Base Shifting

Base shifting is a mathematical concept used in various fields, including computer science, digital electronics, and number systems. It involves representing a number in a different base or radix. The most common number system is the decimal system (base 10), where each digit can have ten possible values (0 to 9). However, in base shifting, we represent numbers in other bases, such as binary (base 2), hexadecimal (base 16), or any other chosen base.

Converting a number from one base to another:

1. Decimal to Binary:

To convert a decimal number to binary, we repeatedly divide the number by 2 and note the remainders from bottom to top, forming the binary representation.

     Example:

Decimal number: 27

Step 1: 27 / 2 = 13 remainder 1

Step 2: 13 / 2 = 6 remainder 1

Step 3: 6 / 2 = 3 remainder 0

Step 4: 3 / 2 = 1 remainder 1

Step 5: 1 / 2 = 0 remainder 1

Binary representation: 11011

2. Decimal to Hexadecimal:

To convert a decimal number to hexadecimal, we repeatedly divide the number by 16 and note the remainders from bottom to top, forming the hexadecimal representation. For remainders greater than 9, we use the corresponding letters A, B, C, D, E, and F.

   Example:

Decimal number: 255

Step 1: 255 / 16 = 15 remainder 15 (F in hexadecimal)

Step 2: 15 / 16 = 0 remainder 15 (F in hexadecimal)

Hexadecimal representation: FF

3. Binary to Decimal:

To convert a binary number to decimal, we multiply each digit by its corresponding power of 2 and then sum all the results.

  Example:

Binary number: 11011

Decimal representation: (1 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 16 + 8 + 0 + 2 + 1 = 27

4. Binary to Hexadecimal:

To convert a binary number to hexadecimal, first, group the binary digits into sets of four (starting from the right) and then convert each group into its hexadecimal equivalent.

  Example:

Binary number: 11011010

Grouping: 1101 1010

Hexadecimal representation: DA

Base shifting is essential in computing because it allows more efficient storage and representation of numbers in various systems. It is commonly used in programming, digital logic design, and data representation. Understanding base shifting helps in grasping the underlying principles of different number systems and facilitates smooth conversions between them.