# List of numbers

This is a list of articles about numbers (not about numerals).

## Rational numbers

### Natural numbers

 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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 230 240 250 260 270 280 290 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000 50000 60000 70000 80000 90000 100k–1M 1M–10M 10M–100M 100M–1G 1G–10G Larger numbers

### Integers

#### Notable integers

Other numbers that are notable for their mathematical properties or cultural meanings include:

### Prime numbers

A prime number is a positive integer which has exactly two divisors: one and itself.

 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541

### Highly composite numbers

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used in geometry, grouping and time measurement.

The first 20 highly composite numbers (the seven values with more divisors than any lesser number twice itself are in bold):

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1 260, 1 680, 2 520, 5 040, 7 560

### Perfect numbers

A perfect number is an integer that is the sum of its positive proper divisors (all divisors except itself).

The first 10 perfect numbers:

1 6
2 28
3 496
4 8 128
5 33 550 336
6 8 589 869 056
7 137 438 691 328
8 2 305 843 008 139 952 128
9 2 658 455 991 569 831 744 654 692 615 953 842 176
10    191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216

### Cardinal numbers

In the following tables, [and] indicates that the word and is used in some dialects (such as British English), and omitted in other dialects (such as American English).

#### Small numbers

This table demonstrates the standard English construction of small cardinal numbers up to one hundred million—names for which all variants of English agree.

Value Name Alternate names, and names for sets of the given size
0 Zero aught, cipher, cypher, donut, goose egg, love, nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip
1 One ace, individual, single, singleton, unary, unit, unity
2 Two binary, brace, couple, couplet, distich, deuce, double, doubleton, duad, duality, duet, duo, dyad, pair, span, twain, twosome, yoke
3 Three deuce-ace, leash, set, tercet, ternary, ternion, terzetto, threesome, tierce, trey, triad, trine, trinity, trio, triplet, troika, hat-trick
4 Four foursome, quadruplet, quatern, quaternary, quaternion, quaternity, quartet, tetrad
5 Five cinque, fin, fivesome, pentad, quint, quintet, quintuplet
6 Six half dozen, hexad, sestet, sextet, sextuplet, sise
7 Seven heptad, septet, septuple
8 Eight octad, octave, octet, octonary, octuplet, ogdoad
10 Ten deca, decade
11 Eleven onze, ounze, ounce
12 Twelve dozen
13 Thirteen baker's dozen, long dozen[1]
14 Fourteen
15 Fifteen
16 Sixteen
17 Seventeen
18 Eighteen
19 Nineteen
20 Twenty score
21 Twenty-one long score[1]
22 Twenty-two Deuce-deuce
23 Twenty-three
24 Twenty-four two dozen
25 Twenty-five
26 Twenty-six
27 Twenty-seven
28 Twenty-eight
29 Twenty-nine
30 Thirty
31 Thirty-one
40 Forty
50 Fifty half-century
60 Sixty three-score
70 Seventy three-score and ten
80 Eighty four-score
87 Eighty-seven four-score and seven
90 Ninety
100 One hundred centred, century, ton, short hundred
101 One hundred [and] one
110 One hundred [and] ten
111 One hundred [and] eleven
120 One hundred [and] twenty long hundred,[1] great hundred, (obsolete) hundred
121 One hundred [and] twenty-one
144 One hundred [and] forty-four gross, dozen dozen, small gross
169 One hundred [and] sixty-nine baker's gross[citation needed]
200 Two hundred
300 Three hundred
400 Four hundred
500 Five hundred
600 Six hundred
666 Six hundred [and] sixty-six Number of the Beast
700 Seven hundred
777 Seven hundred [and] seventy-seven Number of Luck
800 Eight hundred
900 Nine hundred
1 000 One thousand chiliad, grand, G, thou, yard, kilo, k
1 001 One thousand [and] one
1 010 One thousand [and] ten
1 011 One thousand [and] eleven
1 024 One thousand [and] twenty-four kibi or kilo in computing, see binary prefix (kilo is shortened to K, Kibi to Ki)
1 100 One thousand one hundred Eleven hundred
1 101 One thousand one hundred [and] one
1 728 One thousand seven hundred [and] twenty-eight great gross, long gross, dozen gross
2 000 Two thousand
3 000 Three thousand
10 000 Ten thousand myriad, wan (China)
100 000 One hundred thousand lakh
500 000 Five hundred thousand crore (Iranian)
1 000 000 One million Mega, meg, mil, (often shortened to M)
1 048 576 One million forty-eight thousand five hundred [and] seventy-six Mibi or Mega in computing, see binary prefix (Mega is shortened to M, Mibi to Mi)
10 000 000 Ten million crore (Bhartia)
100 000 000 One hundred million yi (China)

#### English names for powers of 10

This table compares the English names of cardinal numbers according to various American, British, and Continental European conventions. See English numerals or names of large numbers for more information on naming numbers.

