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# 10

Please do not rely on any information it contains.

Since we have ten fingers, which are convenient for counting, 10 is the base of our numeral system. At various times in our history, 5 (five fingers), 12 (first abundant number), 20 (perhaps for fingers and toes) and 60 (an abundant number) were contenders for the base of numeration, but in the end, the decimal numeral system won out. Almost all computers carry out their computations in binary, but the vast majority of the time they take input and give output in decimal.

## Membership in core sequences

 Even numbers 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... A005843(5) Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ... A002808(5) Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, ... A001358(4) Squarefree numbers 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, ... A005117(7) Triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ... A000217(4)

In Pascal's triangle, 10 occurs four times: firstly the two binomial coefficients in the middle of row 5

${\displaystyle {\binom {5}{2}},{\binom {5}{3}},}$

and secondly twice in row 10, in the second and next to last positions

${\displaystyle {\binom {10}{1}},{\binom {10}{9}}.}$

## Core sequences modulo 10

Given that 10 is the base of our preferred numeral system, it is only natural that the OEIS has entries for several core sequences modulo 10.

 Integers modulo 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, ... A010879 Prime numbers modulo 10 2, 3, 5, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, ... A007652 Fibonacci numbers modulo 10 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, ... A003893 Squares modulo 10 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, ... A008959 Powers of 2 modulo 10 1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, ... A000689

## Sequences pertaining to 10

 Multiples of 10 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, ... A008592 Decagonal numbers 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, ... A001107 Decagonal pyramidal numbers 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375, 1826, ... A007585 ${\displaystyle 3x+1}$ sequence beginning at 3 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ... A033478 ${\displaystyle 3x-1}$ sequence beginning at 36 36, 18, 9, 26, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, 14, ... A008894

## Partitions of 10

There are forty-two[1] partitions of 10.

The numbers of partitions of 10 with largest part {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} are respectively {1, 5, 8, 9, 7, 5, 3, 2, 1, 1}. (There are 20 partitions of 10 with largest part odd and 22 partitions of 10 with largest part even.)

The numbers of partitions of 10 with number of parts {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} are respectively {1, 5, 8, 9, 7, 5, 3, 2, 1, 1}. (There are 20 partitions of 10 into an odd number of parts and 22 partitions of 10 into an even number of parts.)

The four partitions of 10 into equal parts (factorization partitions) are

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2} , {5, 5} and {10}.

The ten partitions of 10 into distinct parts are

{4, 3, 2, 1}, {5, 3, 2}, {5, 4, 1}, {6, 3, 1}, {6, 4}, {7, 2, 1}, {7, 3}, {8, 2}, {9, 1}, {10},

where {4, 3, 2, 1} is a triangular partition.

The seven partitions of 10 into even parts are

{2, 2, 2, 2, 2}, {4, 2, 2, 2}, {4, 4, 2}, {6, 2, 2}, {6, 4}, {8, 2}, {10}.

The ten partitions of 10 into odd parts are

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 1, 1, 1, 1}, {3, 3, 3, 1}, {5, 1, 1, 1, 1, 1}, {5, 3, 1, 1}, {5, 5}, {7, 1, 1, 1}, {7, 3}, {9, 1}.

The five partitions of 10 into prime parts are

{2, 2, 2, 2, 2}, {3, 3, 2, 2}, {5, 3, 2}, {5, 5} and {7, 3},

where {5, 5} and {7, 3} are Goldbach partitions of 10.

The two partitions of 10 into distinct prime parts are

{5, 3, 2} and {7, 3}.

The four partitions of 10 into square parts are

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 1, 1, 1, 1, 1, 1}, {4, 4, 1, 1} and {9, 1},

where {4, 4, 1, 1} is a "sum of four squares" partition of 10.

