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|00000000| 5c 61 70 70 65 6e 64 69 | 78 0a 25 5c 73 65 63 74 |\appendi|x.%\sect|
|00000010| 69 6f 6e 7b 45 73 74 69 | 6d 61 74 69 6e 67 20 50 |ion{Esti|mating P|
|00000020| 79 74 68 61 67 6f 72 65 | 61 6e 20 53 71 75 61 72 |ythagore|an Squar|
|00000030| 65 2d 72 6f 6f 74 7d 0a | 5c 6d 65 64 73 6b 69 70 |e-root}.|\medskip|
|00000040| 5c 6e 6f 69 6e 64 65 6e | 74 0a 7b 5c 4c 61 72 67 |\noinden|t.{\Larg|
|00000050| 65 5c 62 66 20 41 70 70 | 65 6e 64 69 78 20 41 5c |e\bf App|endix A\|
|00000060| 20 5c 20 5c 20 45 73 74 | 69 6d 61 74 69 6e 67 20 | \ \ Est|imating |
|00000070| 50 79 74 68 61 67 6f 72 | 65 61 6e 20 53 71 75 61 |Pythagor|ean Squa|
|00000080| 72 65 2d 72 6f 6f 74 7d | 0a 0a 5c 6d 65 64 73 6b |re-root}|..\medsk|
|00000090| 69 70 5c 6e 6f 69 6e 64 | 65 6e 74 0a 46 6f 72 20 |ip\noind|ent.For |
|000000a0| 74 68 65 20 6c 69 6e 65 | 20 64 72 61 77 69 6e 67 |the line| drawing|
|000000b0| 20 63 6f 6d 6d 61 6e 64 | 73 20 64 65 73 63 72 69 | command|s descri|
|000000c0| 62 65 64 20 69 6e 20 74 | 68 65 20 6d 61 69 6e 20 |bed in t|he main |
|000000d0| 73 65 63 74 69 6f 6e 73 | 20 6f 66 20 74 68 69 73 |sections| of this|
|000000e0| 0a 64 6f 63 75 6d 65 6e | 74 2c 20 77 65 20 6e 65 |.documen|t, we ne|
|000000f0| 65 64 20 74 6f 20 65 73 | 74 69 6d 61 74 65 20 74 |ed to es|timate t|
|00000100| 68 65 20 50 79 74 68 61 | 67 6f 72 65 61 6e 20 73 |he Pytha|gorean s|
|00000110| 71 75 61 72 65 2d 72 6f | 6f 74 20 69 6e 20 6f 72 |quare-ro|ot in or|
|00000120| 64 65 72 20 74 6f 0a 64 | 65 74 65 72 6d 69 6e 65 |der to.d|etermine|
|00000130| 20 74 68 65 20 6c 65 6e | 67 74 68 20 6f 66 20 74 | the len|gth of t|
|00000140| 68 65 20 6c 69 6e 65 20 | 28 61 6c 6f 6e 67 20 69 |he line |(along i|
|00000150| 74 73 20 73 6c 6f 70 65 | 29 2e 20 4d 6f 72 65 20 |ts slope|). More |
|00000160| 70 72 65 63 69 73 65 6c | 79 2c 20 77 65 20 6e 65 |precisel|y, we ne|
|00000170| 65 64 0a 74 6f 20 65 73 | 74 69 6d 61 74 65 20 74 |ed.to es|timate t|
|00000180| 68 65 20 6e 75 6d 62 65 | 72 20 6f 66 20 73 65 67 |he numbe|r of seg|
|00000190| 6d 65 6e 74 73 20 6f 66 | 20 61 20 67 69 76 65 6e |ments of| a given|
|000001a0| 20 6c 65 6e 67 74 68 20 | 6e 65 65 64 65 64 20 74 | length |needed t|
|000001b0| 6f 20 64 72 61 77 20 61 | 20 6c 69 6e 65 2e 0a 5c |o draw a| line..\|
|000001c0| 54 65 58 5c 20 64 6f 65 | 73 20 6e 6f 74 20 70 72 |TeX\ doe|s not pr|