Short scale Long scale Power
Value American British
(Nicolas Chuquet)
Continental European
(Jacques Peletier du Mans)
of a thousand of a million
100 One 1000−1+1 10000000
101 Ten
102 Hundred
103 Thousand 10000+1 10000000.5
106 Million 10001+1 10000001
109 Billion Thousand million Milliard 10002+1 10000001.5
1012 Trillion Billion 10003+1 10000002
1015 Quadrillion Thousand billion Billiard 10004+1 10000002.5
1018 Quintillion Trillion 10005+1 10000003
1021 Sextillion Thousand trillion Trilliard 10006+1 10000003.5
1024 Septillion Quadrillion 10007+1 10000004
1027 Octillion Thousand quadrillion Quadrilliard 10008+1 10000004.5
1030 Nonillion Quintillion 10009+1 10000005
1033 Decillion Thousand quintillion Quintilliard 100010+1 10000005.5
1036 Undecillion Sextillion 100011+1 10000006
1039 Duodecillion Thousand sextillion Sextilliard 100012+1 10000006.5
1042 Tredecillion Septillion 100013+1 10000007
1045 Quattuordecillion Thousand septillion Septilliard 100014+1 10000007.5
1048 Quindecillion Octillion 100015+1 10000008
1051 Sexdecillion Thousand octillion Octilliard 100016+1 10000008.5
1054 Septendecillion Nonillion 100017+1 10000009
1057 Octodecillion Thousand nonillion Nonilliard 100018+1 10000009.5
1060 Novemdecillion Decillion 100019+1 100000010
1063 Vigintillion Thousand decillion Decilliard 100020+1 100000010.5
1066 Unvigintillion Undecillion 100021+1 100000011
1069 Duovigintillion Thousand undecillion Undecilliard 100022+1 100000011.5
1072 Trevigintillion Duodecillion 100023+1 100000012
1075 Quattuorvigintillion Thousand duodecillion Duodecilliard 100024+1 100000012.5
... ... ... ... ...
1078 Quinvigintillion Tredecillion 100025+1 100000013
1093 Trigintillion Thousand quindecillion Quindecilliard 100030+1 100000015.5
... ... ... ... ...
10120 Novemtrigintillion Vigintillion 100039+1 100000020
10123 Quadragintillion Thousand vigintillion Vigintilliard 100040+1 100000020.5
... ... ... ... ...
10153 Quinquagintillion Thousand quinvigintillion Quinvigintilliard 100050+1 100000025.5
... ... ... ... ...
10180 Novemquinquagintillion Trigintillion 100059+1 100000030
10183 Sexagintillion Thousand trigintillion Trigintilliard 100060+1 100000030.5
... ... ... ... ...
10213 Septuagintillion Thousand quintrigintillion Quintrigintilliard 100070+1 100000035.5
... ... ... ... ...
10240 Novemseptuagintillion Quadragintillion 100079+1 100000040
10243 Octogintillion Thousand quadragintillion Quadragintilliard 100080+1 100000040.5
... ... ... ... ...
10273 Nonagintillion Thousand quinquadragintillion Quinquadragintilliard 100090+1 100000045.5
... ... ... ... ...
10300 Novemnonagintillion Quinquagintillion 100099+1 100000050
10303 Centillion Thousand quinquagintillion Quinquagintilliard 1000100+1 100000050.5
... ... ... ...
10360 Sexagintillion 1000119+1 100000060
10420 Septuagintillion 1000139+1 100000070
10480 Octogintillion 1000159+1 100000080
10540 Nonagintillion 1000179+1 100000090
10600 Centillion 1000199+1 1000000100
10603 Ducentillion Thousand Centillion Centilliard 1000200+1 1000000100.5

There is no consistent and widely accepted way to extend cardinals beyond centillion (centilliard).

