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The method of 9's complement works because it efficiently simplifies the process of subtraction in binary systems. By converting a number into its 9's complement, you effectively represent the value needed to reach the next higher base (10 for decimal). When you add this complement to the original number, it allows you to perform subtraction as an addition operation, and any carry generated can be discarded, yielding the correct result. This is particularly useful in digital systems and calculators for faster arithmetic operations.

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