Inter-Conversions in Various Number Systems:

1. Binary to Decimal Conversion:
   • To convert a binary number to decimal, multiply each digit of the binary number by 2 raised to the power of its position from right to left, starting from 0.
   • Add the results together to obtain the decimal equivalent. Example: Convert binary number 1011 to decimal: 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0 = 8 + 0 + 2 + 1 = 11 (decimal)
2. Decimal to Binary Conversion:
   • To convert a decimal number to binary, repeatedly divide the decimal number by 2 and record the remainders in reverse order.
   • The binary equivalent is obtained by reading the remainders from bottom to top. Example: Convert decimal number 13 to binary: 13 ÷ 2 = 6 remainder 1 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1 Binary equivalent: 1101
3. Binary to Octal Conversion:
   • To convert a binary number to octal, group the binary digits into groups of three from right to left. Add leading zeros if necessary.
   • Convert each group of three binary digits to its octal equivalent. Example: Convert binary number 1011101 to octal: 101 110 1 (grouped into three) Octal equivalent: 5 6 1 Octal representation: 561
4. Octal to Binary Conversion:
   • To convert an octal number to binary, convert each octal digit to its binary equivalent. Example: Convert octal number 46 to binary: 4 -> 100 6 -> 110 Binary equivalent: 100110
5. Binary to Hexadecimal Conversion:
   • To convert a binary number to hexadecimal, group the binary digits into groups of four from right to left. Add leading zeros if necessary.
   • Convert each group of four binary digits to its hexadecimal equivalent. Example: Convert binary number 10110101 to hexadecimal: 1011 0101 (grouped into four) Hexadecimal equivalent: B5 Hexadecimal representation: B5
6. Hexadecimal to Binary Conversion:
   • To convert a hexadecimal number to binary, convert each hexadecimal digit to its binary equivalent. Example: Convert hexadecimal number 2E9A to binary: 2 -> 0010 E -> 1110 9 -> 1001 A -> 1010 Binary equivalent: 0010111010011010

Binary Arithmetic:

1. Addition:
   • Perform binary addition similar to decimal addition, starting from the rightmost bit.
   • Carry over if the sum exceeds 1. Example: 1011

• 0110

10001 (carry over 1)

2. Subtraction:
   • Perform binary subtraction similar to decimal subtraction, using the concept of borrowing.
   • Borrow from higher bits if necessary. Example: 1011

• 0110

0101 (borrow 1)

3. Multiplication:
   • Perform binary multiplication using the same process as decimal multiplication, multiplying each bit of one number with each bit of the other and shifting appropriately. Example: 1011 x 110

1011 (shifted) 1011 (shifted) +1011 (shifted)

1000010

4. Division:
   • Perform binary division using the same process as decimal division, using repeated subtraction and shifting. Example: 1011 ÷ 10 Quotient: 101 Remainder: 1

Understanding inter-conversions between binary, octal, and hexadecimal number systems, as well as binary arithmetic operations, is essential for various computer-related tasks, including programming, digital logic design, and data manipulation.