I recently came across an interesting mathematical knowledge while reading 'Chaotic Thoughts from the Old Millenium'. It is a book written by the founder of Creative Technology, the iconic homegrown technology firm, the late Mr Sim Wong Hoo. I will touch more on the book at the end of the article.

In one of his short stories in the book, he shared a number trick involving the number 9. Well, it is not exactly a trick, but a mathematical relationship, which is that the sum of digits of any number multiplied by 9 will be equal to 9 or its multiple.

        1 x 9 = 9
        2 x 9 = 18 => 1 + 8 = 9
        3 x 9 = 27 => 2 + 7 = 9
        4 x 9 = 36 => 3 + 6 = 9
        .
        .
        11 x 9 = 99 => 9 + 9 = 18 (multiple of 9)
        .
        .
        23 x 9 = 207 => 2 + 0 + 7 = 9
        

One who knows this relationship could ask another person to choose any number which is not made known to the person asking. Get the person to perform a few intermediate arithmetic computations. It wouldn't matter what these computations were as long as the final step involved multiplying the value by 9. Finally, get the person to recite all digits in the resulting value except the last digit. The person asking would be able to guess the last digit easily as the digits should sum to 9 or its multiple. I tried it out with my wife and it worked beautifully. As she is a Math educator, she easily guessed that there was a relationship involving the number 9.

Being mathematical people ourselves, we were not satisfied with merely knowing the relationship. We needed to find a proof. So we set out to find the reason behind this relationship online. The algebraic approach to proving it might be easier to understand.

        Let 9k be an integer which is a multiple of 9.
        9k = n1 + 10n2 + 100n3 + ...
        where n1, n2 ... are the digits of 9k.

        9k = (n1 + n2 +   n3 + ...) +
                    (                      9n2 + 99n3 + ...)
                  = (n1 + n2 +   n3 + ...) +
                    9(n2 + 11n3 + 111n3 + ...)
        
        n1 + n2 +   n3 + ... are the sum of digits of the integer 9k.

        Let 9(n2 + 11n3 + 111n3 + ...) = 9x. 

        ∴ 9k = (Σ digits of 9k) + 9x
                9k - 9x  = Σ digits of 9k
                9(k - x) = Σ digits of 9k

        Since (k - x) ∈ ℤ (element of set of integers), we have the relationship that any number multipled by 9 is equal to Σ digits of 9k.
                    

The relationship presented above is one of the many interesting mathematical relationships which mathematicians have discovered. There are many more relationships which have been discovered,

Coming back to the late Mr Sim's book, it is an engaging read filled with many anecdotes. Mr Sim shares many of his observations, childhood and non-childhood stories in a light-hearted and entertaining way, peppering in some wisdom and life philosophy along the way. A common thread in the book is his ability to win against the odds, even when it is stacked against him. We see this in how he grew Creative from a 2-man-run computer repair shop to the technological great it is today. Interestingly, 'Chaotic Thoughts from the Old Millenium' was written in a span of 8 weeks before the turn of the previous millenium, a challenge which he had set for himself and once again, which showed his ability to win against all odds.