The number system makes up the spine of digital electronics, as we have studied previously. Number system can also be transformed into different number systems via different methods in this blog we will go over the methods to convert one number system into another one.

### Binary (Base-2)

At the core of digital communication lies the binary system, where everything is represented using only two digits: 0 and 1. Computers, being the binary wizards they are, use this system to process and store data in the form of bits (binary digits). Understanding binary is like deciphering the secret code that powers the digital world.

### Decimal (Base-10)

Decimal is the number system we use in our day-to-day lives. It's a base-10 system, meaning it has ten digits (0-9). When you count from 0 to 9, you start over with a new place value, and that's what makes it base 10. Decimal is familiar to us, making it an excellent bridge between our human understanding and the digital language of computers.

### Hexadecimal (Base-16)

Hexadecimal is like a hybrid between binary and human-readable notation. It uses 16 digits, including numbers 0-9 and letters A-F. Hexadecimal is often employed in programming and digital design because it provides a concise representation of binary data. Each hexadecimal digit corresponds to a group of four binary digits, making it a handy shorthand.

### Binary to Decimal Conversion**:**

Let's start with the basics. To convert a binary number (base-2) to decimal (base-10), follow these steps:

Write down the binary number.

Assign powers of 2 to each digit, starting from the right with 2^0.

Multiply each binary digit by its corresponding power of 2.

Sum up the results to get the decimal equivalent.

**Example:** Convert 1011 (Binary) to Decimal.

(1×2^3)+(0×2^2)+(1×2^1)+(1×2^0)=8+0+2+1=11

### Decimal to Binary Conversion:

Converting a decimal number to a binary involves repeated division by 2. Here's a simplified guide:

Divide the decimal number by 2.

Note the remainder (it will be either 0 or 1).

Repeat the process with the quotient until the quotient is 0.

Read the remainder in reverse order to get the binary equivalent.

__Example:__

Convert 14 (decimal) to binary.

14÷2=7 remainder 0

7÷2=3 remainder 1

3÷2=1 remainder 1

1÷2=0 remainder 1

Reading the remainders backward gives 1110 in binary.

### Binary to Hexadecimal and Vice Versa:

When it comes to binary-to-hexadecimal conversion (and vice versa), it's all about grouping binary digits into sets of four and mapping them to their hexadecimal equivalents.

### Converting any number system to decimal:

Identify the base of the original number system (e.g., 2 for binary, 8 for octal, 16 for hexadecimal).

Multiply each digit in the original number by the base raised to the power of its position (rightmost digit is position 0).

Add the products of these multiplications together. The result is the decimal equivalent.

__Example____:__ Binary to decimal:

1101 binary = (1*2^3) + (1*2^2) + (0*2^1) + (1*2^0) = 8 + 4 + 0 + 1 = 13 decimal

### Converting decimal to any other number system:

Repeatedly divide the decimal number by the base of the target number system. Record the remainder from each division in reverse order. These remainders form the digits in the target number system.

__Example:__

Decimal to binary:

13 decimal / 2 = 6 remainder 1

6 / 2 = 3 remainder 0

3 / 2 = 1 remainder 1

1 / 2 = 0 remainder 1 (reading remainders in reverse order)

13 decimal = 1101 binary

### Direct conversions:

__Octal to binary:__ Group octal digits into groups of 3 and convert each group to 3-bit binary

(**e.g.**, 472 octal = 100 111 010 binary).

__Binary to octal: __Group binary digits into groups of 3 and convert each group to an octal digit (**e.g**., 101110 binary = 56 octal).

__Hexadecimal to binary__: Convert each hexadecimal digit to its 4-bit binary equivalent.

(**e.g**., A5 hexadecimal = 1010 0101 binary).

__Binary to hexadecimal:__ Group binary digits into groups of 4 and convert each group to a hexadecimal digit (**e.g**., 11010110 binary = DA hexadecimal).

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