|000001d0| 6f 76 69 64 65 20 66 6f | 72 20 66 6c 6f 61 74 69 |ovide fo|r floati|
|000001e0| 6e 67 20 70 6f 69 6e 74 | 20 63 61 6c 63 75 6c 61 |ng point| calcula|
|000001f0| 74 69 6f 6e 73 2c 20 61 | 6e 64 20 74 68 75 73 20 |tions, a|nd thus |
|00000200| 74 68 65 72 65 20 61 72 | 65 20 6e 6f 0a 64 69 72 |there ar|e no.dir|
|00000210| 65 63 74 20 6d 65 61 6e | 73 20 6f 66 20 63 61 6c |ect mean|s of cal|
|00000220| 63 75 6c 61 74 69 6e 67 | 20 74 68 65 20 61 62 6f |culating| the abo|
|00000230| 76 65 20 73 71 75 61 72 | 65 2d 72 6f 6f 74 2e 20 |ve squar|e-root. |
|00000240| 4d 6f 73 74 20 73 74 61 | 6e 64 61 72 64 20 6e 75 |Most sta|ndard nu|
|00000250| 6d 65 72 69 63 61 6c 0a | 74 65 63 68 6e 69 71 75 |merical.|techniqu|
|00000260| 65 73 20 61 72 65 20 69 | 74 65 72 61 74 69 76 65 |es are i|terative|
|00000270| 20 61 6e 64 20 77 6f 75 | 6c 64 20 62 65 20 74 6f | and wou|ld be to|
|00000280| 6f 20 73 6c 6f 77 20 77 | 68 65 6e 20 75 73 65 64 |o slow w|hen used|
|00000290| 20 77 69 74 68 20 5c 54 | 65 58 5c 20 66 6f 72 20 | with \T|eX\ for |
|000002a0| 6c 61 63 6b 0a 6f 66 20 | 66 6c 6f 61 74 69 6e 67 |lack.of |floating|
|000002b0| 20 70 6f 69 6e 74 20 63 | 61 6c 63 75 6c 61 74 69 | point c|alculati|
|000002c0| 6f 6e 73 2c 20 61 6e 64 | 20 69 6e 20 70 61 72 74 |ons, and| in part|
|000002d0| 69 63 75 6c 61 72 2c 20 | 72 65 61 6c 20 64 69 76 |icular, |real div|
|000002e0| 69 73 69 6f 6e 2c 20 73 | 69 6e 63 65 0a 63 61 6c |ision, s|ince.cal|
|000002f0| 63 75 6c 61 74 69 6f 6e | 20 6f 66 20 73 75 63 68 |culation| of such|
|00000300| 20 61 20 73 71 75 61 72 | 65 2d 72 6f 6f 74 20 69 | a squar|e-root i|
|00000310| 73 20 6e 65 65 64 65 64 | 20 76 65 72 79 20 66 72 |s needed| very fr|
|00000320| 65 71 75 65 6e 74 6c 79 | 2e 0a 0a 41 20 73 69 6d |equently|...A sim|
|00000330| 70 6c 65 20 6e 6f 6e 2d | 69 74 65 72 61 74 69 76 |ple non-|iterativ|
|00000340| 65 20 66 6f 72 6d 75 6c | 61 20 66 6f 72 20 65 73 |e formul|a for es|
|00000350| 74 69 6d 61 74 69 6e 67 | 20 74 68 65 20 73 71 75 |timating| the squ|
|00000360| 61 72 65 2d 72 6f 6f 74 | 20 69 73 20 64 65 72 69 |are-root| is deri|
|00000370| 76 65 64 20 61 6e 64 0a | 64 65 73 63 72 69 62 65 |ved and.|describe|
|00000380| 64 20 62 65 6c 6f 77 2e | 0a 0a 5c 62 69 67 73 6b |d below.|..\bigsk|
|00000390| 69 70 5c 6e 6f 69 6e 64 | 65 6e 74 0a 7b 5c 62 66 |ip\noind|ent.{\bf|