#### Proposed systematic names for powers of 10

Value Name Notation
100 One 1
101 Ten 10
102 Hundred 100
103 Ten hundred 1000
105 Ten myriad 10,0000
106 Hundred myriad 100,0000
107 Ten hundred myriad 1000,0000
108 Myllion 1;0000,0000
1012 Myriad myllion 1,0000;0000,0000
1016 Byllion 1:0000,0000;0000,0000
1024 Myllion byllion 1;0000,0000:0000,0000;0000,0000
1032 Tryllion 1'0000,0000;0000,0000:0000,0000;0000,0000
10128 Quintyllion
10256 Sextyllion
10512 Septyllion
101024 Octyllion
102048 Nonyllion
104096 Decyllion
108192 Undecyllion
1016,384 Duodecyllion
1032,768 Tredecyllion
1065,536 Quattuordecyllion
10131,072 Quindecyllion
10262,144 Sexdecyllion
10524,288 Septendecyllion
101,048,576 Octodecyllion
102,097,152 Novemdecyllion
${10}^{\,\! 4\cdot 2^{20}}$ Vigintyllion
${10}^{\,\! 4\cdot 2^{30}}$ Trigintyllion
${10}^{\,\! 4 \cdot 2^{40}}$ Quadragintyllion
${10}^{\,\! 4 \cdot 2^{50}}$ Quinquagintyllion
${10}^{\,\! 4 \cdot 2^{60}}$ Sexagintyllion
${10}^{\,\! 4 \cdot 2^{70}}$ Septuagintyllion
${10}^{\,\! 4 \cdot 2^{80}}$ Octogintyllion
${10}^{\,\! 4 \cdot 2^{90}}$ Nonagintyllion
${10}^{\,\! 4 \cdot 2^{100}}$ Centyllion
${10}^{\,\! 4 \cdot 2^{1000}}$ Millyllion
${10}^{\,\! 4 \cdot 2^{10,000}}$ Myryllion
##### SI-derived
Value 1000m SI prefix Name Binary prefix 1024m=210m Value
1 000 10001 k Kilo Ki 10241 1 024
1 000 000 10002 M Mega Mi 10242 1 048 576
1 000 000 000 10003 G Giga Gi 10243 1 073 741 824
1 000 000 000 000 10004 T Tera Ti 10244 1 099 511 627 776
1 000 000 000 000 000 10005 P Peta Pi 10245 1 125 899 906 842 624
1 000 000 000 000 000 000 10006 E Exa Ei 10246 1 152 921 504 606 846 976
1 000 000 000 000 000 000 000 10007 Z Zetta Zi 10247 1 180 591 620 717 411 303 424
1 000 000 000 000 000 000 000 000 10008 Y Yotta Yi 10248 1 208 925 819 614 629 174 706 176

### Fractional numbers

This is a table of English names for positive rational numbers less than or equal to 1. It also lists alternative names, but there is no widespread convention for the names of extremely small positive numbers.

Keep in mind that rational numbers like 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), twelve percent (12%), three twenty-fifths $\left({3 \over 25}\right)$, nine seventy-fifths $\left({9 \over 75} \right)$, six fiftieths $\left({6 \over 50}\right)$, twelve hundredths $\left({12 \over 100}\right)$, twenty-four two-hundredths $\left({24 \over 200}\right)$, etc.

