Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. 9 is a Motzkin number. It is the first composite lucky number.
Nine is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p2) and the first that is odd. All subsequent squares of this form are odd. It has a unique aliquot sum 4 which is itself a square prime. 9 is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of 9 has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree.
There are nine Heegner numbers.[1]
Since 9 = 3^{2^1}, 9 is an exponential factorial.
8 and 9 form a Ruth-Aaron pair under the second definition that counts repeated prime factors as often as they occur.
A polygon with nine sides is called an enneagon[2] A group of nine of anything is called an ennead. (technically) or nonagon (in common usage).
In base 10 a number is evenly divisible by nine if and only if its digital root is 9.[3] That is, if you multiply nine by any whole number (except zero), and repeatedly add the digits of the answer until it is just one digit, you will end up with nine:
* 2 × 9 = 18 (1 + 8 = 9)
* 3 × 9 = 27 (2 + 7 = 9)
* 9 × 9 = 81 (8 + 1 = 9)
* 121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
* 234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)
* 578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27 (2 + 7 = 9))
* 482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45 (4 + 5 = 9))
* (Exception) 0 x 9 = 0 (0 is not equal to 9)(though it can be considered -1 in the tens place and 10 in the ones place(10-1=9))
The only other number with this property is three. In base N, the divisors of N − 1 have this property. Another consequence of 9 being 10 − 1, is that it is also a Kaprekar number.
The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:
* The sum of the digits of 41 is 5, and 41-5 = 36. The digital root
of 36 is 3+6 = 9, which, as explained above, demonstrates that it is
evenly divisible by nine.
* The sum of the digits of 35967930 is 3+5+9+6+7+9+3+0 = 42, and
35967930-42 = 35967888. The digital root of 35967888 is 3+5+9+6+7+8+8+8
= 54, 5+4 = 9.
Subtracting two base-10 positive integers that are transpositions of
each other yields a number that is a whole multiple of nine. Some
examples:
* 41-14 = 27. The digital root of 27 is 2+7 = 9.
* 36957930-35967930 = 990000, which is obviously a multiple of nine.
This works regardless of the number of digits that are transposed.
For example, the largest transposition of 35967930 is 99765330 (all
digits in descending order) and its smallest transposition is 03356799
(all digits in ascending order); subtracting pairs of these numbers
produces:
* 99765330-35967930 = 63797400; 6+3+7+9+7+4+0+0 = 36, 3+6 = 9.
* 99765330-03356799 = 96408531; 9+6+4+0+8+5+3+1 = 36.
* 35967930-03356799 = 32611131; 3+2+6+1+1+1+3+1 = 18, 1+8 = 9.
Casting out nines
is a quick way of testing the calculations of sums, differences,
products, and quotients of integers, known as long ago as the 12th
Century.[4]
The Nine Chapters on the Mathematical Art
is a Chinese mathematics book, composed by scholars between the 10th
century BC, and the 1st century AD; it is the one of the earliest
surviving mathematical text from China.
Every prime in a Cunningham chain of the first kind with a length of 4 or greater is congruent to 9 mod 10 (the only exception being the chain 2, 5, 11, 23, 47).
Six recurring nines appear in the decimal places 762 through 767 of pi. This is known as the Feynman point.
If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.[5]
Nine is the binary complement of number six:
The decimal expansion of p begins
3.14159265358 … and the number is defined as the ratio of a circle’s
circumference to its diameter. If you fast forward to about the 760th
digit in the decimal expansion of p, you will see the string “…34
999999 83…” That group of six repeating 9s in the center is known as
the ‘Feynman Point.’
p is a real number that is irrational and transcendental, meaning it
is not the ratio of two integers and it is not the solution of a
polynomial with rational coefficients. Although it hasn’t been
proved, it’s generally believed that any finite set of digits will
eventually occur in the decimal expansion of any irrational number due
to their non-repeating behavior. For example, it’s thought that at some
point in the decimal expansion of, say, v666 that the string
‘6660066600666′ will occur, and the string ‘12345432123454321′ will
occur, and any other finite string you come up with. Thus, the
appearance of six 9s in the decimal expansion of p isn’t that unusual.
But the fact that it occurs after only 762 digits is what makes it a
genuine curiosity. (For comparison, the earliest location of any four
repeating digits is at position 1589, in which four 7s appear.)
The group of six 9s in p referred to as the Feynman Point is named
after Nobel Prize winning physicist Richard Feynman (1918 - 1988), who
was awarded the prize in 1965 for his work on quantum electrodynamics.
Feynman once stated in a lecture that he wished he could memorize all
the digits of p up to the group of six 9s so he could continue: “nine,
nine, nine, nine, nine, nine, and so on.” Of course that would be
misleading because if the digits went on as repeating 9s, p would cease
being an irrational number.
Quite a bit of research has been done on patterns that occur in the
decimal expansions of irrational numbers. But if one becomes too
intrigued with the digits for their own sake, it is not looked upon
favorably by most mathematicians. In Petr Beckmann’s book, A History of
p, he has a chapter titled “Digit Hunters,” and uses the phrase
disparagingly since there is no practical use for computing more and
more digits of p, or for locating patterns in their decimal expansions.
(Only 10 digits are needed for any real-world problem).
Nevertheless, when you have a computer it’s fun to search for
patterns in anything. So taking inspiration from the Feynman Point I
thought of this sequence: The first position in the digits of p (not
counting the initial 3) at which a string of n copies of the digit n
occurs. For example, the first occurance of four 4s is at position
54525, “…11793 4444 82014…” and that is the fourth term of our
sequence. Here are all the known terms: 1, 135, 1698, 54525, 24466,
252499, 3346228, 46663520, 564665206. Because all finite patterns of
digits are believed to eventually appear in irrational numbers, surely
this sequence is infinite. But where will the string of one hundred
100s occur?
I’ll close this article with two strange mathematical curiosities only loosely related to the Feynman Point.
Consider v666 = 25.8069758011278803151884206 …Â The first ‘666′
string occurs in its decimal expansion at position 503, and the
locations for the next 666s are 928, 975, 2705, 4229, 7021, 9195, 9338,
and 9349. Those are all positions of 666s among the first 10,000
digits, and I have dubbed it ‘Lucifer’s sequence’ because it is
believed that if a person recites the terms backwards while spinning
counterclockwise six times, upon their final revolution they will
either become totally omniscient, or their head will burst into flames.
Only kidding.Â
The second curiosity is based on primes, my favorite type of number,
and we’ll use some information from the Feynman Point. The six
repeating 9s start at the 762nd place and end at the 767th place. If we
concatenate those three numbers we get this prime: 762999999767. And if
we continuously add zeros between those three numbers we’ll continue to
find more. The next one is 76200009999990000767. Let f(n) = (762 *
10n+6 + 999999) * 10n+3Â + 767. It is prime for n = 0, 4, 11, 15, 32,
215, 408, 461, 1489, 9349, 9470 and no more up to 10000. The last two
have 18710 and 18952 digits, respectively (but they’re actually only
probable primes since their form is not easily provable). Surely there
must be infinitely many Feynman Point primes.