Appendix C: Formulas, Conversions & Prime Catalogs

C.1 Elementary Math & Geometry Word-Rule Directory

In many school textbooks, geometry rules and unit conversions are written with lots of letter variables and parentheses:

\[\text{Triangle Area} = (\text{Base} \times \text{Height}) \div 2 \qquad \text{Trapezoid Area} = (\text{Top} + \text{Bottom}) \div 2 \times \text{Height} \qquad \text{Celsius} = (\text{Fahrenheit} - 32) \times 5 \div 9\]

When students try to type these rules into an ordinary calculator, it is easy to miss a parenthesis or lose track of a middle step.

On the StackCalc SC-6, the four-shelf stack ($X, Y, Z, T$) needs zero parentheses (Łukasiewicz, 1924; Hamrick, 1996). Just like looking inside a cutaway drawing of a machine (Macaulay, 2016), every middle step stays visible on Waiting Shelf $Y$ and Upper Shelf $Z$ and drops into place right when you tap an operation key. This appendix gives you step-by-step RPN keystrokes, unit conversion factors, divisibility tests, and prime-number catalogs for every elementary and middle-school topic. This directory pairs core number rules, 2D flat shape rules, and 3D solid box and cylinder rules with their physical block models and SC-6 keystroke sequences.

C.1.1 Foundational Number Rules & Block Models

Elementary math is built on simple, reliable patterns. The table below pairs each rule with its physical block proof, chapter link, and SC-6 stack check:

Math Rule	Rule in Words & Numbers	Physical Block Model	Chapter Link	SC-6 Stack Check (`Example`)
Turn-Around Rule (Addition)	$\text{First} + \text{Second} = \text{Second} + \text{First}$ ($3 + 5 = 5 + 3$)	Sliding Number Rack beads left vs. right	Chapter 4	`3 [ENTER] 5 [+]` equals `5 [ENTER] 3 [+]`
Turn-Around Rule (Multiplication)	$\text{Rows} \times \text{Columns} = \text{Columns} \times \text{Rows}$ ($4 \times 6 = 6 \times 4$)	Turning a 24-tile rectangle $90^\circ$ on the desk	Chapter 4	`4 [ENTER] 6 [×]` equals `6 [ENTER] 4 [×]`
Grouping Rule (Addition)	$(\text{First} + \text{Second}) + \text{Third} = \text{First} + (\text{Second} + \text{Third})$	Joining 3 color trains of snap cubes	Chapter 4	`2 [ENTER] 3 [+] 4 [+]`
Grouping Rule (Multiplication)	$(\text{Length} \times \text{Width}) \times \text{Height}$ in any order	Stacking 3D layers of unit cubes	Chapter 4	`2 [ENTER] 3 [×] 4 [×]`
Split-Apart Rule	$4 \times (10 + 3) = (4 \times 10) + (4 \times 3)$	Splitting a 4-color tile rectangle along a color seam	Chapter 4	`10 [ENTER] 3 [+] 4 [×]`
Zero and One Rules	Adding $0$ or multiplying by $1$ leaves a number unchanged	Adding zero tiles; taking 1 group of tiles	Chapter 4	`7 [ENTER] 0 [+]` $\to 7$; `7 [ENTER] 1 [×]` $\to 7$
Division with Remainder Check	$\text{Total} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$	Sharing counters equally into cups with leftovers	Chapter 4	`29 [ENTER] 4 [g] [÷R]` $\to$ Remainder `1` in $Y$, Quotient `7` in $X$
GCF and LCM Partner Rule	$\text{GCF} \times \text{LCM} = \text{First Number} \times \text{Second Number}$	Comparing Prime Factor Studio towers side by side	Chapter 7	`12 [ENTER] 18 [GCF] 12 [ENTER] 18 [f] [LCM] [×]` $\to 216$
Fraction-to-Decimal Bridge	$\text{Top Number} \div \text{Bottom Number} = \text{Decimal}$	Comparing Fraction Towers to the $1\text{-Whole}$ tower	Chapter 5	`3 [ENTER] 4 [÷]` $\to 0.75$

C.1.2 Two-Dimensional Perimeter & Area Word Rules

The table below summarizes 2D flat shape rules in plain words alongside their chapter links and SC-6 RPN keystroke sequences:

Flat Shape	Word Rule	Chapter Link	SC-6 RPN Keystroke Sequence	Stack Shelves Used
Square (Perimeter)	$\text{Perimeter} = 4 \times \text{Side}$	Chapter 6	`[Side] [ENTER] 4 [×]`	$Y, X$
Square (Area)	$\text{Area} = \text{Side} \times \text{Side}$	Chapter 6	`[Side] [x²]`	$X$
Rectangle (Perimeter)	$\text{Perimeter} = 2 \times (\text{Length} + \text{Width})$	Chapter 6	`[Length] [ENTER] [Width] [+] 2 [×]`	$Y, X$
Rectangle (Area)	$\text{Area} = \text{Length} \times \text{Width}$	Chapter 6	`[Length] [ENTER] [Width] [f] [RECT]`	$Y, X$
Triangle (Perimeter)	$\text{Perimeter} = \text{Side 1} + \text{Side 2} + \text{Side 3}$	Chapter 6	`[Side 1] [ENTER] [Side 2] [+] [Side 3] [+]`	$Z, Y, X$
Triangle (Area)	$\text{Area} = (\text{Base} \times \text{Height}) \div 2$	Chapter 6	`[Base] [ENTER] [Height] [×] 2 [÷]`	$Y, X$
Parallelogram (Area)	$\text{Area} = \text{Base} \times \text{Height}$	Chapter 6	`[Base] [ENTER] [Height] [×]`	$Y, X$
Trapezoid (Area)	$\text{Area} = (\text{Top} + \text{Bottom}) \div 2 \times \text{Height}$	Chapter 6	`[Top] [ENTER] [Bottom] [+] 2 [÷] [Height] [×]`	$Z, Y, X$
Circle (Circumference)	$\text{Distance Around} = \pi \times \text{Width (Diameter)}$	Chapter 6	`[Diameter] [π] [×]`	$Y, X$
Circle (Area)	$\text{Area} = \pi \times \text{Radius} \times \text{Radius}$	Chapter 6	`[Radius] [x²] [π] [×]`	$Y, X$

C.1.3 Flat Shapes on the Stack

To see how Shelves $T, Z, Y,$ and $X$ update step by step when measuring flat shapes, let's trace five classic two-dimensional shapes on the stack. Suppose you start with a triangle whose base is $14\text{ cm}$ and height is $9\text{ cm}$. Because any triangle is half of a bounding rectangle, the triangle word rule tells us that $\text{Area} = (\text{Base} \times \text{Height}) \div 2 = (14 \times 9) \div 2 = 63\text{ cm}^2$:

Keystroke	Register Y	Register X	What Happens on the Stack
`14`	0	14	Enter the $14\text{ cm}$ triangle base onto Shelf $X$
`[ENTER]`	14	14	Copy the $14\text{ cm}$ base up onto Shelf $Y$
`9`	14	9	Enter the $9\text{ cm}$ height onto Shelf $X$
`[×]`	0	126	Multiply base by height ($14 \times 9 = 126\text{ cm}^2$ rectangle)
`2`	126	2	Type 2 to halve the rectangle; $126$ lifts to Shelf $Y$
`[÷]`	0	63	Divide by 2 for the triangle area: 63 cm²

Notice how you first multiply $14 \times 9 = 126$ for the full bounding rectangle, and then type 2 and tap [÷] to slice that rectangle in half: $126 \div 2 = 63$.

What if the top of the shape isn't a sharp point, but a flat roof—making a trapezoid with a top width of $8\text{ in}$, a bottom width of $14\text{ in}$, and a height of $6\text{ in}$? Averaging the top and bottom widths first gives the middle width ($(8 + 14) \div 2 = 11\text{ in}$), and multiplying by the height gives $\text{Area} = (\text{Top} + \text{Bottom}) \div 2 \times \text{Height} = 11 \times 6 = 66\text{ sq in}$:

Keystroke	Register Y	Register X	What Happens on the Stack
`8`	0	8	Enter the $8\text{ in}$ top width onto Shelf $X$
`[ENTER]`	8	8	Copy $8\text{ in}$ up onto Shelf $Y$
`14`	8	14	Enter the $14\text{ in}$ bottom width onto Shelf $X$
`[+]`	0	22	Add top and bottom widths ($8 + 14 = 22\text{ in}$)
`2`	22	2	Type 2 to average the two widths
`[÷]`	0	11	Divide by 2 for the middle width ($11\text{ in}$)
`6`	11	6	Enter the $6\text{ in}$ straight-up height
`[×]`	0	66	Multiply middle width by height: 66 sq in

Why does averaging the top and bottom widths turn any trapezoid into an equivalent rectangle? If you snip off the two slanted triangular wings on the left and right sides of the trapezoid at the halfway height mark and flip them upside down into the top corners, the slanted sides lock together into a straight-sided $11\text{ in} \times 6\text{ in}$ rectangle with zero leftover scraps.

When straight edges curve into a circle—such as a circular splash pad with a radius of $3.5\text{ m}$—imagine slicing the circle into thin pie wedges and alternating them teeth-together like a zipper. As the wedges get thinner, they form a nearly flat rectangle whose height is the radius ($r$) and whose width is half the circumference ($\pi \times r$). Squaring the radius first (3.5 [x²] $\to 12.25\text{ m}^2$, the area of a $3.5\text{ m} \times 3.5\text{ m}$ corner square) and scaling by [π] ($\approx 3.1415926$) yields $\text{Area} = \pi \times \text{Radius} \times \text{Radius} = \pi \times 3.5^2 \approx 38.48451\text{ m}^2$:

Keystroke	Register Y	Register X	What Happens on the Stack
`3.5`	0	3.5	Enter the $3.5\text{ m}$ radius onto Shelf $X$
`[x²]`	0	12.25	Square the radius ($3.5 \times 3.5 = 12.25\text{ m}^2$)
`[π]`	12.25	3.1415926	Enter $\pi$; $12.25$ lifts to Shelf $Y$
`[×]`	0	38.484510	Multiply by $\pi$ for circle area: 38.48 m²

Notice how squaring the radius first ([x²]) turns a 1D spoke length ($3.5\text{ m}$) into a 2D corner square ($12.25\text{ m}^2$) on Ground Shelf $X$, and tapping [π] [×] scales that single radius-square by slightly more than three to fill all four quadrants of the circle.

Whenever we want the distance around the outside of a shape instead of the space inside it, we walk its boundary perimeter in 1D length units. Around a rectangular garden of length $14\text{ m}$ and width $9\text{ m}$, walking down one long side and across one short side covers an L-shaped half-lap ($14 + 9 = 23\text{ m}$). Doubling that half-lap covers all four sides without having to type each measurement twice, giving $\text{Perimeter} = 2 \times (\text{Length} + \text{Width}) = 2 \times 23 = 46\text{ m}$:

Keystroke	Register Y	Register X	What Happens on the Stack
`14`	0	14	Enter the $14\text{ m}$ garden length onto Shelf $X$
`[ENTER]`	14	14	Copy $14\text{ m}$ up onto Shelf $Y$
`9`	14	9	Enter the $9\text{ m}$ garden width onto Shelf $X$
`[+]`	0	23	Add length and width for the $23\text{ m}$ half-lap
`2`	23	2	Type 2 to double the half-lap
`[×]`	0	46	Multiply by 2 for the full perimeter: 46 m

And to find how far a $24\text{-inch}$ bicycle wheel rolls along the pavement in one full turn ($\text{Circumference} = \pi \times 24$), typing 24 [π] [×] on the stack immediately displays $75.39822\text{ inches}$ per revolution—showing that every wheel travels a little more than three times its own diameter on every spin.