|000003a0| 20 50 72 6f 62 6c 65 6d | 3a 20 7d 20 47 69 76 65 | Problem|: } Give|
|000003b0| 6e 20 24 61 24 20 61 6e | 64 20 24 62 24 2c 20 74 |n $a$ an|d $b$, t|
|000003c0| 6f 20 66 69 6e 64 20 24 | 63 24 20 3d 20 24 5c 73 |o find $|c$ = $\s|
|000003d0| 71 72 74 7b 61 5e 32 20 | 2b 20 62 5e 32 7d 24 20 |qrt{a^2 |+ b^2}$ |
|000003e0| 75 73 69 6e 67 0a 6f 6e | 6c 79 20 6f 70 65 72 61 |using.on|ly opera|
|000003f0| 74 69 6f 6e 73 20 69 6e | 20 5c 7b 24 2b 2c 2d 2c |tions in| \{$+,-,|
|00000400| 2a 2c 2f 24 5c 7d 2e 0a | 0a 57 65 20 63 61 6e 20 |*,/$\}..|.We can |
|00000410| 67 65 74 20 76 65 72 79 | 20 74 69 67 68 74 20 62 |get very| tight b|
|00000420| 6f 75 6e 64 73 20 6f 6e | 20 74 68 65 20 73 71 75 |ounds on| the squ|
|00000430| 61 72 65 2d 72 6f 6f 74 | 20 61 73 20 66 6f 6c 6c |are-root| as foll|
|00000440| 6f 77 73 2e 0a 57 69 74 | 68 6f 75 74 20 6c 6f 73 |ows..Wit|hout los|
|00000450| 73 20 6f 66 20 67 65 6e | 65 72 61 6c 69 74 79 2c |s of gen|erality,|
|00000460| 20 6c 65 74 20 24 61 20 | 5c 67 65 20 62 24 2e 20 | let $a |\ge b$. |
|00000470| 57 65 20 73 65 65 6b 20 | 61 20 73 69 6d 70 6c 65 |We seek |a simple|
|00000480| 20 24 6e 24 0a 73 75 63 | 68 20 74 68 61 74 3a 0a | $n$.suc|h that:.|
|00000490| 5c 5b 5c 73 71 72 74 7b | 61 5e 32 20 2b 20 62 5e |\[\sqrt{|a^2 + b^|
|000004a0| 32 7d 20 5c 67 65 20 61 | 20 2b 20 5c 66 72 61 63 |2} \ge a| + \frac|
|000004b0| 7b 62 7d 7b 6e 7d 5c 5d | 0a 0a 53 71 75 61 72 69 |{b}{n}\]|..Squari|
|000004c0| 6e 67 20 62 6f 74 68 20 | 73 69 64 65 73 2c 20 77 |ng both |sides, w|
|000004d0| 65 20 68 61 76 65 0a 5c | 5b 5c 62 65 67 69 6e 7b |e have.\|[\begin{|
|000004e0| 61 72 72 61 79 7d 7b 6c | 72 63 6c 7d 0a 5c 4c 65 |array}{l|rcl}.\Le|
|000004f0| 66 74 72 69 67 68 74 61 | 72 72 6f 77 20 26 20 61 |ftrighta|rrow & a|
|00000500| 5e 32 2b 62 5e 32 20 26 | 5c 67 65 26 20 61 5e 32 |^2+b^2 &|\ge& a^2|
|00000510| 20 2b 20 5c 64 69 73 70 | 6c 61 79 73 74 79 6c 65 | + \disp|laystyle|
|00000520| 5c 66 72 61 63 7b 62 5e | 32 7d 7b 6e 5e 32 7d 20 |\frac{b^|2}{n^2} |
|00000530| 2b 0a 5c 64 69 73 70 6c | 61 79 73 74 79 6c 65 5c |+.\displ|aystyle\|
|00000540| 66 72 61 63 7b 32 61 62 | 7d 7b 6e 7d 20 5c 5c 5b |frac{2ab|}{n} \\[|
|00000550| 32 6d 6d 5d 0a 5c 4c 65 | 66 74 72 69 67 68 74 61 |2mm].\Le|ftrighta|
|00000560| 72 72 6f 77 20 26 20 28 | 31 20 2d 20 5c 64 69 73 |rrow & (|1 - \dis|
