10

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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

$(\genfrac{}{}{0ex}{}{5}{2}),(\genfrac{}{}{0ex}{}{5}{3}),$

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

$(\genfrac{}{}{0ex}{}{10}{1}),(\genfrac{}{}{0ex}{}{10}{9}).$

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
$3x+1$ sequence beginning at 3	3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...	A033478
$3x-1$ 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 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}	Y			Y		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}	Y		Y		Y	5
7	{3, 1, 1, 1, 1, 1, 1, 1}				Y		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}				Y		6
12	{3, 3, 2, 1, 1}						5
13	{3, 3, 2, 2}					Y	4
14	{3, 3, 3, 1}				Y		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}			Y			4
19	{4, 3, 1, 1, 1}						5
20	{4, 3, 2, 1}		Y				4
21	{4, 3, 3}						3
22	{4, 4, 1, 1}						4
23	{4, 4, 2}			Y			3
24	{5, 1, 1, 1, 1, 1}				Y		6
25	{5, 2, 1, 1, 1}						5
26	{5, 2, 2, 1}						4
27	{5, 3, 1, 1}				Y		4
28	{5, 3, 2}		Y			Y	3
29	{5, 4, 1}		Y				3
30	{5, 5}	Y			Y	Y	2
31	{6, 1, 1, 1, 1}						5
32	{6, 2, 1, 1}						4
33	{6, 2, 2}			Y			3
34	{6, 3, 1}		Y				3
35	{6, 4}		Y	Y			2
36	{7, 1, 1, 1}				Y		4
37	{7, 2, 1}		Y				3
38	{7, 3}		Y		Y	Y	2
39	{8, 1, 1}						3
40	{8, 2}		Y	Y			2
41	{9, 1}		Y		Y		2
42	{10}	Y	Y	Y			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, $\log x$ refers to the natural logarithm of $x$, 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.

${\log }_{10}2$	0.30102999	A007524	${\log }_{2}10$	3.32192809	A020862	2 10	1024
${\log }_{10}e$	0.43429448	A002285	$\log 10$	2.30258509	A002392	$e^{10}$	22026.46579480
${\log }_{10}3$	0.47712125	A114490	${\log }_{3}10$	2.09590327	A152566	3 10	59049
${\log }_{10}\pi$	0.49714987	A053511	${\log }_{\pi }10$	2.01146586	A235955	${\pi }^{10}$	93648.04747608
${\log }_{10}4$	0.60205999	A114493	${\log }_{4}10$	1.66096404	A154155	4 10	1048576
${\log }_{10}5$	0.69897000	A153268	${\log }_{5}10$	1.43067655	A154156	5 10	9765625
${\log }_{10}6$	0.77815125	A153496	${\log }_{6}10$	1.28509720	A154157	6 10	60466176
${\log }_{10}7$	0.84509804	A153620	${\log }_{7}10$	1.18329466	A154158	7 10	282475249
${\log }_{10}8$	0.90308998	A153790	${\log }_{8}10$	1.10730936	A154159	8 10	1073741824
${\log }_{10}9$	0.95424250	A104139	${\log }_{9}10$	1.04795163	A154160	9 10	3486784401
${\log }_{10}10$			1.00000000			10 10	10000000000

(See A008454 for the tenth powers of integers).

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

Values for number theoretic functions with 10 as an argument

$\mu (10)$	1
$M(10)$	–1
$\pi (10)$	4
${\sigma }_1(10)$	18
${\sigma }_0(10)$	4
$\varphi (10)$	4
$\mathrm{\Omega }(10)$	2
$\omega (10)$	2
$\lambda (10)$	4	This is the Carmichael lambda function.
$\lambda (10)$	1	This is the Liouville lambda function.
$\zeta (10)=\frac{{\pi }^{10}}{93555}$	1.0009945751278180853371459589... (see A013668).
10!	3628800
$\mathrm{\Gamma }(10)$	362880

Factorization of 10 in some quadratic integer rings

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

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

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

Nor is $(-1)(2-\sqrt{14})(2+\sqrt{14})$ 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 $\mathbb{\mathbb{Z}}[\sqrt{10}]$, with units of the form $\pm (3+\sqrt{10}{)}^n$ ($n\in \mathbb{\mathbb{Z}}$), is not a unique factorization domain, having class number 2. If an odd prime $p\in \mathbb{\mathbb{Z}}$ is of the form $10x^2-y^2$, it is composite in $\mathbb{\mathbb{Z}}[\sqrt{10}]$ (see A141180).

$\mathbb{\mathbb{Z}}[\sqrt{-10}]$ 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.

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

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

$p$	Factorization of $⟨p⟩$
	In $\mathbb{\mathbb{Z}}[\sqrt{-10}]$	In $\mathbb{\mathbb{Z}}[\sqrt{10}]$
2	$⟨2,\sqrt{-10}{⟩}^2$	$⟨2,\sqrt{10}{⟩}^2$
3	Prime	$⟨3,1-\sqrt{10}⟩⟨3,1+\sqrt{10}⟩$
5	$⟨5,\sqrt{-10}{⟩}^2$	$⟨5,\sqrt{10}{⟩}^2$
7	$⟨7,2-\sqrt{-10}⟩⟨7,2+\sqrt{-10}⟩$	Prime
11	$⟨1-\sqrt{-10}⟩⟨1+\sqrt{-10}⟩$
13	$⟨13,4-\sqrt{-10}⟩⟨13,4+\sqrt{-10}⟩$	$⟨13,6-\sqrt{10}⟩⟨13,6+\sqrt{10}⟩$
17	Prime
19	$⟨3-\sqrt{-10}⟩⟨3+\sqrt{-10}⟩$	Prime
23	$⟨23,6-\sqrt{-10}⟩⟨23,6+\sqrt{-10}⟩$
29	Prime
31		$⟨31,14-\sqrt{10}⟩⟨31,14+\sqrt{10}⟩$
37	$⟨37,8-\sqrt{-10}⟩⟨37,8+\sqrt{-10}⟩$	$⟨37,11-\sqrt{10}⟩⟨37,11+\sqrt{10}⟩$
41	$⟨1-2\sqrt{-10}⟩⟨1+2\sqrt{-10}⟩$	$⟨9-2\sqrt{10}⟩⟨9+2\sqrt{10}⟩$
43	Prime	$⟨43,15-\sqrt{10}⟩⟨43,15+\sqrt{10}⟩$
47	$⟨47,15-\sqrt{-10}⟩⟨47,15+\sqrt{-10}⟩$	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 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 $3^2+3^0$ (it has the same representation in the "normal" ternary system). In negabinary, 10 is 11110, since $(-2{)}^4+(-2{)}^3+(-2{)}^2+(-2{)}^1=16-8+4-2=10$. In quater-imaginary base, 10 is 10202. In the factorial numeral system, 10 is 120, since $3!+2\times 2!=10$.

See also

• 10-gonal numbers (decagonal numbers)

Some integers

							$-1$
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