Partitions of 10 (in lexicographic order)
Ordinal Partition Equal
parts
Distinct
parts
Even
parts
Odd
parts
Prime
parts
Number
of parts
1 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1} 10
2 {2, 1, 1, 1, 1, 1, 1, 1, 1} 9
3 {2, 2, 1, 1, 1, 1, 1, 1} 8
4 {2, 2, 2, 1, 1, 1, 1} 7
5 {2, 2, 2, 2, 1, 1} 6
6 {2, 2, 2, 2, 2} 5
7 {3, 1, 1, 1, 1, 1, 1, 1} 8
8 {3, 2, 1, 1, 1, 1, 1} 7
9 {3, 2, 2, 1, 1, 1} 6
10 {3, 2, 2, 2, 1} 5
11 {3, 3, 1, 1, 1, 1} 6
12 {3, 3, 2, 1, 1} 5
13 {3, 3, 2, 2} 4
14 {3, 3, 3, 1} 4
15 {4, 1, 1, 1, 1, 1, 1} 7
16 {4, 2, 1, 1, 1, 1} 6
17 {4, 2, 2, 1, 1} 5
18 {4, 2, 2, 2} 4
19 {4, 3, 1, 1, 1} 5
20 {4, 3, 2, 1} 4
21 {4, 3, 3} 3
22 {4, 4, 1, 1} 4
23 {4, 4, 2} 3
24 {5, 1, 1, 1, 1, 1} 6
25 {5, 2, 1, 1, 1} 5
26 {5, 2, 2, 1} 4
27 {5, 3, 1, 1} 4
28 {5, 3, 2} 3
29 {5, 4, 1} 3
30 {5, 5} 2
31 {6, 1, 1, 1, 1} 5
32 {6, 2, 1, 1} 4
33 {6, 2, 2} 3
34 {6, 3, 1} 3
35 {6, 4} 2
36 {7, 1, 1, 1} 4
37 {7, 2, 1} 3
38 {7, 3} 2
39 {8, 1, 1} 3
40 {8, 2} 2
41 {9, 1} 2
42 {10} 1

## Roots and powers of 10

In the following tables, irrational numbers are given truncated to eight decimal places.

 n√  10
A-numbers
 2√  10
3.16227766 A010467
 3√  10
2.15443469 A010582
 4√  10
1.77827941 A011007
 5√  10
1.58489319 A011095
 6√  10
1.46779926 A011275
 7√  10
1.38949549 A011276
 8√  10
1.33352143 A011277
 9√  10
1.29154966 A011278
 10√  10
1.25892541 A011279
 10 n
A011557
 10 2
100
 10 3
1000
 10 4
10000
 10 5
100000
 10 6
1000000
 10 7
10000000
 10 8
100000000
 10 9
1000000000
 10 10
10000000000

## Logarithms and tenth powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript. But in many contexts, base 10 logarithms (common logarithms) are meant when there is no subscript.

As above, irrational numbers in the following tables are truncated to eight decimal places.

 ${\displaystyle \log _{10}2}$ 0.30102999 A007524 ${\displaystyle \log _{2}10}$ 3.32192809 A020862 2 10 1024 ${\displaystyle \log _{10}e}$ 0.43429448 A002285 ${\displaystyle \log 10}$ 2.30258509 A002392 ${\displaystyle e^{10}}$ 22026.5 ${\displaystyle \log _{10}3}$ 0.47712125 A114490 ${\displaystyle \log _{3}10}$ 2.09590327 A152566 3 10 59049 ${\displaystyle \log _{10}\pi }$ 0.49714987 A053511 ${\displaystyle \log _{\pi }10}$ 2.01146586 A235955 ${\displaystyle \pi ^{10}}$ 93648 ${\displaystyle \log _{10}4}$ 0.60205999 A114493 ${\displaystyle \log _{4}10}$ 1.66096404 A154155 4 10 1.04858e+06 ${\displaystyle \log _{10}5}$ 0.69897000 A153268 ${\displaystyle \log _{5}10}$ 1.43067655 A154156 5 10 9.76562e+06 ${\displaystyle \log _{10}6}$ 0.77815125 A153496 ${\displaystyle \log _{6}10}$ 1.28509720 A154157 6 10 6.04662e+07 ${\displaystyle \log _{10}7}$ 0.84509804 A153620 ${\displaystyle \log _{7}10}$ 1.18329466 A154158 7 10 2.82475e+08 ${\displaystyle \log _{10}8}$ 0.90308998 A153790 ${\displaystyle \log _{8}10}$ 1.10730936 A154159 8 10 1.07374e+09 ${\displaystyle \log _{10}9}$ 0.95424250 A104139 ${\displaystyle \log _{9}10}$ 1.04795163 A154160 9 10 3.48678e+09 ${\displaystyle \log _{10}10}$ 1.00000000 10 10 1e+10

(See A008454 for the tenth powers of integers).

It follows from Fermat's little theorem that if an integer ${\displaystyle n}$ is not a multiple of 11, then ${\displaystyle n^{10}-1}$ is.

