MOTION MOUNTAIN

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270 8 thought and language . . 0 1 –1 1 smaller 11 2 1 1 1/2 –1/2 1 1 –2 11 earlier 111 3 1 11 3/2 1 1 1 11 3/4 1 1/4 –1/4 –3/4 –3/2 –3 111 1111 4 –4 1111 ω=simplestinfinite 1=1111... ω π 111 1 8/3 1 1 2/3 –1/3 1=1111 ... 11111 ω+4 1 ω−4 1111 1 1 1111 ι 4ι 1 1 1 1 11 1 1 11 1 11 11 1 1 1 –4/5 11 1 1 11111 11... −√2 −ω F I G U R E 166 The surreal numbers in conventional and in bit notation. 1 1 2ω 1 1 1 1 1 ω/2 ω/4 1 1 1 1 2/3+2ι/3 1 1 1 1 1 √ω 1=1 1 1 ι=1/ω=simplestinfinitesimal 1 1 1 −ω/2 −2ω write them are given in Figure 166. The empty sequence is the number zero. A finite sequence ofnups is the integer numbern, andafinite sequenceofndownsis theinteger−n. Finite sequencesofmixed ups and downs give the dyadic rational numbers. Examples are 1, 2, 3,−7, 19/4,37/256, etc. They all have denominators with a power of 2.The otherrationalnumbers are those that end in an infinitely repeating string of ups and downs, such as thereals, theinfinitesimals and simple infinite numbers. Longer countably infinite series give even more crazy numbers. The complete class of numbers that is defined by a sequence of bits is called the class ofsurreal numbers.* There is a second way to write surreal numbers. The first is the just mentioned sequence of bits. But in order to define addition and multiplication, another notation is usually used, deduced from Figure 166. A surrealαis defined as the earliestnumber of * The surreal numbers do not form a set since they contain all ordinal numbers,which themselves do not form a set, even though they of course contain sets. In short, ordinals and surreals are classes which are largerthan sets. 1 1 √ι ι 2 1 −ω 2 1 ω 2 e ω −e ω 1 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net
thought and language 271 Challenge 269 s Ref. 250 Challenge 270 s Vol. IV, page 222 all thosebetweentwoseriesofearliersurreals,theleftandtherightseries: Forexample, wehave α={a,b,c,...|A,B,C,...} with a,b,c,
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ChristophSchiller MOTION MOUNTAIN t
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Editiovicesima octava. Proprietassc
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DieMenschenstärken,die Sachenklär
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8 preface PHYSICS: Describing motio
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10 preface Your donation to the cha
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12 contents Do we see what exists?
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Light, Charges and Brains Inourques
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16 1 electricity and fields F I G U
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18 1 electricityandfields F I G U R
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20 1 electricity and fields nylon r
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24 1 electricity and fields TA B L
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28 1 electricity and fields Challen
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30 1 electricity and fields If elec
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34 1 electricityandfields F I G U R
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42 1 electricity and fields Oersted
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44 1 electricity and fields Vol. IV
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48 1 electricity and fields Ref. 24
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50 1 electricity and fields Page 21
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52 1 electricity and fields Ref. 29
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Vol. IV, page 231 Chapter 2 THE DES
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76 2 the description of electromagn
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Chapter 3 WHAT IS LIGHT? Ref. 57 Pa
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96 3 what is light? Electric field
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98 3 what is light? spark transmitt
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100 3 what is light? Ref. 59 Page 1
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102 3 what is light? primary infrar
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104 3 what is light? Page 84 gases
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106 3 what is light? Fre - Waveleng
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108 3 what is light? Challenge 110
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110 3 what is light? incoming light
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112 3 what is light? Ref. 70 Challe
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114 3 what is light? F I G U R E 69
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116 3 what is light? light Ref. 72
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118 3 what is light? light Challeng
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120 3 what is light? momentumLis gi
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122 3 what is light? Ref. 80 Challe
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126 3 what is light? Ref. 87 Page 9
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130 3 what is light? v ph v So F I
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132 3 what is light? The forerunner
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134 3 what is light? vocabulary. On
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136 3 what is light? source beam sp
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138 3 what is light? B B Beam in A
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Chapter 4 IMAGES AND THE EYE - OPTI
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142 4 images and the eye - optics 2
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144 4 images and the eye - optics C
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146 4 images and the eye - optics R
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148 4 images and the eye - optics T
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168 4 imagesandthe eye-optics F I G
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Chapter 5 ELECTROMAGNETIC EFFECTS R
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206 5 electromagneticeffects graupe
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208 5 electromagnetic effects Ref.
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214 5 electromagnetic effects F I G
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216 5 electromagnetic effects TA B
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218 5 electromagnetic effects TA B
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Page 236 and 237: 236 7 the story of the brain F I G
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Chapter 10 CLASSICAL PHYSICS IN A N
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322 10 classical physics in a nutsh
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328 a units, measurements and const
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342 challenge hints and solutions n
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360 challenge hints and solutions V
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362 bibliography 10 Thiseffect hasf
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364 bibliography 37 A discussionof
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366 bibliography M.B. van der Mark
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368 bibliography Vol. I, page 96 19
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370 bibliography S.W. McCuskey, F.C
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372 bibliography 119 A complete lis
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374 bibliography F I G U R E 178 Th
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376 bibliography in unserer Zeit33,
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378 bibliography private communicat
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382 bibliography Vol. IV, page 135
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384 bibliography and 304. 260 For a
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386 bibliography Springer, 1996. wh
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CREDITS Acknowledgements Many peopl
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390 credits tute of Molecular Patho
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NAME INDEX A Abbott A Abbott,T.A.37
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396 name index D Democritus Democri
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398 name index H Holmes Holmes, C.D
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400 name index M Mohr Mohr, P.J. 38
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402 name index S Sheldon Sheldon, E
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404 name index Z Zybin Motion Mount
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406 subject index A units 328 Aplys
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408 subject index C classical class
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410 subject index E Earth Earth age
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412 subject index F finite finite 2
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414 subject index I infrared photog
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416 subject index L Lorentz Lorentz
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418 subject index M motion motion a
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422 subject index S Scarabeus Scara
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424 subject index T table d’Unit
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426 subject index W Wien’s Wien
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