|00000570| 70 6c 61 79 73 74 79 6c | 65 5c 66 72 61 63 7b 31 |playstyl|e\frac{1|
|00000580| 7d 7b 6e 5e 32 7d 29 20 | 62 5e 32 20 26 5c 67 65 |}{n^2}) |b^2 &\ge|
|00000590| 26 0a 5c 64 69 73 70 6c | 61 79 73 74 79 6c 65 5c |&.\displ|aystyle\|
|000005a0| 66 72 61 63 7b 32 61 62 | 7d 7b 6e 7d 5c 5c 5b 32 |frac{2ab|}{n}\\[2|
|000005b0| 6d 6d 5d 0a 5c 4c 65 66 | 74 72 69 67 68 74 61 72 |mm].\Lef|trightar|
|000005c0| 72 6f 77 20 26 20 5c 64 | 69 73 70 6c 61 79 73 74 |row & \d|isplayst|
|000005d0| 79 6c 65 5c 66 72 61 63 | 7b 62 7d 7b 61 7d 20 26 |yle\frac|{b}{a} &|
|000005e0| 5c 67 65 26 20 5c 64 69 | 73 70 6c 61 79 73 74 79 |\ge& \di|splaysty|
|000005f0| 6c 65 5c 66 72 61 63 7b | 32 6e 7d 7b 28 6e 5e 32 |le\frac{|2n}{(n^2|
|00000600| 0a 2d 31 29 7d 5c 5c 5b | 32 6d 6d 5d 0a 5c 6d 62 |.-1)}\\[|2mm].\mb|
|00000610| 6f 78 7b 6f 72 20 7d 20 | 20 20 20 20 20 26 20 28 |ox{or } |     & (|
|00000620| 5c 64 69 73 70 6c 61 79 | 73 74 79 6c 65 5c 66 72 |\display|style\fr|
|00000630| 61 63 7b 62 7d 7b 61 7d | 29 6e 5e 32 20 2d 20 32 |ac{b}{a}|)n^2 - 2|
|00000640| 6e 20 2d 28 5c 64 69 73 | 70 6c 61 79 73 74 79 6c |n -(\dis|playstyl|
|00000650| 65 5c 66 72 61 63 7b 62 | 7d 7b 61 7d 29 20 26 5c |e\frac{b|}{a}) &\|
|00000660| 67 65 26 20 30 0a 5c 65 | 6e 64 7b 61 72 72 61 79 |ge& 0.\e|nd{array|
|00000670| 7d 5c 68 66 69 6c 6c 5c | 5d 0a 0a 3e 46 72 6f 6d |}\hfill\|]..>From|
|00000680| 20 74 68 65 20 71 75 61 | 64 72 61 74 69 63 20 65 | the qua|dratic e|
|00000690| 71 75 61 74 69 6f 6e 20 | 61 62 6f 76 65 2c 20 77 |quation |above, w|
|000006a0| 65 20 66 69 6e 61 6c 6c | 79 20 67 65 74 20 61 6e |e finall|y get an|
|000006b0| 20 65 78 70 72 65 73 73 | 69 6f 6e 20 66 6f 72 20 | express|ion for |
|000006c0| 24 6e 24 2c 0a 5c 5b 20 | 6e 20 5c 3b 3d 5c 3b 20 |$n$,.\[ |n \;=\; |
|000006d0| 5c 66 72 61 63 7b 32 20 | 5c 70 6d 20 5c 73 71 72 |\frac{2 |\pm \sqr|
|000006e0| 74 7b 34 20 2b 20 34 28 | 5c 66 72 61 63 7b 62 7d |t{4 + 4(|\frac{b}|
|000006f0| 7b 61 7d 29 5e 32 7d 7d | 7b 5c 66 72 61 63 7b 32 |{a})^2}}|{\frac{2|
|00000700| 62 7d 7b 61 7d 7d 0a 20 | 20 20 20 20 20 5c 3b 3d |b}{a}}. |     \;=|
|00000710| 5c 3b 20 5c 66 72 61 63 | 7b 31 20 5c 70 6d 20 5c |\; \frac|{1 \pm \|
|00000720| 73 71 72 74 7b 31 20 2b | 20 28 5c 66 72 61 63 7b |sqrt{1 +| (\frac{|
|00000730| 62 7d 7b 61 7d 29 5e 32 | 7d 7d 7b 5c 66 72 61 63 |b}{a})^2|}}{\frac|
