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@ -31,7 +31,9 @@ Rewrite the following binary decimals in base 10: \
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#definition(label: "floatbits")
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#definition(label: "floatbits")
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Another way we can interpret a bit string is as a _signed floating-point decimal_, or a `float` for short. \
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Another way we can interpret a bit string is as a _signed floating-point decimal_, or a `float` for short. \
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Floats represent a subset of the real numbers, and are interpreted as follows: \
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Floats represent a subset of the real numbers, and are interpreted as follows: \
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#note([The following only applies to floats that consist of 32 bits. We won't encounter any others today.])
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#note(
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[The following only applies to floats that consist of 32 bits. We won't encounter any others today.],
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)
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#align(center, box(inset: 2mm, cetz.canvas({
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#align(center, box(inset: 2mm, cetz.canvas({
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import cetz.draw: *
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import cetz.draw: *
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@ -134,14 +134,7 @@ Fill the following tropical addition and multiplication tables
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table(
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table(
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columns: (col, col, col, col, col, col),
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columns: (col, col, col, col, col, col),
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align: center,
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align: center,
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table.header(
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table.header([$#tp$], [$1$], [$2$], [$3$], [$4$], [$#sym.infinity$]),
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[$#tp$],
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[$1$],
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[$2$],
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[$3$],
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[$4$],
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[$#sym.infinity$],
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),
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box(inset: 3pt, $1$), [], [], [], [], [],
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box(inset: 3pt, $1$), [], [], [], [], [],
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box(inset: 3pt, $2$), [], [], [], [], [],
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box(inset: 3pt, $2$), [], [], [], [], [],
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@ -152,14 +145,7 @@ Fill the following tropical addition and multiplication tables
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table(
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table(
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columns: (col, col, col, col, col, col),
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columns: (col, col, col, col, col, col),
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align: center,
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align: center,
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table.header(
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table.header([$#tm$], [$0$], [$1$], [$2$], [$3$], [$4$]),
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[$#tm$],
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[$0$],
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[$1$],
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[$2$],
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[$3$],
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[$4$],
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),
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box(inset: 3pt, $0$), [], [], [], [], [],
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box(inset: 3pt, $0$), [], [], [], [], [],
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box(inset: 3pt, $1$), [], [], [], [], [],
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box(inset: 3pt, $1$), [], [], [], [], [],
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@ -178,14 +164,7 @@ Fill the following tropical addition and multiplication tables
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table(
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table(
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columns: (col, col, col, col, col, col),
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columns: (col, col, col, col, col, col),
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align: center,
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align: center,
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table.header(
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table.header([$#tp$], [$1$], [$2$], [$3$], [$4$], [$#sym.infinity$]),
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[$#tp$],
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[$1$],
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[$2$],
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[$3$],
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[$4$],
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[$#sym.infinity$],
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),
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box(inset: 3pt, $1$),
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box(inset: 3pt, $1$),
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box(inset: 3pt, $1$),
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box(inset: 3pt, $1$),
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@ -225,14 +204,7 @@ Fill the following tropical addition and multiplication tables
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table(
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table(
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columns: (col, col, col, col, col, col),
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columns: (col, col, col, col, col, col),
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align: center,
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align: center,
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table.header(
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table.header([$#tm$], [$0$], [$1$], [$2$], [$3$], [$4$]),
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[$#tm$],
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[$0$],
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[$1$],
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[$2$],
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[$3$],
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[$4$],
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),
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box(inset: 3pt, $0$),
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box(inset: 3pt, $0$),
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box(inset: 3pt, $0$),
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box(inset: 3pt, $0$),
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@ -281,8 +253,7 @@ Adjacent parenthesis imply tropical multiplication
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#solution([
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#solution([
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$
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$
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(x #tp 2)(x #tp 3)
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(x #tp 2)(x #tp 3) & = x^2 #tp 2x #tp 3x #tp (2 #tm 3) \
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&= x^2 #tp 2x #tp 3x #tp (2 #tm 3) \
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& = x^2 #tp (2 #tp 3)x #tp (2 #tm 3) \
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& = x^2 #tp (2 #tp 3)x #tp (2 #tm 3) \
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& = x^2 #tp 2x #tp 5
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& = x^2 #tp 2x #tp 5
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$
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$
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@ -12,7 +12,9 @@ There are four classes of Euclidean isometries:
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- reflections
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- reflections
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- rotations
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- rotations
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- glide reflections
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- glide reflections
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#note([We can prove there are no others, but this is beyond the scope of this handout.]) \
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#note(
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[We can prove there are no others, but this is beyond the scope of this handout.],
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) \
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A simple example of each isometry is shown below:
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A simple example of each isometry is shown below:
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#let demo(c) = {
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#let demo(c) = {
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@ -17,7 +17,9 @@ Maximize the product $a_1 #sym.times a_2 #sym.times ... #sym.times a_k$
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Of course, all $a_i$ should be greater than $1$. \
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Of course, all $a_i$ should be greater than $1$. \
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Also, all $a_i$ should be smaller than four, since $x <= x(x-2)$ if $x >= 4$. \
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Also, all $a_i$ should be smaller than four, since $x <= x(x-2)$ if $x >= 4$. \
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Thus, we're left with sequences that only contain 2 and 3. \
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Thus, we're left with sequences that only contain 2 and 3. \
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#note([Note that two twos are the same as one four, but we exclude fours for simplicity.])
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#note(
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[Note that two twos are the same as one four, but we exclude fours for simplicity.],
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)
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#v(2mm)
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#v(2mm)
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