Number 9

[Proudclod229]

9=9

18= 1+8 = 9

27= 2+7 = 9

36= 3+6 = 9

45

54

63

72

81

90

...and here's the kicker:

 

99= 9+9= 18= 1+8= 9

 

this can be done to infinity...

[sanctus]

I never saw it, but this is just due to working in the decimal system and using the highest digit.

 

If you go to base 5 and do the same with the highest digit there (i.e. 4) you get the same:

 

[math]

4_{10} =4_5=4

[/math]

[math]

8_{10} =13_5\to 1+3=4

[/math]

[math]

12_{10}=22_5 \to 2+2=4

[/math]

[math]

16_{10}=31_5 \to 4

[/math]

[math]

52_{10}=202_5 \to 4

[/math]

[math]

\;\;\; \vdots

[/math]

[Ben]

9=9

18= 1+8 = 9

27= 2+7 = 9

36= 3+6 = 9

45

54

63

72

81

90

...and here's the kicker:

 

99= 9+9= 18= 1+8= 9

 

this can be done to infinity...

It gets worse!! Try this party trick, amaze your friends....

 

1. Take any number at all with more than 1 digit and add the digits.

 

2. If the result has more than one digit, add again

 

3. Now subtract this single-digit number from your starting number (not added)

 

4. Add the resulting digits again.

 

5. I bet it will be the number 9

 

Can you see why? Once you do, you will see that the restriction in (1) to more than a single digit is not required.

 

Hint: What Sanctus calls "base 10 arithmetic" could equally well be called "modulo 9 arithmetic" - they are very very intimately related. In fact there is a sense in which they are the same thing

[sanctus]

Ok Ben, I think I see why. But this is not intended as nice mathematical proof ;-).

Staying in base 10 now. Let's say I have a number with 2 digits which I write as xy (not multiplication!). One can then decompose this given number as xy= x*10+y=x*9+x+y. If I now substract (step3) I have xy-x-y which can be written as x*9+x+y-x-y=x*9.

Which is 9 if x==1.

If x is bigger than 1 then x*9 can be written as x'y' and then repeat the above to get x'*9, if x' etc...

 

But with only 1 digit I obtain zero...

[Turtle]

9=9

18= 1+8 = 9

27= 2+7 = 9

36= 3+6 = 9

45

54

63

72

81

90

...and here's the kicker:

 

99= 9+9= 18= 1+8= 9

 

this can be done to infinity...

 

this is called the "digital root". any natural number - to infinity & beyond - that has a digital root of 9 divides by 9 & 3 and also 6 if it is even.

 

Digital Sum

...The concept of a decimal digit sum is closely related to, but not the same as, the digital root, which is the result of repeatedly applying the digit sum operation until the remaining value is only a single digit. The digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively. ...

 

I never saw it, but this is just due to working in the decimal system and using the highest digit.

 

If you go to base 5 and do the same with the highest digit there (i.e. 4) you get the same:

 

[math]

4_{10} =4_5=4

[/math]

[math]

8_{10} =13_5\to 1+3=4

[/math]

[math]

12_{10}=22_5 \to 2+2=4

[/math]

[math]

16_{10}=31_5 \to 4

[/math]

[math]

52_{10}=202_5 \to 4

[/math]

[math]

\;\;\; \vdots

[/math]

 

correct. any natural number in base 5 that has a digital root of 4 divides by 4 & 2. moreover, this relation* is true for any base. i read a proof online a couple years back authored by some asian-name-sounding folks, but i have never been able to find it again. :doh:

 

that's all.

 

edit: * to clarify, "this relation" is that which for any base b, a natural number written in that base that has a digital sum of b-1, divides evenly by b-1. if b-1 is composite, you get other even-divisibilities as seen with base 10 as above.

 

additionally when b-1 is composite, staying in base 10 for example, a natural number with a digital root of 6 divides evenly by 3 and also 6 if it is even, and a natural number with digital root 3 divides evenly by 3.

