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9 number - In mathematics

9 number - In mathematics: Encyclopedia II - 9 number - In mathematics

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. In base 10 a number is evenly divisible by nine if and only if the iterative sum of its digits reduces to 9. A simple way of saying this is that if you multiply nine by any whole number, and repeatedly add the digits of the answer until it's 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) 234*9=2106 (2+1+0+6=9) 578329* ...

9 number: Encyclopedia II - 9 number - In mathematics

9 number - In mathematics

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.

In base 10 a number is evenly divisible by nine if and only if the iterative sum of its digits reduces to 9. A simple way of saying this is that if you multiply nine by any whole number, and repeatedly add the digits of the answer until it's 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)
234*9=2106 (2+1+0+6=9)
578329*9=5204961 (5+2+0+4+9+6+1=27 (2+7=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.

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).

Since , 9 is an exponential factorial.

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 −log₁₀ 0.01 = 2 nines of probability. Zero probability gives zero nines (−log₁₀ 1 = 0). The purity of chemicals (see Nine (purity)), the effectivity of processes, the availability of systems etc. can similarly be expressed in nines. For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year. This measure can be confusing, a fact which is discussed in The Myth of the Nines.

Six recurring nines appear in the decimal places 762 through 767 of pi. This is known as the Feynman point (see also MathWorld).

A polygon with nine sides is called an enneagon (technically) or nonagon (in common usage). A group of nine of anything is called an ennead.