4.5.1 FractionsHEADmain

2 files changed, 568 insertions, 5 deletions

diff --git a/vol2/4.5.1.lyx b/vol2/4.5.1.lyx
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+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input defs
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children no
+\language english
+\language_package default
+\inputencoding utf8
+\fontencoding auto
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+\cite_engine basic
+\cite_engine_type default
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+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
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+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
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+\quotes_style english
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+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+exerc5[10]
+\end_layout
+
+\end_inset
+
+Compute 
+\begin_inset Formula $(17/120)+(-27/70)$
+\end_inset
+
+ by the method recommended in the text.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+We have 
+\begin_inset Formula $d_{1}=\gcd\{120,70\}=10$
+\end_inset
+
+,
+ 
+\begin_inset Formula $t=17\cdot\frac{70}{10}-27\cdot\frac{120}{10}=119-324=-205$
+\end_inset
+
+,
+ 
+\begin_inset Formula $d_{2}=\gcd\{-205,10\}=5$
+\end_inset
+
+,
+ and so the answer is 
+\begin_inset Formula $-\frac{205}{5}\Big/\left(\frac{120}{10}\frac{70}{5}\right)=-\frac{41}{168}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+rexerc6[M23]
+\end_layout
+
+\end_inset
+
+Show that 
+\begin_inset Formula $u\bot u'$
+\end_inset
+
+ and 
+\begin_inset Formula $v\bot v'$
+\end_inset
+
+ implies 
+\begin_inset Formula $\gcd\{uv'+vu',u'v'\}=d_{1}d_{2}$
+\end_inset
+
+,
+ where 
+\begin_inset Formula $d_{1}=\gcd\{u',v'\}$
+\end_inset
+
+ and 
+\begin_inset Formula $d_{2}=\gcd\{d_{1},u(v'/d_{1})+v(u'/d_{1})\}$
+\end_inset
+
+.
+ (Hence if 
+\begin_inset Formula $d_{1}=1$
+\end_inset
+
+ we have 
+\begin_inset Formula $(uv'+vu')\bot u'v'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Greyedout
+status open
+
+\begin_layout Plain Layout
+(I had to look up the solution.) 
+\end_layout
+
+\end_inset
+
+Let 
+\begin_inset Formula $u''\coloneqq u'/d_{1}$
+\end_inset
+
+ and 
+\begin_inset Formula $v''\coloneqq v'/d_{1}$
+\end_inset
+
+,
+ we just need to prove that
+\begin_inset Formula 
+\[
+\gcd\{uv''+vu'',u''v''d_{1}\}=d_{2}=\gcd\{uv''+vu'',d_{1}\},
+\]
+
+\end_inset
+
+as multiplying the first equality by 
+\begin_inset Formula $d_{1}$
+\end_inset
+
+ gives us the required answer.
+ Obviously 
+\begin_inset Formula $d_{2}\mid uv''+vu'',u''v''d_{1}$
+\end_inset
+
+,
+ and we have to see that any integer 
+\begin_inset Formula $d$
+\end_inset
+
+ that divides both 
+\begin_inset Formula $uv''+vu''$
+\end_inset
+
+ and 
+\begin_inset Formula $u''v''d_{1}$
+\end_inset
+
+ also divides 
+\begin_inset Formula $d_{1}$
+\end_inset
+
+ and therefore 
+\begin_inset Formula $d_{2}$
+\end_inset
+
+.
+ Let 
+\begin_inset Formula $p$
+\end_inset
+
+ be a prime factor of 
+\begin_inset Formula $d$
+\end_inset
+
+,
+ because 
+\begin_inset Formula $u\bot u'$
+\end_inset
+
+ and therefore 
+\begin_inset Formula $u\bot d_{1},u''$
+\end_inset
+
+,
+ if 
+\begin_inset Formula $p\mid u''$
+\end_inset
+
+ then 
+\begin_inset Formula $p\mid uv''$
+\end_inset
+
+ but 
+\begin_inset Formula $p\nmid u$
+\end_inset
+
+,
+ so 
+\begin_inset Formula $p\mid v''$
+\end_inset
+
+ and 
+\begin_inset Formula $u''$
+\end_inset
+
+ and 
+\begin_inset Formula $v''$
+\end_inset
+
+ are not coprime,
+\begin_inset Formula $\#$
+\end_inset
+
+ and similarly 
+\begin_inset Formula $p\nmid v''$
+\end_inset
+
+.
+ That means that 
+\begin_inset Formula $d$
+\end_inset
+
+ doesn't have any common factors with either 
+\begin_inset Formula $u''$
+\end_inset
+
+ and 
+\begin_inset Formula $v''$
+\end_inset
+
+,
+ so 
+\begin_inset Formula $d\mid d_{1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+rexerc8[22]
+\end_layout
+
+\end_inset
+
+Discuss using 
+\begin_inset Formula $(1/0)$