[Proudclod229]

Exponents:

 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

 

rows by power, columns by base...only a small sampling ofcourse; and if you look at the diagonal!

[Turtle]

Exponents:

 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

 

rows by power, columns by base...only a small sampling of course; and if you look at the diagonal!

 

 

i worked this out many years ago, though i called taking the "digital root" the "katabatak function". here is my graph of your list; note it has a repeating block of rows as well as columns. i have added strikeouts to the unecessary elements in your list. below my graph i have some other links to the threads where i have discussed my findings as well as a variety of my graphs in the gallery. enjoy.

 

 

Katabatak Powers

 

katabatak graphs in gallery:

Studio K

 

katabatak thread (many of the images were lost in the gallery change, but i have them all archived. if you want to see some particular one or ones, just ask.)

Katabatak Math-An Exploration In Pure Number Theory

[Qfwfq]

Once you do, you will see that the restriction in (1) to more than a single digit is not required.

Let's say, once you see that 9 = 0.

 

What Sanctus calls "base 10 arithmetic" could equally well be called "modulo 9 arithmetic" - they are very very intimately related. In fact there is a sense in which they are the same thing

A sense! Call them the same thing!

 

Actually Sanctus didn't call it exactly "base 10 arithmetic" and the real point is that the homomorphism of [imath]\mathbb{Z}[/imath] onto [imath]\mathbb{Z}_9[/imath] is made easy in base ten becasue, modulo 9, 10 = 1.

[Ben]

OK, let's do it! First I apologize to anyone who find the following patronizing, but it illustrates an important general point in mathematics.

 

If, for integers [math]x,\,y,\,a[/math] I say that [math]x = y \mod a[/math] I mean simply that [math]x[/math] and [math]y[/math] differ by the integer [math]a[/math]. That is [math]x-y=a[/math], say. This is called a "congruence" and has the following properties.

 

[math]x=x \mod a[/math] (reflexivity)

 

[math] x= y \mod a \Rightarrow y=x \mod a[/math] (symmetry)

 

[math]x = y \mod a,\,\,\, y = z \mod a \Rightarrow x = z \mod a[/math] (transitivity)

 

These three properties define the modulo relation as an equivalence relation, and these crop up all over the place in mathematics (usually masquerading as isomorphisms or even equalities!).

 

So now let's set [math]a = 9[/math], and we easily see that any member of the set [math]\{....,-18,-9,0,9,18,.....\}[/math] can be written as [math]x=y \mod 9[/math]. This set is called an "equivalence class" and it is customary to elect a class representative, and write [math]\{....-18,-9,0,9,18,....\} \equiv [0][/math] although this choice is entirely arbitrary.

 

Likewise we will have that [math]\{....-17,-8,1,10,19,....\} \equiv [1][/math] and so on up to the class [math][8][/math].

 

Right, so the number, say [math]456[/math] is really just shorthand for [math]400+50+6[/math], and since we easily see [math]400 =4 \mod 9,\,\,\,50 = 5 \mod 9,\,\,\,6=6 \mod 9[/math] then it is no great shock to learn that [math]4+5+6 = 456 \mod 9[/math].

 

Likewise it is expected that [math]4+5+6 = 15 = 1+5 = 6[/math] is in the same equivalence class as [math]456[/math] so that [math]6 = 456 \mod 9[/math], it doesn't matter which class we are in.

 

Now by the definition of this particular sort of equivalence relation, we must have that subtracting one member of an equivalence class from another member of the same class sends the result to the class [math][0][/math] of which [math]9 [/math] is a member. So from the above we will have, no matter how many digits we start with, proceeding recursively we will always have the result to be 0 or 9.

 

There are any number of "mind reading" party tricks you can devise using this sort of arithmetic, say mod 2, since it will be a closed book to most of your awe-struck audience

[Turtle]

bin[/-]ben']Once you do, you will see that the restriction in (1) to more than a single digit is not required.