|00000740| 7b 62 7d 7b 61 7d 7d 20 | 5c 5d 0a 0a 4f 6e 6c 79 |{b}{a}} |\]..Only|
|00000750| 20 74 68 65 20 24 2b 24 | 76 65 20 72 6f 6f 74 20 | the $+$|ve root |
|00000760| 69 6e 74 65 72 65 73 74 | 73 20 75 73 20 73 69 6e |interest|s us sin|
|00000770| 63 65 20 24 6e 24 20 68 | 61 73 20 74 6f 20 62 65 |ce $n$ h|as to be|
|00000780| 20 70 6f 73 69 74 69 76 | 65 2e 0a 4e 6f 74 65 20 | positiv|e..Note |
|00000790| 74 68 61 74 20 74 68 65 | 20 74 65 72 6d 20 75 6e |that the| term un|
|000007a0| 64 65 72 20 74 68 65 20 | 72 6f 6f 74 20 69 73 20 |der the |root is |
|000007b0| 20 62 6f 75 6e 64 65 64 | 20 61 62 6f 76 65 20 61 | bounded| above a|
|000007c0| 6e 64 20 62 65 6c 6f 77 | 20 28 73 69 6e 63 65 0a |nd below| (since.|
|000007d0| 24 5c 66 72 61 63 7b 62 | 7d 7b 61 7d 20 5c 6c 65 |$\frac{b|}{a} \le|
|000007e0| 20 31 24 29 3a 0a 5c 5b | 31 20 5c 3b 5c 6c 65 5c | 1$):.\[|1 \;\le\|
|000007f0| 3b 20 5c 73 71 72 74 7b | 31 20 2b 20 28 5c 66 72 |; \sqrt{|1 + (\fr|
|00000800| 61 63 7b 62 7d 7b 61 7d | 29 5e 32 7d 20 5c 3b 5c |ac{b}{a}|)^2} \;\|
|00000810| 6c 65 5c 3b 20 5c 73 71 | 72 74 7b 32 7d 5c 5d 0a |le\; \sq|rt{2}\].|
|00000820| 0a 48 65 6e 63 65 2c 20 | 77 65 20 68 61 76 65 20 |.Hence, |we have |
|00000830| 74 77 6f 20 76 61 6c 75 | 65 73 20 66 6f 72 20 24 |two valu|es for $|
|00000840| 6e 24 2c 0a 5c 5b 20 6e | 5f 6c 5c 3b 3d 5c 3b 20 |n$,.\[ n|_l\;=\; |
|00000850| 5c 66 72 61 63 7b 31 2b | 31 7d 7b 5c 66 72 61 63 |\frac{1+|1}{\frac|
|00000860| 7b 62 7d 7b 61 7d 7d 20 | 5c 3b 3d 5c 3b 20 5c 66 |{b}{a}} |\;=\; \f|
|00000870| 72 61 63 7b 32 61 7d 7b | 62 7d 3b 5c 3b 5c 3b 5c |rac{2a}{|b};\;\;\|
|00000880| 3b 5c 3b 5c 3b 5c 3b 5c | 3b 5c 3b 5c 3b 5c 3b 0a |;\;\;\;\|;\;\;\;.|
|00000890| 6e 5f 75 20 5c 3b 3d 5c | 3b 20 5c 66 72 61 63 7b |n_u \;=\|; \frac{|
|000008a0| 31 2b 20 5c 73 71 72 74 | 7b 32 7d 7d 7b 5c 66 72 |1+ \sqrt|{2}}{\fr|
|000008b0| 61 63 7b 62 7d 7b 61 7d | 7d 20 5c 3b 3d 5c 3b 20 |ac{b}{a}|} \;=\; |
|000008c0| 5c 66 72 61 63 7b 28 31 | 2b 20 5c 73 71 72 74 7b |\frac{(1|+ \sqrt{|
|000008d0| 32 7d 29 61 7d 7b 62 7d | 0a 5c 5d 0a 25 0a 77 68 |2})a}{b}|.\].%.wh|
|000008e0| 69 63 68 20 66 69 6e 61 | 6c 6c 79 20 67 69 76 65 |ich fina|lly give|
|000008f0| 73 20 75 73 20 61 20 6c | 6f 77 65 72 20 61 6e 64 |s us a l|ower and|
|00000900| 20 61 6e 20 75 70 70 65 | 72 20 62 6f 75 6e 64 20 | an uppe|r bound |
