Number System Conversions in Computing

The hexadecimal number system is a base-16 system, meaning it uses 16 distinct symbols to represent values. These symbols include the digits 0-9 and the letters A-F, where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. This system is commonly used in computing and digital electronics because it can represent large binary numbers in a more compact and readable format. Each digit in hexadecimal corresponds to a 4-bit binary sequence, making it efficient for binary-to-hexadecimal conversions.

To convert the binary number 10011011₂ to decimal, we can break it down by its place values. Starting from the right, each digit represents a power of 2:

1 (2^7) + 0 (2^6) + 0 (2^5) + 1 (2^4) + 1 (2^3) + 0 (2^2) + 1 (2^1) + 1 (2^0) = 128 + 0 + 0 + 16 + 8 + 0 + 2 + 1 = 155.

Thus, the decimal equivalent of the binary number 10011011₂ is 155.

When converting a decimal integer to binary, the process involves dividing the number by 2 repeatedly. Each division yields a quotient and a remainder. The remainder reflects the binary digits, starting from the least significant bit to the most significant. This method is based on the binary number system, which is base-2, meaning it uses only two digits: 0 and 1. By continuing this division until the quotient is zero, you can construct the binary representation of the original decimal integer.

To convert the decimal number 155 to binary, we divide the number by 2 repeatedly and record the remainders. Starting with 155, we divide by 2 to get 77 with a remainder of 1. Continuing this process, we get 38 (0), 19 (0), 9 (1), 4 (0), 2 (0), and 1 (1). Reading the remainders from bottom to top gives us 10011011, which is the binary representation of 155.

To convert a binary number to octal, the binary digits are grouped into sets of three, starting from the right. This is because each octal digit corresponds to three binary digits (2^3 = 8). By grouping the binary bits in threes, each group can be directly translated into its octal equivalent, simplifying the conversion process. If the leftmost group has fewer than three bits, it can be padded with zeros to ensure it has three digits, maintaining the integrity of the conversion.

To convert the decimal number 946 to hexadecimal, divide it by 16 and record the remainder. 946 divided by 16 gives a quotient of 59 and a remainder of 2. Next, divide 59 by 16, which gives a quotient of 3 and a remainder of 11 (which is represented as 'B' in hexadecimal). Finally, the quotient of 3 gives a remainder of 3. Reading the remainders from last to first, the hexadecimal representation of 946 is 3B2.

In the hexadecimal number system, which is base 16, the digits range from 0 to 9 and then continue with letters A to F, where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. Therefore, the letter 'B' corresponds to the decimal value of 11, as it is the second letter in the sequence after A.

To convert the binary fraction 110.1011₂ to decimal, break it down into two parts: the integer part (110₂) and the fractional part (0.1011₂). The integer part equals 4 + 2 + 0 = 6. For the fractional part, calculate each bit's value: 1/2 + 0/4 + 1/8 + 1/16 = 0.5 + 0 + 0.125 + 0.0625 = 0.6875. Adding both parts together gives 6 + 0.6875 = 6.6875 in decimal, confirming the statement is true.

To convert the octal number 4503₈ to decimal, each digit is multiplied by 8 raised to the power of its position, starting from 0 on the right. This gives:

4 × 8³ + 5 × 8² + 0 × 8¹ + 3 × 8⁰ = 4 × 512 + 5 × 64 + 0 × 8 + 3 × 1 = 2048 + 320 + 0 + 3 = 2371.

Thus, the decimal equivalent of the octal number 4503₈ is 2371.