Value Fraction Common names Alternative names
1 $1 \over 1$ One 0.999..., Unity
0.9 $9 \over 10$ Nine tenths, [zero] point nine
0.8 $4 \over 5$ Four fifths, eight tenths, [zero] point eight
0.7 $7 \over 10$ Seven tenths, [zero] point seven
0.6 $3 \over 5$ Three fifths, six tenths, [zero] point six
0.5 $1 \over 2$ One half, five tenths, [zero] point five
0.4 $2 \over 5$ Two fifths, four tenths, [zero] point four
0.3 (333 333)... $1 \over 3$ One third
0.3 $3 \over 10$ Three tenths, [zero] point three
0.25 $1 \over 4$ One quarter, one fourth, twenty-five hundredths, [zero] point two five
0.2 $1 \over 5$ One fifth, two tenths, [zero] point two
0.16 (666 666)... $1 \over 6$ One sixth
0.142 857 (142 857)... $1 \over 7$ One seventh
0.125 $1 \over 8$ One eighth, one-hundred-[and-]twenty-five thousandths, [zero] point one two five
0.1 (111 111)... $1 \over 9$ One ninth
0.1 $1 \over 10$ One tenth, [zero] point one One perdecime, one perdime
0.090 (909 090)... $1 \over 11$ One eleventh
0.09 $9 \over 100$ Nine hundredths, [zero] point zero nine
0.083 (333 333)... $1 \over 12$ One twelfth
0.08 $2 \over 25$ Two twenty-fifths, eight hundredths, [zero] point zero eight
0.0625 $1 \over 16$ One sixteenth, six-hundred-[and-]twenty-five ten-thousandths, [zero] point zero six two five
0.05 $1 \over 20$ One twentieth, [zero] point zero five
0.047 619 (047 619)... $1 \over 21$ One twenty-first
0.045 (454 545)... $1 \over 22$ One twenty-second
0.043 478 260 869 565 217 3913 (043 478)... $1 \over 23$ One twenty-third
0.03 (333 333)... $1 \over 30$ One thirtieth
0.016 (666 666)... $1 \over 60$ One sixtieth One minute
0.012345679 (012345679)... $1 \over 81$ One eighty-first
0.01 $1 \over 100$ One hundredth, [zero] point zero one One percent
0.001 $1 \over 1000$ One thousandth, [zero] point zero zero one One permille
0.000 27 (777 777)... $1 \over 3600$ One thirty-six hundredth One second
0.000 1 $1 \over 10000$ One ten-thousandth, [zero] point zero zero zero one One myriadth, one permyria, one permyriad, one basis point
0.000 01 $1 \over 10^5$ One hundred-thousandth One lakhth, one perlakh
0.000 001 $1 \over 10^6$ One millionth One perion, one ppm
0.000 000 1 $1 \over 10^7$ One ten-millionth One crorth, one percrore
0.000 000 01 $1 \over 10^8$ One hundred-millionth One awkth, one perawk
0.000 000 001 $1 \over 10^9$ One billionth (in some dialects) One ppb
0 $0 \over 1$ Zero Nil