|00000910| 66 6f 72 20 24 63 24 2c | 20 74 68 65 20 50 79 74 |for $c$,| the Pyt|
|00000920| 68 61 67 6f 72 65 61 6e | 0a 73 71 75 61 72 65 2d |hagorean|.square-|
|00000930| 72 6f 6f 74 2c 0a 5c 5b | 20 61 20 2b 20 5c 66 72 |root,.\[| a + \fr|
|00000940| 61 63 7b 62 5e 32 7d 7b | 28 31 2b 20 5c 73 71 72 |ac{b^2}{|(1+ \sqr|
|00000950| 74 7b 32 7d 29 61 7d 20 | 5c 3b 5c 6c 65 5c 3b 20 |t{2})a} |\;\le\; |
|00000960| 63 20 5c 3b 5c 6c 65 5c | 3b 20 61 20 2b 20 5c 66 |c \;\le\|; a + \f|
|00000970| 72 61 63 7b 62 5e 32 7d | 7b 32 61 7d 5c 5d 0a 0a |rac{b^2}|{2a}\]..|
|00000980| 54 68 65 73 65 20 61 72 | 65 20 76 65 72 79 20 74 |These ar|e very t|
|00000990| 69 67 68 74 20 62 6f 75 | 6e 64 73 2e 20 44 65 6e |ight bou|nds. Den|
|000009a0| 6f 74 69 6e 67 20 74 68 | 65 20 6c 6f 77 65 72 20 |oting th|e lower |
|000009b0| 62 6f 75 6e 64 20 61 73 | 20 24 63 5f 6c 24 20 61 |bound as| $c_l$ a|
|000009c0| 6e 64 20 75 70 70 65 72 | 0a 6f 6e 65 20 24 63 5f |nd upper|.one $c_|
|000009d0| 75 24 2c 20 62 65 6c 6f | 77 20 61 72 65 20 73 6f |u$, belo|w are so|
|000009e0| 6d 65 20 6e 75 6d 65 72 | 69 63 61 6c 20 72 65 73 |me numer|ical res|
|000009f0| 75 6c 74 73 20 28 24 63 | 24 20 3d 20 65 78 61 63 |ults ($c|$ = exac|
|00000a00| 74 20 73 71 75 61 72 65 | 2d 72 6f 6f 74 29 3a 0a |t square|-root):.|
|00000a10| 0a 5c 62 65 67 69 6e 7b | 63 65 6e 74 65 72 7d 0a |.\begin{|center}.|
|00000a20| 5c 62 65 67 69 6e 7b 74 | 61 62 75 6c 61 72 7d 7b |\begin{t|abular}{|
|00000a30| 7c 63 7c 63 7c 63 7c 63 | 7c 63 7c 7d 0a 5c 68 6c ||c|c|c|c||c|}.\hl|
|00000a40| 69 6e 65 0a 61 20 26 20 | 62 20 26 20 63 20 26 20 |ine.a & |b & c & |
|00000a50| 24 63 5f 6c 24 20 26 20 | 24 63 5f 75 24 5c 5c 0a |$c_l$ & |$c_u$\\.|
|00000a60| 5c 68 6c 69 6e 65 0a 31 | 30 30 2e 30 20 26 20 31 |\hline.1|00.0 & 1|
|00000a70| 30 30 2e 30 20 26 20 31 | 34 31 2e 34 32 31 33 20 |00.0 & 1|41.4213 |
|00000a80| 26 20 31 34 31 2e 34 32 | 31 33 20 26 20 31 35 30 |& 141.42|13 & 150|
|00000a90| 2e 30 5c 20 5c 20 5c 20 | 5c 20 5c 20 5c 20 5c 5c |.0\ \ \ |\ \ \ \\|
|00000aa0| 0a 31 30 30 2e 30 20 26 | 20 5c 20 5c 2c 38 30 2e |.100.0 &| \ \,80.|
|00000ab0| 30 20 20 26 20 31 32 38 | 2e 30 36 34 32 20 26 20 |0  & 128|.0642 & |
|00000ac0| 31 32 36 2e 35 30 39 36 | 20 26 20 31 33 32 2e 30 |126.5096| & 132.0|
|00000ad0| 5c 20 5c 20 5c 20 5c 20 | 5c 20 5c 20 20 5c 5c 0a |\ \ \ \ |\ \  \\.|
