The magical nine is a concept that has been seen throughout history and across various cultures. It refers to the belief that the number nine possesses special qualities and has a mystical significance. In many ancient civilizations, the number nine was considered a sacred number and held great importance in religious and spiritual practices. For example, in Norse mythology, there were nine worlds connected by the World Tree. In Hinduism, there are nine forms of the deity Durga, which are worshiped during the Navaratri festival. In Chinese culture, the number nine is associated with longevity and immortality, and it is considered lucky.
In Chinese culture, the number nine is associated with longevity and immortality, and it is considered lucky. One of the reasons why the number nine is seen as magical is its mathematical properties. When multiplied by any single-digit number and the resulting digits are added together, the sum is always nine.
The Number 9, Not So Magic After All
This blog is not about signal processing. Rather, it discusses an interesting topic in number theory, the magic of the number 9. As such, this blog is for people who are charmed by the behavior and properties of numbers.
For decades I've thought the number 9 had tricky, almost magical, qualities. Many people feel the same way. I have a book on number theory, whose chapter 8 is titled "Digits — and the Magic of 9", that discusses all sorts of interesting mathematical characteristics of the number 9 [1]. That book is not alone in its fascination with the number 9. If you search the Internet for the phrase "magic number 9" you'll receive dozens of relevant "hits."
A CHALLENGING ARITHMETIC PROBLEM
I first began thinking the number 9 was special years ago when I encountered a straightforward math problem alleged to test a person's intelligence. The problem is; given
you are required, using pencil and paper, to find the digit A within 60 seconds.
Back then, of course, I couldn't solve that problem in 60 seconds. Later I learned the solution requires us to know the curious property that when you multiply a natural number by 9, the sum of the product's digits are a whole multiple of 9. (By "natural number" I mean a positive whole number, what mathematicians call "positive integers.")
For example, 762 x 9 = 6858, and the sum of 6+8+5+8 is 27 which is a whole multiple of 9. That is, the whole number 3 times 9 equals 27. Try this yourself: multiply a natural number by 9 and add the product's digits to see that their sum is always a whole multiple of 9.
So to quickly solve Eq. (1) for A, we view that equation as:
Because (523 + A)2 is a natural number, after multiplying it by 9, the sum of the digits on the left side of Eq. (2) must be a whole multiple of 9. That is:
where 36 is a whole multiple of 9. Because 36 – 32 = 4, A = 4 is the problem's solution. (For Matlab aficionados, Appendix A gives a Matlab software method of finding A in Eq. (1)).
ANOTHER CURIOUS PROPERTY OF 9
If we sum the digits of any natural number and subtract that sum from the original number, the result is a whole multiple of nine. As an example, for any devil worshippers among us, the sum of the digits in 666 is 18. And 666-18 = 648, which is a whole multiple of 9 (9 x 72 = 648). How remarkable!
THE MAGIC OF MULTIPLYING BY 9
Multiplying natural numbers by 9 leads to some interesting results. While once playing around with my hand calculator I discovered the products shown in Table 1 of Figure 1.
Multiplying particular sequential natural numbers by 9 produce interesting numerical patterns. For example, the noteworthy Tables 2 through 5 can be found in Reference [1].
Reinforcing my notion of the special nature of the number 9, a neat parlor trick employing the magic of 9 that you can use to amaze your friends can be found at: http://www.youtube.com/watch?v=nd_Z_jZdzP4
DIVISION BY 9
Dividing a natural number by 9 also produces some peculiar results. Appendix B presents a few examples of those results.
IS THE NUMBER 9 REALLY MAGICAL?
Thinking about the apparent magical properties of the number 9, I recalled a quote from a dead mathematician. The 19th century German mathematician Leopold Kronecker, a pioneer in the field of number theory, believed "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk." ("Natural numbers were made by our dear God, all else is the work of men.")
Now if God (or Mother Nature, if you prefer) created the natural numbers I wondered, "Why would God give the number 9 magical properties? Doing so seems prejudicial, downright unreasonable." Then it hit me, 9 is one less than the 10 in our base-10 (decimal) number system. Next I wondered, "Does the digit 9 also have special properties in number systems having a base other than 10? Or could there be other digits that are magical in other number systems?"
Exploring the natural numbers in a base-7 number system (whose digits are 0,1,2,3,4,5, and 6) I created the tables in Figure 2.
So there you have it. In the base-7 number system, the number 6 is magical!
For those familiar with computer programming's hexadecimal (base-16) number system multiplying a natural number by the digit F, a decimal 15, the sum of the product's digits will be a whole multiple of decimal 15. Thus, in the hexadecimal number system the hexadecimal digit F (decimal 15) is magical.
Being in the DSP field I, of course, wondered if there was any special behavior when we multiply numbers in our familiar binary number system. The only mildly interesting multiplication pattern I found in our base-2 binary number system is shown in Figure 3.
CONCLUSION
So after all these years, I now realize the number 9 is not a magic number. In a base-B number system, the number B-1 is the digit with magical properties.
If we want to call anything "magic", we might generally agree with Herr Kronecker and merely say, "All natural numbers can be magical."