C.1.4 Three-Dimensional Solid Volume & Surface Area Word Rules

3D Solid Shape	Volume Word Rule	Total Surface Area Word Rule	Chapter Link	SC-6 Volume Keystrokes
Cube	$\text{Volume} = \text{Side} \times \text{Side} \times \text{Side}$	$\text{Surface Area} = 6 \times (\text{Side} \times \text{Side})$	Chapter 6	`[Side] [ENTER] 3 [g] [yˣ]`
Rectangular Box (Prism)	$\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$	Add all $6$ rectangular faces ($2\text{ Top/Bottom} + 2\text{ Front/Back} + 2\text{ Sides}$)	Chapter 6	`[Length] [ENTER] [Width] [×] [Height] [×]`
Triangular Tent (Prism)	$\text{Volume} = (\text{Base} \times \text{Height} \div 2) \times \text{Length}$	Add the $2$ triangle ends plus the $3$ rectangular side panels	Chapter 6	`[Base] [ENTER] [Height] [×] 2 [÷] [Length] [×]`
Cylinder (Can)	$\text{Volume} = (\pi \times \text{Radius}^2) \times \text{Height}$	$\text{Curved Side} \ (2 \times \pi \times \text{Radius} \times \text{Height}) + 2\text{ Circle Ends} \ (2 \times \pi \times \text{Radius}^2)$	Chapter 6	`[Radius] [x²] [π] [×] [Height] [×]`

C.1.5 Solid Shapes on the Stack

When a flat base is lifted upward into three dimensions, the stack lets you find both volume and surface area without parentheses or scratch paper. Every prism or cylinder volume works like stacking identical coasters or pancakes: first calculate the flat 2D footprint on the floor, and then multiply by the height to sweep that footprint upward into 3D space. Consider a cylindrical farm storage silo with a radius of $4\text{ m}$ and a height of $10\text{ m}$. Its volume stacks the circular base area across the height: $(\pi \times 4^2) \times 10 = 160\pi \approx 502.6548\text{ m}^3$:

Keystroke	Register Y	Register X	What Happens on the Stack
`4`	0	4	Enter the $4\text{ m}$ silo radius onto Shelf $X$
`[x²]`	0	16	Square the radius ($4 \times 4 = 16$)
`[π]`	16	3.1415926	Enter $\pi$; $16$ lifts to Shelf $Y$
`[×]`	0	50.265482	Multiply for the circular floor area ($50.27\text{ m}^2$)
`10`	50.265482	10	Enter the $10\text{ m}$ silo height
`[×]`	0	502.654824	Stack the circle floor by height: 502.65 m³

What if we want to paint that same silo's outer shell and both circular lids? If you unroll the curved metal wall of a cylinder like a soup-can label, it flattens into a rectangle whose width is the circle's circumference ($2 \times \pi \times 4\text{ m} \approx 25.13\text{ m}$) and whose height is $10\text{ m}$. On the stack, you can calculate that unrolled side wrapper first ($251.33\text{ m}^2$) and let automatic stack lift hold it safely on Waiting Shelf $Y$ while you build the two circular end caps ($2 \times \pi \times 4^2 \approx 100.53\text{ m}^2$) on Ground Shelf $X$, then unite both parts with [+] for $\approx 351.8584\text{ m}^2$:

Keystroke	Register Y	Register X	What Happens on the Stack
`2`	0	2	Start the $2\pi r h$ curved side wall
`[π] [×]`	0	6.2831853	Multiply $2 \times \pi \approx 6.2832$
`4 [×]`	0	25.132741	Multiply by the $4\text{ m}$ radius ($25.13\text{ m}$ rim)
`10 [×]`	0	251.327412	Multiply by $10\text{ m}$ height ($251.33\text{ m}^2$ side wall)
`2`	251.327412	2	Start the 2 circular lids; side wall lifts to Shelf $Y$
`[π] [×]`	251.327412	6.2831853	Multiply $2 \times \pi$ on Shelf $X$
`4 [x²] [×]`	251.327412	100.530964	Multiply by $4^2$ for both lids ($100.53\text{ m}^2$)
`[+]`	0	351.858376	Add side wall and lids: 351.86 m² total surface

The same layer-stacking idea works for rectangular boxes and cubes. Imagine filling the bottom of a square-base storage box ($6\text{ cm}$ by $6\text{ cm}$ base, $8\text{ cm}$ tall) with $1\text{ cm}$ unit cubes: squaring the $6\text{ cm}$ side covers the bottom floor with a single layer of $36$ unit cubes ($36\text{ cm}^2$), and multiplying by $8$ stacks eight identical layers on top of one another ($36 \times 8 = 288\text{ cm}^3$):

Keystroke	Register Y	Register X	What Happens on the Stack
`6 [x²]`	0	36	Square the $6\text{ cm}$ side for the $36\text{ cm}^2$ floor
`8`	36	8	Enter the $8\text{ cm}$ height; $36$ lifts to Shelf $Y$
`[×]`	0	288	Multiply floor area by height: 288 cm³ volume

And when all three dimensions match—such as a $9\text{ cm}$ cube—it holds $\text{Side}^3 = 9^3 = 729\text{ cm}^3$ of space inside (9 [ENTER] 3 [g] [yˣ] $\to \mathbf{729}\text{ cm}^3$) and is wrapped by six identical $9 \times 9$ square faces (9 [x²] 6 [×] $\to \mathbf{486}\text{ cm}^2$).

