9 is considered a "magic number" due to some peculiar mathematical properties it possesses, as illustrated by the references. These properties make it stand out in calculations.
The Magic of Nine
The properties that make 9 a "magic number" are primarily related to multiplication and digit manipulation.
Multiplication Magic
According to reference (1), a key property of 9 is:
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When you multiply nine by any number and add up the digits of the answer, you get 9.
For example:
- 9 x 3 = 27 -> 2 + 7 = 9
- 9 x 12 = 108 -> 1 + 0 + 8 = 9
- 9 x 45 = 405 -> 4 + 0 + 5 = 9
This property is consistent and works for any multiplication of 9 and any integer, showing that the sum of digits is always reducible to 9.
Digital Root and Multiples of Nine
Reference (2) explains another interesting attribute of 9:
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The difference between a positive integer and the sum of its digits (or digital root) is a multiple of nine.
For example, as mentioned in the reference:
- The sum of the digits of 32 is 5, and 32 - 5 = 27, which is a multiple of 9.
Other examples: - The sum of the digits of 15 is 6, and 15 - 6 = 9, a multiple of 9.
- The sum of the digits of 81 is 9, and 81 - 9 = 72, a multiple of 9.
- The sum of the digits of 123 is 6, and 123 - 6 = 117, a multiple of 9 (13 * 9).
- The sum of the digits of 32 is 5, and 32 - 5 = 27, which is a multiple of 9.
This property is a crucial link between the numbers, their digit sums and multiples of 9.
Summary
These mathematical properties are not commonly found in other numbers, making the number 9 a "magic number." The consistent patterns and relationships with its multiples contribute to the perception that nine has some special properties. It’s the consistent and unique results related to digit sums and multiples that have given rise to the term ‘magic number’ associated with 9.