## Irrational and suspected irrational numbers

### Algebraic numbers

Expression Approximate value Notes
$\frac{\sqrt{3}}{4}$ 0.433 012 701 892 219 323 381 861 585 376 Area of a triangle with sides of length one and half its height.
${\sqrt{5} - 1} \over 2$ 0.618 033 988 749 894 848 204 586 834 366 Golden ratio conjugate $\Phi\,$, reciprocal of and one less than the golden ratio.
$\sqrt[12]{2}$ 1.059 463 094 359 295 264 561 825 294 946 Twelfth root of two.
Proportion between the frequencies of adjacent semitones in the equal temperament scale.
$\frac{3 \sqrt{2}}{4}$ 1.060 660 171 779 821 286 601 266 543 157 The size of the cube that satisfies Prince Rupert's cube.
$\sqrt[3]{2}$ 1.259 921 049 894 873 164 767 210 607 278 Cube root of two.
Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.
n/a 1.303 577 269 034 296 391 257 099 112 153 Conway's constant, defined as the unique positive real root of a certain polynomial of degree 71.
$\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+$
$\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}$
1.324 717 957 244 746 025 960 908 854 478 Plastic number, the unique real root of the cubic equation $x^3=x+1\, .$
$\sqrt{2}$ 1.414 213 562 373 095 048 801 688 724 210 $\sqrt{2} = 2 \sin 45^\circ = 2 \cos 45^\circ$
Square root of two a.k.a. Pythagoras' constant.
Ratio of diagonal to side length in a square.
Proportion between the sides of paper sizes in the ISO 216 series (originally DIN 476 series).
$\frac{\sqrt{17}-1}{2}$ 1.561 552 812 808 830 274 910 704 927 987 The Triangular root of 2.
${\sqrt{5} + 1} \over 2$ 1.618 033 988 749 894 848 204 586 834 366 Golden ratio $\left(\phi\right)$, the larger of the two real roots of $x^2=x+1\, .$
$\sqrt{3}$ 1.732 050 807 568 877 293 527 446 341 506 $\sqrt{3} = 2 \sin 60^\circ = 2 \cos 30^\circ$
Square root of three a.k.a. the measure of the fish.
Length of the space diagonal of a cube with edge length 1.
Length of the diagonal of a $1 \times \sqrt{2}$ rectangle.
Altitude of an equilateral triangle with side length 2.
Twice the altitude of an equilateral triangle with side length 1.
Altitude of a regular hexagon with side length 1 and diagonal length 2.
$\frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}$ 1.839 286 755 214 161 132 551 852 564 653 The Tribonacci constant.
Used in the formula for the volume of the snub cube and properties of some of its dual polyhedrons.
It satisfies the equation x + x−3 = 2.
$\sqrt{5}$ 2.236 067 977 499 789 696 409 173 668 731 Square root of five.
Length of the diagonal of a $1 \times 2$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{3}$ rectangle.
Length of the space diagonal of a $1 \times \sqrt{2} \times \sqrt{2}$ rectangular box.
$\sqrt{2} + 1$ 2.414 213 562 373 095 048 801 688 724 210 Silver ratio $\left(\delta_S\right)$, the larger of the two real roots of $x^2=2x+1\, .$
$\sqrt{6}$ 2.449 489 742 783 178 098 197 284 074 706 $\sqrt{2} \cdot \sqrt{3}$ = area of a $\sqrt{2} \times \sqrt{3}$ rectangle.
Length of the space diagonal of a $1 \times 1 \times 2$ rectangular box.
Length of the diagonal of a $1 \times \sqrt{5}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{2}$ rectangle.
Length of the diagonal of a square with side length $\sqrt{3}$.
$\sqrt{7}$ 2.645 751 311 064 590 590 501 615 753 639 Length of the space diagonal of a $1 \times 2 \times \sqrt{2}$ rectangular box.
Length of the diagonal of a $1 \times \sqrt{6}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{3}$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{5}$ rectangle.
$\sqrt{8}$ 2.828 427 124 746 190 097 603 377 448 419 $2 \sqrt{2}$
Volume of a cube with edge length $\sqrt{2}$.
Length of the diagonal of a square with side length 2.
Length of the diagonal of a $1 \times \sqrt{7}$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{6}$ rectangle.
Length of the diagonal of a $\sqrt{3} \times \sqrt{5}$ rectangle.
$\sqrt{10}$ 3.162 277 660 168 379 331 998 893 544 433 $\sqrt{2} \cdot \sqrt{5}$ = area of a $\sqrt{2} \times \sqrt{5}$ rectangle.
Length of the diagonal of a $1 \times 3$ rectangle.
Length of the diagonal of a $2 \times \sqrt{6}$ rectangle.
Length of the diagonal of a $\sqrt{3} \times \sqrt{7}$ rectangle.
Length of the diagonal of a square with side length $\sqrt{5}$.
$\sqrt{11}$ 3.316 624 790 355 399 849 114 932 736 671 Length of the space diagonal of a $1 \times 1 \times 3$ rectangular box.
Length of the diagonal of a $1 \times \sqrt{10}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{7}$ rectangle.
Length of the diagonal of a $3 \times \sqrt{2}$ rectangle.
Length of the diagonal of a $\sqrt{3} \times \sqrt{8}$ rectangle.
Length of the diagonal of a $\sqrt{5} \times \sqrt{6}$ rectangle.
$\sqrt{12}$ 3.464 101 615 137 754 587 054 892 683 012 $2 \sqrt{3}$
Length of the space diagonal of a cube with edge length 2.
Length of the diagonal of a $1 \times \sqrt{11}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{8}$ rectangle.
Length of the diagonal of a $3 \times \sqrt{3}$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{10}$ rectangle.
Length of the diagonal of a $\sqrt{5} \times \sqrt{7}$ rectangle.
Length of the diagonal of a square with side length $\sqrt{6}$.

## Hypercomplex numbers

### Algebraic complex numbers

• Imaginary unit: $i = \sqrt{-1}$
• nth roots of unity: $\xi^k_n = \cos\left(2\pi \tfrac{k}{n}\right)+i\sin\left(2\pi \tfrac{k}{n}\right)$

## Transfinite numbers

• Infinity in general: $\infty$
• Aleph-null: $\aleph_0$
• Aleph-one: $\aleph_1$
• Beth-one: ($\beth_1$) is the cardinality of the continuum $(2^{\aleph_0})$
• ℭ or $\mathfrak c$: the cardinality of the continuum $(2^{\aleph_0})$
• omega: ω, the smallest infinite ordinal

## Notes

1. ^ a b c The shipmaster's assistant, and commercial digest
2. ^ a b c "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 27.
3. ^ a b "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
4. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 30.
5. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33.
6. ^ http://www.qbyte.org/puzzles/p029s.html
7. ^
8. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
9. ^
10. ^
11. ^
12. ^
13. ^
14. ^
15. ^ http://mathworld.wolfram.com/ContinuedFractionConstant.html
16. ^
17. ^
18. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
19. ^
20. ^
21. ^
22. ^
23. ^