POSTSCRIPT - THE SPECIAL NUMBER 42
Thinking about numbers, something has just occurred to me. Millions of technically astute people consider the decimal number 42 to be a truly extraordinary number. They believe 42 is, literally, the Answer to the Ultimate Question of Life, the Universe, and Everything. To understand this belief, search the Internet for the phrase "the answer to life the universe and everything".
APPENDIX A: A HIGH-TECH METHOD OF SOLVING EQ.(1)
Here's one way to solve Eq. (1) for A. Given:
Squaring (523 + A) and collecting non-zero terms on one side of the equation we can write:
Using Matlab's symbolic math to solve the 2nd-order quadratic Eq. (A-2), we enter:
Giving us two possibilities for the value of A:
APPENDIX B: FUN WITH DIVISION BY 9
Dividing a natural number by 9 also yields what I think is an interesting property. That is, dividing a natural number by 9 produces a decimal quotient having a positive integer I, to the left of the decimal point, and an endlessly repeating single decimal fractional digit F, to the right of the decimal point, as
Examples of this behavior are:
OK, these division-by-9 examples may not seem too exciting, but I noticed something about division by 9 that seems almost magic. There's a way to determine the fraction digit F without performing any division.
If you add the digits of an integer dividend N you'll obtain a natural number P.
- If P is a single digit less than 9, then the fraction digit F = P.
Looking at the above Eq. (B-3), the sum of the dividend N = 134 digits is P = 1+3+4 = 8, so F = P = 8.
- If P is more than one digit, we merely add P's digits to obtain the single digit Q, in which case, the fraction digit F = Q.
Looking at the above Eq. (B-4), the sum of the dividend N = 76241 digits is P = 20. Because 20 has two digits, we add them to yield Q = 2+0 = 2, and our fraction digit F = Q = 2.
So the point of this N/9 = I.FFFFF. discussion is that you can determine the fraction digit F by inspecting N, no actual division is necessary. (A week or so after I wrote this blog, I learned that the above iterative adding-of-digits operation is referred to as "finding the digital root" of a natural number. See references [2] and [3].)
Other interesting 'divide by 9' results can be seen. Grab your hand calculator and divide a small natural number N by 99, then divide N by 999, and finally divide N by 9999 and see the peculiar results.
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For example, 762 x 9 = 6858, and the sum of 6+8+5+8 is 27 which is a whole multiple of 9. That is, the whole number 3 times 9 equals 27. Try this yourself: multiply a natural number by 9 and add the product's digits to see that their sum is always a whole multiple of 9.
This property is known as the "digital root" or "casting out nines" method. For example, if you multiply nine by three, you get twenty-seven, and if you add the digits of twenty-seven together (2 + 7), the sum is nine. This property has fascinated mathematicians for centuries and adds to the allure of the number nine. The magical qualities of the number nine can also be seen in everyday life. For instance, many people believe that the ninth year of a relationship or marriage is a significant milestone. It is often seen as a time of reflection, growth, and transformation. Additionally, in numerology, the number nine is associated with completion, fulfillment, and spiritual awakening. Overall, the magical nine is a concept that encompasses the mystical and mathematical qualities of the number nine. Whether seen in ancient mythologies, religious practices, or everyday life, the number nine holds a special place in human history and continues to fascinate and intrigue people to this day..
Reviews for "Exploring the Magical Nine in Music and Art: A Journey of Inspiration"
1) Sarah - 2 stars - I was really looking forward to reading "The Magical Nine" based on all the hype, but I was so disappointed. The story felt incredibly underdeveloped and rushed. The characters lacked depth and I found it hard to connect with any of them. The plot was predictable and lacked originality. Overall, it was a forgettable read for me.
2) Michael - 1 star - "The Magical Nine" was a complete waste of time. The writing was subpar, filled with clichés and awkward dialogue. The pacing was all over the place, with the story dragging in some parts and completely jumping over important events in others. The magic system was poorly explained and inconsistent. I would not recommend this book to anyone.
3) Emily - 2 stars - I had high hopes for "The Magical Nine" as I'm a fan of fantasy novels, but sadly, this one fell flat for me. The worldbuilding was weak and lacked detail, leaving a lot of unanswered questions. The characters were one-dimensional and their actions felt forced and unnatural. The plot was predictable, lacking any surprises or twists. Overall, it was an unremarkable book that failed to deliver.
4) John - 2.5 stars - While "The Magical Nine" had an interesting premise, the execution was lacking. The pacing was off, with slow moments that dragged on for too long and action-packed scenes that felt rushed. The plot had potential, but it felt like the author didn't know where to take it. The dialogue was awkward at times and the characters lacked depth. Overall, it was an average read that left me feeling disappointed.
5) Megan - 2 stars - I found "The Magical Nine" to be a mediocre read. The writing style was simplistic and lacked any sort of elegance. The characters were forgettable and lacked development. The plot was predictable and unoriginal, following the same tired tropes we've seen in countless other fantasy novels. Overall, it was a disappointing read that failed to live up to its potential.