Number Systems: Binary, Octal, Decimal & Hex

In the hexadecimal numbering system, which is base 16, the digits include 0-9 and the letters A-F. Each letter corresponds to a decimal value: A is 10, B is 11, C is 12, D is 13, E is 14, and F is 15. Therefore, the letter B specifically represents the decimal value 11, as it is the second letter in the sequence of hexadecimal digits.

The hexadecimal number system is a base-16 numeral system. It uses sixteen distinct symbols: the numbers 0-9 represent values zero to nine, and the letters A-F represent values ten to fifteen. This system is commonly used in computing and digital electronics because it can represent binary values more compactly, allowing for easier human interpretation of large binary numbers. Each digit in a hexadecimal number represents a power of 16, making it efficient for various applications in programming and data representation.

The octal number system is a base-8 numeral system, which means it uses eight distinct digits to represent values. These digits range from 0 to 7. Any digit outside this range, such as 8 or 9, is not valid in octal notation. Therefore, the only valid digits in the octal system are 0, 1, 2, 3, 4, 5, 6, and 7.

To convert the binary number 101101₂ to decimal, we evaluate each bit's value based on its position. Starting from the right, the bits represent 2^0, 2^1, 2^2, 2^3, and 2^4. Thus, 1×2^5 + 0×2^4 + 1×2^3 + 1×2^2 + 0×2^1 + 1×2^0 equals 32 + 0 + 8 + 4 + 0 + 1, which sums to 45. Hence, the decimal equivalent of 101101₂ is 45.

To convert the decimal number 45 to binary, we repeatedly divide the number by 2 and record the remainders. Starting with 45, we divide by 2, which gives a quotient of 22 and a remainder of 1. Continuing this process, we divide 22 by 2 to get 11 (remainder 0), then 11 by 2 to get 5 (remainder 1), followed by 5 by 2 to get 2 (remainder 1), and finally 2 by 2 to get 1 (remainder 0) and 1 by 2 to get 0 (remainder 1). Reading the remainders from bottom to top gives us 101101.

To convert the octal number 572₈ to decimal, each digit is multiplied by 8 raised to the power of its position, starting from the right (0 index). Thus, 5 × 8² + 7 × 8¹ + 2 × 8⁰ equals 5 × 64 + 7 × 8 + 2 × 1, which simplifies to 320 + 56 + 2. Adding these values gives 378 in decimal.

To convert the hexadecimal number 2AF₁₆ to decimal, we break it down by its positional values. The hexadecimal digits represent powers of 16: 2 is in the 16² place, A (which is 10) is in the 16¹ place, and F (which is 15) is in the 16⁰ place. Therefore, the calculation is: (2 × 16²) + (10 × 16¹) + (15 × 16⁰) = (2 × 256) + (10 × 16) + (15 × 1) = 512 + 160 + 15 = 687.

To convert the decimal number 254 to hexadecimal, divide the number by 16. The quotient is 15 and the remainder is 14. In hexadecimal, 15 is represented as 'F' and 14 as 'E'. Therefore, when combining these values, 254 in decimal is represented as FE in hexadecimal. Thus, the answer is FE₁₆.

When converting binary to octal, binary digits are grouped in sets of three because octal is a base-8 numbering system, which corresponds to \(2^3\). Each octal digit represents three binary digits. By grouping binary digits in sets of three, each group can be directly converted to its corresponding octal value, simplifying the conversion process. For example, the binary number 101110 can be grouped as 101 and 110, which translates to the octal digits 5 and 6, respectively. This method ensures an accurate and efficient conversion from binary to octal.

To convert the binary number 1101011₂ to octal, group the binary digits into sets of three, starting from the right: 1 101 011. Add leading zeros if necessary: 001 101 011. Then convert each group to its octal equivalent: 001 (1), 101 (5), and 011 (3). Combining these gives the octal number 153₈.

To convert the octal number 347₈ to binary, each octal digit is represented by its 3-bit binary equivalent. The digit '3' in octal is '011' in binary, '4' is '100', and '7' is '111'. Combining these, we get: 011 (for 3), 100 (for 4), and 111 (for 7), which results in 011100111. However, leading zeros can be omitted, giving us 11100111₂ as the final binary representation.

To convert the binary number 111011101₂ to hexadecimal, group the binary digits into sets of four, starting from the right: 0011 1011 1101. Then convert each group to its hexadecimal equivalent: 0011 (3), 1011 (B), and 1101 (D). Thus, the hexadecimal representation is 3BD₁₆. However, it seems there was a miscalculation in the grouping. If we consider 1110 (E) and 111 (7), we get 1DD₁₆. Therefore, the correct conversion yields 1DD₁₆.

To convert the hexadecimal number A6₁₆ to octal, first convert it to decimal. A6₁₆ equals \(10 \times 16^1 + 6 \times 16^0 = 160 + 6 = 166_{10}\). Next, convert 166 from decimal to octal by dividing by 8. The first division gives 20 with a remainder of 6, and the second division gives 2 with a remainder of 4. Reading the remainders from bottom to top, the octal representation is 246₈.

In positional numeral systems, the value of a digit is determined by its position within a number. Each position corresponds to a power of the base of the system. For example, in base 10, the rightmost digit represents \(10^0\), the next \(10^1\), and so on. Thus, the value of a digit is calculated by multiplying the digit by the base raised to the power of its position. This principle is fundamental to understanding how numbers are represented and manipulated in various numeric systems.