|00000ae0| 5c 20 5c 2c 33 30 2e 30 | 20 20 26 20 5c 20 5c 2c |\ \,30.0|  & \ \,|
|00000af0| 32 30 2e 30 20 20 26 20 | 5c 20 5c 2c 33 36 2e 30 |20.0  & |\ \,36.0|
|00000b00| 35 35 35 20 20 26 20 5c | 20 33 35 2e 35 32 32 38 |555  & \| 35.5228|
|00000b10| 20 20 26 20 5c 20 33 36 | 2e 36 36 36 37 20 5c 5c |  & \ 36|.6667 \\|
|00000b20| 0a 5c 68 6c 69 6e 65 0a | 5c 65 6e 64 7b 74 61 62 |.\hline.|\end{tab|
|00000b30| 75 6c 61 72 7d 0a 5c 65 | 6e 64 7b 63 65 6e 74 65 |ular}.\e|nd{cente|
|00000b40| 72 7d 0a 0a 57 69 74 68 | 20 74 68 65 20 61 62 6f |r}..With| the abo|
|00000b50| 76 65 20 62 6f 75 6e 64 | 73 2c 20 6f 6e 65 20 63 |ve bound|s, one c|
|00000b60| 61 6e 20 64 6f 20 61 20 | 6c 69 6e 65 61 72 20 69 |an do a |linear i|
|00000b70| 6e 74 65 72 70 6f 6c 61 | 74 69 6f 6e 20 74 6f 20 |nterpola|tion to |
|00000b80| 67 65 74 20 65 78 61 63 | 74 20 76 61 6c 75 65 73 |get exac|t values|
|00000b90| 2e 0a 49 6e 20 6f 75 72 | 20 63 61 73 65 2c 20 73 |..In our| case, s|
|00000ba0| 69 6e 63 65 20 69 74 20 | 69 73 20 6e 6f 74 20 72 |ince it |is not r|
|00000bb0| 65 71 75 69 72 65 64 20 | 74 6f 20 62 65 20 7b 5c |equired |to be {\|
|00000bc0| 69 74 20 65 78 74 72 65 | 6d 65 6c 79 5c 2f 7d 20 |it extre|mely\/} |
|00000bd0| 61 63 63 75 72 61 74 65 | 2c 20 66 6f 72 20 0a 65 |accurate|, for .e|
|00000be0| 73 74 69 6d 61 74 69 6e | 67 20 74 68 65 20 73 71 |stimatin|g the sq|
|00000bf0| 75 61 72 65 2d 72 6f 6f | 74 20 69 6e 20 74 68 65 |uare-roo|t in the|
|00000c00| 20 6c 69 6e 65 20 64 72 | 61 77 69 6e 67 20 63 6f | line dr|awing co|
|00000c10| 6d 6d 61 6e 64 73 2c 0a | 77 65 20 73 69 6d 70 6c |mmands,.|we simpl|
|00000c20| 79 20 74 61 6b 65 20 74 | 68 65 20 6d 69 64 70 6f |y take t|he midpo|
|00000c30| 69 6e 74 20 6f 66 20 74 | 68 65 20 74 77 6f 20 62 |int of t|he two b|
|00000c40| 6f 75 6e 64 73 2e 20 46 | 6f 72 20 73 6d 61 6c 6c |ounds. F|or small|
|00000c50| 0a 6e 75 6d 62 65 72 73 | 2c 20 77 68 69 63 68 20 |.numbers|, which |
|00000c60| 69 73 20 65 78 70 65 63 | 74 65 64 20 74 6f 20 62 |is expec|ted to b|
|00000c70| 65 20 74 68 65 20 63 61 | 73 65 20 6d 6f 73 74 20 |e the ca|se most |
|00000c80| 6f 66 20 74 68 65 20 74 | 69 6d 65 2c 0a 74 68 65 |of the t|ime,.the|
|00000c90| 20 65 72 72 6f 72 20 69 | 73 20 76 65 72 79 20 73 | error i|s very s|
|00000ca0| 6d 61 6c 6c 2e 0a 0a 57 | 69 74 68 20 73 6f 6d 65 |mall...W|ith some|
|00000cb0| 20 61 6c 67 65 62 72 61 | 2c 20 77 65 20 67 65 74 | algebra|, we get|