+\end_inset
+
+ and 
+\begin_inset Formula $(-1/0)$
+\end_inset
+
+ as representations for 
+\begin_inset Formula $\infty$
+\end_inset
+
+ and 
+\begin_inset Formula $-\infty$
+\end_inset
+
+,
+ and/or as representations of overflow.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+answer 
+\end_layout
+
+\end_inset
+
+Mathematically they are not that different,
+ since 
+\begin_inset Formula $\infty$
+\end_inset
+
+,
+ when treated as a number,
+ represents the concept of a number larger than any other,
+ or an arbitrarily large number in an asymptotic way,
+ while overflow represents a number larger than any 
+\emph on
+representable
+\emph default
+ number.
+ Mediant rounding would round to 
+\begin_inset Formula $\pm1/0$
+\end_inset
+
+ for numbers with 
+\begin_inset Formula $|x|\geq2^{p}$
+\end_inset
+
+,
+ which makes sense.
+\end_layout
+
+\begin_layout Standard
+Using these representations,
+ multiplying a number by 
+\begin_inset Formula $\infty$
+\end_inset
+
+ gives 
+\begin_inset Formula $\infty$
+\end_inset
+
+ if the number is positive or 
+\begin_inset Formula $-\infty$
+\end_inset
+
+ if it's negative,
+ or vice versa for 
+\begin_inset Formula $-\infty$
+\end_inset
+
+,
+ and 
+\begin_inset Formula $\pm\infty\cdot0=(0/0)$
+\end_inset
+
+,
+ an indeterminate value.
+ Dividing 
+\begin_inset Formula $\pm\infty$
+\end_inset
+
+ by some number yields a similar result,
+ with 
+\begin_inset Formula $\pm\infty/\pm\infty$
+\end_inset
+
+ being indeterminate and 
+\begin_inset Formula $\pm\infty/0=\pm\infty$
+\end_inset
+
+,
+ which works well as a convention,
+ and dividing by 
+\begin_inset Formula $\pm\infty$
+\end_inset
+
+ is equivalent to multiplying by 0.
+\end_layout
+
+\begin_layout Standard
+Addition and subtraction require a bit more attention:
+ if we do 
+\begin_inset Formula $\pm\frac{1}{0}+\frac{a}{b}$
+\end_inset
+
+,
+ we would have 
+\begin_inset Formula $d_{1}=b$
+\end_inset
+
+ (if 
+\begin_inset Formula $b=1$
+\end_inset
+
+ then we shortcut to get 
+\begin_inset Formula $\pm\frac{1}{0}$
+\end_inset
+
+ as the result) and then 
+\begin_inset Formula $t=\pm1$
+\end_inset
+
+,
+ 
+\begin_inset Formula $d_{2}=1$
+\end_inset
+
+,
+ and the result is 
+\begin_inset Formula $\pm\frac{1}{0}$
+\end_inset
+
+,
+ as expected.
+ The exception is if 
+\begin_inset Formula $\frac{a}{b}=\pm\frac{1}{0}$
+\end_inset
+
+;
+ then 
+\begin_inset Formula $d_{1}=0$
+\end_inset
+
+ and calculating 
+\begin_inset Formula $t$
+\end_inset
+
+ would result in a division by 0,
+ and the obvious procedure gives us 
+\begin_inset Formula $\frac{0}{0}$
+\end_inset
+
+ even if both infinities have the same sign,
+ so we have to consider this a special case to get the correct result,
+ namely 
+\begin_inset Formula $\pm\infty\pm\infty=\pm\infty$
+\end_inset
+
+ and 
+\begin_inset Formula $\pm\infty\mp\infty=0/0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+We also need a special case so that additions and subtractions involving 
+\begin_inset Formula $0/0$
+\end_inset
+
+ return 
+\begin_inset Formula $0/0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Note,
+ however,
+ that having 
+\begin_inset Formula $x/(\pm1/0)=0$
+\end_inset
+
+ for 
+\begin_inset Formula $x\neq0$
+\end_inset
+
+ would be a hazard in the case that we are representing overflow,
+ as we may inadvertently discard the overflow and get a potentially very inaccurate result,
+ so sometimes it may be better to use 
+\begin_inset Formula $0/0$
+\end_inset
+
+ for overflow.
+\end_layout
+
+\end_body
+\end_document
diff --git a/vol2/index.lyx b/vol2/index.lyx
index 27530d8..bb4203a 100644
--- a/vol2/index.lyx
+++ b/vol2/index.lyx
@@ -1234,15 +1234,21 @@ Fractions
 \end_layout
 
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+filename "4.5.1.lyx"
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+\end_inset
+
+
 \begin_inset Note Note
 status open
 
 \begin_layout Plain Layout
-3+1;
- 5,
- 6,
- 8 (0:42) -> 2d,
- -2/3
+
+\family typewriter
+A10+R25
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