Many people view spells as something mystical or magical, often associating them with witchcraft or wizardry. Spells are thought to have the power to create desired outcomes or manipulate situations. However, it is important to understand that the genuine nature of spells lies not in their magical properties, but rather in their ability to focus one's intention and energy towards a specific goal. At its core, a spell is simply a ritual or set of actions performed with the intention of manifesting a desired outcome. It is a way of channeling one's energy and intention towards a specific goal, whether it be love, success, healing, or protection. The ingredients and rituals associated with spells are not necessarily magical in and of themselves, but rather serve as symbols or tools to help the practitioner connect with their desired outcome.
The ingredients and rituals associated with spells are not necessarily magical in and of themselves, but rather serve as symbols or tools to help the practitioner connect with their desired outcome. In this sense, spells can be seen as a form of focused intention. By performing a ritual and repeating specific words or actions, the practitioner is able to concentrate their energy and thoughts on their desired goal.
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.
Comments- Comments
- Write a Comment Select to add a comment
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 focused intention can be a powerful tool in manifesting one's desires and creating positive change in their life. For example, a love spell may involve burning candles, reciting certain words or affirmations, and visualizing oneself in a loving and fulfilling relationship. These actions and symbols serve to focus the practitioner's thoughts and intention on attracting love into their life. The candles and words themselves are not inherently magical, but rather serve as a focal point for the practitioner's energy and intention. It is important to note that spells should not be seen as a means to control or manipulate others. The genuine nature of spells lies in their ability to empower the practitioner and align their energy with their desired outcome. Spells should always be performed with positive intentions and with respect for the free will and well-being of others. In conclusion, spells are not inherently magical or mystical, but rather a means for individuals to focus their intention and energy towards a specific goal. The genuine nature of spells lies in their ability to empower individuals and help them manifest positive change in their lives. When performed with positive intentions and respect for others, spells can be a powerful tool for personal growth and transformation..
Reviews for "Unlocking the Secrets of Genuine Spells: A Deep Dive into Their Inner Workings"
1. Jane Doe - 2 stars:
I was really excited to read "The genuine nature of spells," but I ended up being quite disappointed. The characters felt one-dimensional and predictable, and their behavior often didn't make much sense. The plot started off promising, but it quickly became repetitive and lacked enough twists and turns to keep me engaged. Additionally, the writing style felt clunky and the pacing was off. Overall, I found the book to be underwhelming and would not recommend it.
2. John Smith - 2.5 stars:
I had high hopes for "The genuine nature of spells," but it fell short for me. The main character was difficult to connect with, and the way the author portrayed their internal struggles felt forced and unrealistic. The love story that unfolded was predictable and lacked depth, leaving me uninvested in the outcome. While there were some interesting magical elements, they were not fully explored and often took a backseat to the romance. Overall, the book had potential but failed to deliver on its promises.
3. Sarah Thompson - 3 stars:
"The genuine nature of spells" was an okay read for me. While I enjoyed the concept of magic and some of the world-building, the execution fell flat. The writing style was average at best, and the dialogue felt forced and unnatural. The pacing was also inconsistent, with certain parts dragging on while others felt rushed. I appreciate the effort the author put into creating this story, but it just didn't resonate with me personally. I wouldn't actively discourage someone from reading it, but I wouldn't necessarily recommend it either.