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31 Number System

The base-31 number system, also called the untrigesimal system, is a positional numeral system that uses thirty-one symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, and T, where A represents ten, B eleven, C twelve, D thirteen, E fourteen, F fifteen, G sixteen, H seventeen, I eighteen, J nineteen, K twenty, L twenty-one, M twenty-two, N twenty-three, O twenty-four, P twenty-five, Q twenty-six, R twenty-seven, S twenty-eight, and T twenty-nine in decimal. Each digit’s position represents a power of 31, starting from $31^0$ at the rightmost digit and increasing to the left. For example, the base-31 number 3T7 equals $3 \times 31^2 + 29 \times 31^1 + 7 \times 31^0 = 2883 + 899 + 7 = 3789$ in decimal. Base-31 is primarily used in theoretical mathematics, number theory, and educational exercises to explore alternative numeral systems, positional notation, and arithmetic patterns. Understanding base-31 allows learners to perform arithmetic operations, convert numbers between bases, and analyze properties of non-decimal systems. Although rarely used in computing or daily life, studying the untrigesimal system enhances logical reasoning, problem-solving skills, and comprehension of abstract number representations. It also provides a foundation for higher-level numeral systems and theoretical mathematics.

Number System

The octal number system, also known as base-8, is a positional numeral system that uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each digit represents a power of 8, with the rightmost digit representing $8^0$, the next representing $8^1$, and so on. For example, the octal number 157 equals $1 \times 8^2 + 5 \times 8^1 + 7 \times 8^0 = 64 + 40 + 7 = 111$ in decimal. Octal was widely used in early computing systems because it provides a more compact representation of binary numbers: each octal digit corresponds exactly to three binary digits. This makes conversion between binary and octal straightforward and simplifies programming and memory addressing in older computer architectures. Although modern systems primarily use hexadecimal (base-16), octal remains important in some computing contexts, such as Unix file permissions, digital electronics, and certain embedded systems. Understanding octal helps in grasping how different number bases can efficiently represent information and interact with binary systems. It also provides a foundation for learning hexadecimal and other positional numeral systems used in digital technology.