|00000cc0| 20 74 68 65 20 6d 69 64 | 2d 70 6f 69 6e 74 20 65 | the mid|-point e|
|00000cd0| 73 74 69 6d 61 74 65 20 | 6f 66 20 24 63 24 2c 0a |stimate |of $c$,.|
|00000ce0| 5c 5b 63 20 3d 20 5c 66 | 72 61 63 7b 63 5f 6c 2b |\[c = \f|rac{c_l+|
|00000cf0| 63 5f 75 7d 7b 32 7d 20 | 3d 20 61 20 2b 20 5c 66 |c_u}{2} |= a + \f|
|00000d00| 72 61 63 7b 62 5e 32 20 | 2a 20 28 33 20 2b 20 5c |rac{b^2 |* (3 + \|
|00000d10| 73 71 72 74 7b 32 7d 29 | 7d 7b 61 2a 34 2a 28 31 |sqrt{2})|}{a*4*(1|
|00000d20| 20 2b 20 5c 73 71 72 74 | 7b 32 7d 29 7d 0a 3d 20 | + \sqrt|{2})}.= |
|00000d30| 61 20 2b 20 5c 66 72 61 | 63 7b 30 2e 34 35 37 5c |a + \fra|c{0.457\|
|00000d40| 3a 20 62 5e 32 7d 7b 61 | 7d 20 5c 3b 5c 3b 5c 3b |: b^2}{a|} \;\;\;|
|00000d50| 5c 3b 28 61 20 5c 67 65 | 20 62 29 20 5c 5d 0a 0a |\;(a \ge| b) \]..|
|00000d60| 54 68 65 20 6d 61 63 72 | 6f 20 5c 76 65 72 62 7c |The macr|o \verb||
|00000d70| 5c 73 71 72 74 61 6e 64 | 73 74 75 66 66 7c 20 75 |\sqrtand|stuff| u|
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|00000da0| 74 69 6e 67 20 74 68 65 | 0a 6e 75 6d 62 65 72 20 |ting the|.number |
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|00000dc0| 76 65 72 62 7c 5c 64 6f | 74 74 65 64 6c 69 6e 65 |verb|\do|ttedline|
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|00000df0| 28 66 6f 72 0a 5c 76 65 | 72 62 7c 5c 64 61 73 68 |(for.\ve|rb|\dash|
|00000e00| 6c 69 6e 65 7c 20 6d 61 | 63 72 6f 29 2e 20 54 68 |line| ma|cro). Th|
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|00000e20| 73 74 75 66 66 7c 20 6d | 61 63 72 6f 2c 20 69 6e |stuff| m|acro, in|
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|00000fd0| 5c 44 65 6c 74 61 20 79 | 20 3d 20 7c 79 5f 32 20 |\Delta y| = |y_2 |
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|00001050| 54 65 58 5c 20 61 72 65 | 20 69 6e 74 65 67 65 72 |TeX\ are| integer|
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|000010e0| 6c 74 61 20 78 7d 7b 64 | 7d 20 3d 20 24 20 6e 75 |lta x}{d|} = $ nu|
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|00001120| 43 61 76 65 61 74 3a 7d | 20 54 68 65 20 61 70 70 |Caveat:}| The app|
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|00001210| 76 65 72 69 6e 67 20 69 | 74 2e 0a                |vering i|t..     |
+--------+-------------------------+-------------------------+--------+--------+