## Values for number theoretic functions with 10 as an argument

 ${\displaystyle \mu (10)}$ 1 ${\displaystyle M(10)}$ –1 ${\displaystyle \pi (10)}$ 4 ${\displaystyle \sigma _{1}(10)}$ 18 ${\displaystyle \sigma _{0}(10)}$ 4 ${\displaystyle \phi (10)}$ 4 ${\displaystyle \Omega (10)}$ 2 ${\displaystyle \omega (10)}$ 2 ${\displaystyle \lambda (10)}$ 4 This is the Carmichael lambda function. ${\displaystyle \lambda (10)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (10)={\frac {\pi ^{10}}{93555}}}$ 1.0009945751278180853371459589... (see A013668). 10! 3628800 ${\displaystyle \Gamma (10)}$ 362880

## Factorization of 10 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 10 has the prime factorization of 2 × 5. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (-i)(1-i)(1+i)(2+i)(1+2i)}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle (-1)({\sqrt {-2}})^{2}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{2}5}$ ${\displaystyle \mathbb {Z} [\omega ]}$ 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle (-1)2({\sqrt {-5}})^{2}}$ ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle 2(-1+2\phi )^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2-{\sqrt {6}})(2+{\sqrt {6}})(1-{\sqrt {6}})(1+{\sqrt {6}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)5}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3-{\sqrt {7}})(3+{\sqrt {7}})5}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 × 5 OR ${\displaystyle (-1)({\sqrt {-10}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 × 5 OR ${\displaystyle ({\sqrt {10}})^{2}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (-1)(3\pm {\sqrt {11}})(7\pm 2{\sqrt {11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (-1)(4\pm {\sqrt {14}})(3\pm {\sqrt {14}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 × 5 OR ${\displaystyle (5-{\sqrt {15}})(5+{\sqrt {15}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle (-1)\left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)5}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-19}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-19}}{2}}\right)5}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (-1)(13\pm 3{\sqrt {19}})(48\pm 11{\sqrt {19}})}$

Note that ${\displaystyle (-1)(1-{\sqrt {11}})(1+{\sqrt {11}})}$ is not a distinct factorization because both ${\displaystyle (1-{\sqrt {11}})}$ and ${\displaystyle (1+{\sqrt {11}})}$ are divisible by ${\displaystyle (3\pm {\sqrt {11}})}$ and ${\displaystyle (7\pm 2{\sqrt {11}})}$.

Nor is ${\displaystyle (-1)(2-{\sqrt {14}})(2+{\sqrt {14}})}$ is a distinct factorization as the latter two factors are both reducible.

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −10, 10

The commutative quadratic integer ring with unity ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {10}}]\,}$, with units of the form ${\displaystyle \scriptstyle \pm (3+{\sqrt {10}})^{n}\,}$ (${\displaystyle \scriptstyle n\,\in \,\mathbb {Z} \,}$), is not a unique factorization domain, having class number 2. If an odd prime ${\displaystyle p\in \mathbb {Z} }$ is of the form ${\displaystyle 10x^{2}-y^{2}}$, it is composite in ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {10}}]\,}$ (see A141180).

${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ is not a unique factorization domain either, also having class number 2. But in it, there are only two units: 1 and –1. Therefore, we can say with much greater confidence that we have correctly identified instances of multiple factorization. But rather than "prime" we will use the term "irreducible" for those primes which are not composite in these domains.[2]

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 Irreducible Irreducible 3 Prime 4 2 2 5 Irreducible 6 2 × 3 2 × 3 OR ${\displaystyle (-1)(2-{\sqrt {10}})(2+{\sqrt {10}})}$ 7 Irreducible Prime 8 2 3 9 3 2 3 2 OR ${\displaystyle (-1)(1-{\sqrt {10}})(1+{\sqrt {10}})}$ 10 2 × 5 OR ${\displaystyle (-1)({\sqrt {-10}})^{2}}$ 2 × 5 OR ${\displaystyle ({\sqrt {10}})^{2}}$ 11 ${\displaystyle (1-{\sqrt {-10}})(1+{\sqrt {-10}})}$ Prime 12 2 2 × 3 13 Irreducible 14 2 × 7 OR ${\displaystyle (2-{\sqrt {-10}})(2+{\sqrt {-10}})}$ 15 3 × 5 3 × 5 OR ${\displaystyle (5-{\sqrt {10}})(5+{\sqrt {10}})}$ 16 2 4 2 4 17 Prime 18 2 × 3 2 2 × 3 2 OR ${\displaystyle 3(4-{\sqrt {10}})(4+{\sqrt {10}})}$ 19 ${\displaystyle (3-{\sqrt {-10}})(3+{\sqrt {-10}})}$ Prime 20 2 2 × 5 OR ${\displaystyle (-1)2({\sqrt {-10}})^{2}}$ 2 2 × 5 OR ${\displaystyle 2({\sqrt {10}})^{2}}$

