he Number 9, Not So Magic After All

Rick Lyons●October 1, 2014●6 comments

 Tips and Tricks

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

This article is available in PDF format for easy printing

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:

we can write:

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.