Binary Number System Conversion Operations StudySmarter
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binary number system conversion operations: a comprehensive tutorial
the binary number system is the foundation of digital computing. understanding how to convert between binary and other number systems (decimal, hexadecimal, octal) is crucial for anyone working with computers or digital electronics. this tutorial will cover these conversions in detail, providing both conceptual understanding and practical code examples (using python).
*1. understanding number systems*
before diving into conversions, let's clarify the basics of different number systems:
*decimal (base-10):* this is the everyday number system we use. it uses ten digits (0-9) and each position represents a power of 10. for example, 1234 = (1 * 10³) + (2 * 10²) + (3 * 10¹) + (4 * 10⁰).
*binary (base-2):* uses only two digits (0 and 1). each position represents a power of 2. for example, 1011₂ = (1 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰) = 8 + 0 + 2 + 1 = 11₁₀ (the subscript indicates the base).
*octal (base-8):* uses eight digits (0-7). each position represents a power of 8.
*hexadecimal (base-16):* uses sixteen digits (0-9 and a-f, where a=10, b=11, c=12, d=13, e=14, f=15). each position represents a power of 16.
*2. conversion techniques*
*a. decimal to binary:*
there are two primary methods for converting decimal numbers to binary:
*repeated division by 2:* continuously divide the decimal number by 2 and record the remainders. the remainders, read in reverse order, form the binary equivalent.
example: convert 13₁₀ to binary:
13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
reading the remainders from bottom to top: 1101₂. therefore, 13₁₀ = 1101₂.
*positional representation:* determine the largest power of 2 less than or equal to the decimal number. subtract this power of 2 from the decimal number and repeat the process with the remainder until the remainder is 0. a '1' is placed in the p ...
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