Note that ${\displaystyle (38-12{\sqrt {10}})(38+12{\sqrt {10}})}$ is not a distinct factorization of 4, as it is readily obtained by multiplying 2 by ${\displaystyle (3-{\sqrt {10}})^{2}}$. Nor is ${\displaystyle (4-{\sqrt {10}})(4+{\sqrt {10}})}$ a distinct factorization of 6 because ${\displaystyle (2+{\sqrt {10}})(3-{\sqrt {10}})=4-{\sqrt {10}}}$. That ${\displaystyle (10-3{\sqrt {10}})(10+3{\sqrt {10}})}$ is not a distinct factorization of 10 should become obvious upon comparison to the fundamental unit. Similarly for 16, ${\displaystyle (-1)(12-4{\sqrt {10}})(12+4{\sqrt {10}})}$ is not a distinct factorization because it can be obtained by twice doubling the fundamental unit.

Ideals really help us make sense of multiple distinct factorizations in these domains.

 ${\displaystyle p}$ Factorization of ${\displaystyle \langle p\rangle }$ In ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ In ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 ${\displaystyle \langle 2,{\sqrt {-10}}\rangle ^{2}}$ ${\displaystyle \langle 2,{\sqrt {10}}\rangle ^{2}}$ 3 Prime ${\displaystyle \langle 3,1-{\sqrt {10}}\rangle \langle 3,1+{\sqrt {10}}\rangle }$ 5 ${\displaystyle \langle 5,{\sqrt {-10}}\rangle ^{2}}$ ${\displaystyle \langle 5,{\sqrt {10}}\rangle ^{2}}$ 7 ${\displaystyle \langle 7,2-{\sqrt {-10}}\rangle \langle 7,2+{\sqrt {-10}}\rangle }$ Prime 11 ${\displaystyle \langle 1-{\sqrt {-10}}\rangle \langle 1+{\sqrt {-10}}\rangle }$ 13 ${\displaystyle \langle 13,4-{\sqrt {-10}}\rangle \langle 13,4+{\sqrt {-10}}\rangle }$ ${\displaystyle \langle 13,6-{\sqrt {10}}\rangle \langle 13,6+{\sqrt {10}}\rangle }$ 17 Prime 19 ${\displaystyle \langle 3-{\sqrt {-10}}\rangle \langle 3+{\sqrt {-10}}\rangle }$ Prime 23 ${\displaystyle \langle 23,6-{\sqrt {-10}}\rangle \langle 23,6+{\sqrt {-10}}\rangle }$ 29 Prime 31 ${\displaystyle \langle 31,14-{\sqrt {10}}\rangle \langle 31,14+{\sqrt {10}}\rangle }$ 37 ${\displaystyle \langle 37,8-{\sqrt {-10}}\rangle \langle 37,8+{\sqrt {-10}}\rangle }$ ${\displaystyle \langle 37,11-{\sqrt {10}}\rangle \langle 37,11+{\sqrt {10}}\rangle }$ 41 ${\displaystyle \langle 1-2{\sqrt {-10}}\rangle \langle 1+2{\sqrt {-10}}\rangle }$ ${\displaystyle \langle 9-2{\sqrt {10}}\rangle \langle 9+2{\sqrt {10}}\rangle }$ 43 Prime ${\displaystyle \langle 43,15-{\sqrt {10}}\rangle \langle 43,15+{\sqrt {10}}\rangle }$ 47 ${\displaystyle \langle 47,15-{\sqrt {-10}}\rangle \langle 47,15+{\sqrt {-10}}\rangle }$ Prime

## Representation of 10 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 through 36 Representation 1010 101 22 20 14 13 12 11 10 A

For bases 37 and higher, we could theoretically just keep adding symbols. At least one base converter available online[3] adds the lowercase letters a to z in order to accomplish conversion to bases 37 through 62. Thus, a is 36 in base 37, but A is still A in base 37.

In the balanced ternary numeral system, 10 is {1, 0, 1}, meaning ${\displaystyle 3^{2}+3^{0}}$ (it has the same representation in the "normal" ternary system). In negabinary, 10 is 11110, since ${\displaystyle (-2)^{4}+(-2)^{3}+(-2)^{2}+(-2)^{1}=16-8+4-2=10}$. In quater-imaginary base, 10 is 10202. In the factorial numeral system, 10 is 120, since ${\displaystyle 3!+2\times 2!=10}$.

## References

1. A000041(10)
2. See irreducible elements for an explanation of these terminologies.
3. http://convertxy.com/index.php/numberbases/

 ${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729