Let's say, once you see that 9 = 0.

 

just to clarify, the digital root function never returns zero, only the mod function does. if your "=" sign denotes "equivalent", then i agree with you, although it seems "≡" is preferred to reference modular arithmetic equivalernce. for any base b where the mod function returns 0, the digital root returns b-1. because i used digital roots in constructing my graphs, they have no zeros.

 

attached is a table of the natural numbers in base 10 sorted by their digital root equivalence classes. by skewing the table i got columns with the same ending digit, and of course the ending digit in base 10 is the result of that number mod 10. a two-fer! :D note the color coded table elements:

blue=primes

orange=perfect_squares

pink=strange_numbers.

[Turtle]

Exponents:

 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

 

rows by power, columns by base...only a small sampling of course; and if you look at the diagonal!

 

hey P-clo...you still here? didn't want to shortchange your observation; g'donya!! anyway, i edited my strike-through list of your base 10 list to match my graph, however there are some interesting "except fors" when we venture to other bases. while 1 1 1 1 1 1 1 1 1 & 1 2 3 4 5 6 7 8 9 do not occur in the repeating part of the pattern of powers in base 10, they do in other bases. if you want to keep this to base 10, being it's called Number 9 :doh:, i can pick up this discussion in the katabatak powers thread. again; nice observation.

[Qfwfq]

the digital root function never returns zero, only the mod function does.

just to clarify, that's exactly the reason I made the point; bin's party trick also has a subtraction in it...

as for my notation, I was deliberately being stooooopid, as usual

[Ben]

So now both Qfwfq and Turtle refer to me as "bin"? How very rude.

 

Surely there is some forum rule about mutating a member's name in a derogatory sense? There certainly should be....

[Turtle]

So now both Qfwfq and Turtle refer to me as "bin"? How very rude.

 

Surely there is some forum rule about mutating a member's name in a derogatory sense? There certainly should be....

 

:omg: geez ben; i mutate members' names in a playful manner quite often; just ask q-ball. :eight: anyway, there is some intersting math here if you can get past the heart on your sleeve. :unlove:

 

just to clarify, that's exactly the reason I made the point; bin's party trick also has a subtraction in it...

as for my notation, I was deliberately being stooooopid, as usual.

 

ahhhh ha!! there certainly should be a rule against that... :wink: to be sure, my skew-table does not include 0 and the negative numbers and it jolly well could as ben alluded to. nonetheless that doesn't change the residue patterns of powers, which is what we are on about. seems like i better pick this up where i left off in my own thread; i'll get over there & get it revved. better bring your nitro! :Nurse: :D

[Qfwfq]

Sure, toydel always does it. We ain't even from new yoyk either. Anyway osama ain't the only son of, and it ain't necessarily of a gun... it's the same for ben gurion anyway.

 

nonetheless that doesn't change the residue patterns of powers

Yes, but suppose the starting number is single digit. It is equal to the sum of its digits so, when you subtract this from the starting number you get 0 and when you sum its digits you have 0.

So he's a bin gun after all, he looses the bet cuz he said it would be 9, even starting with a single digit number.

:doh:

[Proudclod229]

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

[Turtle]

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

1 2 9 4 5 9 7 8 9 1 2 9 4 5 9 7 8 9 1

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 9 1

1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9 1

1 7 9 4 4 9 7 1 9 1 7 9 4 4 9 7 1 9 1

1 5 9 7 2 9 4 8 9 1 5 9 7 2 9 4 8 9 1

1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1 1 9 1

 

since our system of cells is composed of repeating parts, it should come as no surprise that composites of those parts produce other different repeating parts. i could as easily color up cells in "L's" -as in how a knight moves on a chess board- and get repeating patterns. while it's interesting visually, does it tell us anything mathematically about the system?