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9 (nine ) is the natural number following 8 and preceding 10.
Nine is a number that appears often in Indian Culture and mythology. Some instances are enumerated below.
According to Georges Ifrah, the origin of the 9 integers can be attributed to the ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0.^{[2]}
In the beginning, various Indians wrote 9 similar to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the @ character encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic.
While the shape of the 9 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .
This numeral resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, the 9 is often underlined, as is done for the 6. Another distinction from the 6 is that it is often handwritten with a straight stem.
Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. Nine is a Motzkin number. It is the first composite lucky number, along with the first composite odd number.
9 is the only positive perfect power that is one more than another positive perfect power, by Mihăilescu's Theorem.
Nine is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p^{2}) 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. Nine is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of nine has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree. It is the aliquot sum of only one number the discrete semiprime 15.
There are nine Heegner numbers.^{[6]}
Since 9 = 3^{21}, 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.
In bases 12, 18 and 24, nine is a 1-automorphic number and in base 6 a 2-automorphic number (displayed as '13').
A polygon with nine sides is called a nonagon or enneagon.^{[7]} A group of nine of anything is called an ennead.
In base 10 a positive number is divisible by nine if and only if its digital root is 9.^{[8]} That is, if any natural number is multiplied by nine, and repeatedly add the digits of the answer until it is just one digit, the sum will be nine:
There are other interesting patterns involving multiples of nine:
This works for all the multiples of 9. n = 3 is the only other n > 1 such that a number is divisible by n if and only if its digital root is n. 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:
Subtracting two base-10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Examples:
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:
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.^{[9]}
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.^{[10]}
If dividing a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal. (e.g. 274/999 = 0.274274274274...)
Nine is the binary complement of number six:
9 = 1001 6 = 0110
In probability, the nine is a logarithmic measure of probability of an event, defined as the negative of the base-10 logarithm of the probability of the event's complement. For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to −log_{10} 0.01 = 2 nines of probability. Zero probability gives zero nines (−log_{10} 1 = 0). A 100% probability is considered to be impossible in most circumstances: that results in infinite improbability. The effectivity of processes and the availability of systems can be expressed (as a rule of thumb, not explicitly) as a series of "nines". For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year - typically a very high degree of reliability; but never 100%.
A human pregnancy normally lasts nine months, the basis of Naegele's rule.
The jersey number 9 has been retired by several North American sports teams in honor of past playing greats (or in one case, an owner):
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