equivalent.

Here, you need to know the in hexadecimal number system that alphabet

- A denotes 10
- B denotes 11
- C denotes 12
- D denotes 13
- E denotes 14
- F denotes 15
- G denotes 16

Therefore the equivalent of this decimal number in hexadecimal is (1B2E).

**Let’s learn Conversion from other systems to the Decimal system:**

Converting from other systems to decimal system involves multiplication.

Multiply each digit of the given number by its positional weight. The base of the given number multiplies each positional weight.

Final value is obtained by adding each positional value.

We can perform this process through these steps.

*Step 1: *Multiply each digit of the given number by its base raised to positional power.

*Step 2: *Add all results of multiplication of each position.

## Binary to Decimal Conversion:

We convert binary to decimal by finding the decimal equivalent of the binary array of digits (10011)_{2}and expanding the binary digits into a series with a base of 2 giving an equivalent of (19)

_{10}in decimal or denary.

Note that in number conversion systems “subscripts” are used to indicate the relevant base numbering system, 1001_{2} = 9_{10}. If no subscript is used after a number, then it is generally assumed to be decimal.

Let’s convert decimal number to binary (10011)^{2}:

= 1 x 2^{4} + 0 x 2^{3} + 0 x 2^{2} + 1 x 2^{1} + 1 + 2^{0}

= 1 x 16 + 0 x 8 + 0 x 4 + 1 x 2 + 1 x 1

= 16 + 0 + 0 + 2 + 1

= ( 19 )_{10}

The equivalent decimal number is 19 for binary number 10011.

When converting from Binary to Decimal or even from Decimal to Binary, we need to be careful that we do not mix up the two sets of numbers.

For example, if we write the digits 10 on the page it could mean the number “ten” if we assume it to be a decimal number, or it could equally be a “1” and a “0” together in binary, which is equal to the number two in the weighted decimal format from above. for example, if we were using a binary number string we would add the subscript “2” to denote a base-2 number so the number would be written as 10_{2}. Likewise, if it was a standard decimal number we would add the subscript “10” to denote a base-10 number so the number would be written as 10_{10}.

## Octal to Decimal Conversion:

Octal numbers, therefore, have a range of just “8” digits, (0, 1, 2, 3, 4, 5, 6, 7) making them a Base-8 numbering system and therefore, q is equal to “8”.

Convert the octal number (546)_{8} to its decimal number equivalent, (base-8 to base-10).

Step 1: = 5 x 8^{2} + 4 x 8^{1} + 6 x 8^{0}

= 5 x 64 + 4 x 8 + 6 x 1

Step 2: = 320 + 32 + 6

= (358)_{10}

Converting octal to decimal shows that (546)_{8} in its Octal form is equivalent to (358)_{10} in its Decimal form.

While Octal is another type of digital numbering system, it is little used these days instead of the more commonly used Hexadecimal Numbering System is used as it is more flexible.

## Hexadecimal to Decimal Conversions:

The “Hexadecimal” or simply “Hex” numbering system uses the **Base of 16** system and is a popular choice for representing long binary values because their format is quite compact and much easier to understand.

Then in the Hexadecimal Numbering System, we use the numbers from 0 to 9 and the capital letters A to F to represent it’s Binary or Decimal number equivalent, starting with the least significant digit at the right-hand side.

Convert the following Hexadecimal number (3AC8)_{16} into Decimal or Denary equivalent using subscripts to identify each numbering system.

Step 1: = 3 x 16^{3} + A x 16^{2} + C x 16^{1} + 8 x 16^{0}

= 3 x 4096 + 10 x 256 + 12 x 16 + 8 x 1

*Step 2 = 12288 + 2560 + 192 + 8*

= (15048)_{10}

Then, the Decimal number of **15048** can be represented as **3AC8 **in Hexadecimal.