C.2 Complete Unit Conversions Directory

Why does a meter divide cleanly by $10$, $100$, and $1,000$, while an English mile contains the seemingly strange number of $5,280\text{ feet}$? Every conversion number in the tables below carries a piece of human history. In ancient Rome, soldiers measured long roads by counting $1,000$ double-paces (mille passus, where one left-right stride was about $5\text{ Roman feet}$), giving a Roman mile of $5,000\text{ feet}$. Centuries later in England, farmers measured plowed fields in furlongs ("furrow-longs" of $660\text{ feet}$, the distance an ox team could pull a plow before resting). In 1593, Queen Elizabeth I passed a law making the statute mile equal to 8 furlongs ($8 \times 660 = \mathbf{5,280\text{ feet}}$) so land surveyors and road builders could share one common scale.

Before the French Revolution of the 1790s, almost every town in Europe used a different foot or pound. Between 1792 and 1799, French scientists decided to base a new universal unit—the meter (from the Greek metron, "measure")—on the planet Earth itself, defining $1\text{ meter}$ as one ten-millionth of the distance from the North Pole to the Equator. Two astronomers, Jean-Baptiste Delambre and Pierre Méchain, spent seven years trekking across mountains and church bell towers from Dunkirk to Barcelona with brass surveying triangles to measure the Earth's curve and cast the first platinum meter bar.

Even the numbers on a kitchen or weather thermometer tell an engineering story. In 1724, instrument maker Daniel Gabriel Fahrenheit mixed ice, water, and salt to reach the coldest laboratory temperature he could make ($0^\circ\text{F}$), and placed freezing pure water at $32^\circ\text{F}$ and human body temperature near $96^\circ\text{F}$ so the $64\text{-degree}$ gap between them could be halved six times in a row ($64 = 2^6$) when marking glass thermometer tubes! Eighteen years later, in 1742, Swedish astronomer Anders Celsius divided the span between freezing and boiling water into $100$ equal steps (originally putting $0^\circ$ at boiling and $100^\circ$ at freezing, until botanist Carl Linnaeus flipped the scale upright so warmer days had higher numbers).

Every unit conversion on the SC-6 works two ways: for the most common everyday measurements (grams/ounces, kilograms/pounds, centimeters/inches, milliliters/fluid ounces, liters/gallons, and Celsius/Fahrenheit), you can use the one-tap shifted conversion keys on Rows 5–6 of the keypad ([f] for Metric, [g] for U.S. Customary), or you can multiply and divide directly by the exact conversion factor on the stack so you can see the rate at work.

C.2.1 Length Conversions

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse (Back Again)
Inches (in)	Centimeters (cm)	$2.54$	`[in] [ENTER] 2.54 [×]`	$\div 2.54$
Feet (ft)	Inches (in)	$12.0$	`[ft] [ENTER] 12 [×]`	$\div 12$
Feet (ft)	Meters (m)	$0.3048$	`[ft] [ENTER] [.] 3048 [×]`	$\div 0.3048$
Yards (yd)	Feet (ft)	$3.0$	`[yd] [ENTER] 3 [×]`	$\div 3$
Yards (yd)	Meters (m)	$0.9144$	`[yd] [ENTER] [.] 9144 [×]`	$\div 0.9144$
Miles (mi)	Feet (ft)	$5280.0$	`[mi] [ENTER] 5280 [×]`	$\div 5280$
Miles (mi)	Kilometers (km)	$1.609344$	`[mi] [ENTER] 1.609344 [×]`	$\div 1.609344$
Nautical Miles (NM)	Kilometers (km)	$1.852$	`[NM] [ENTER] 1.852 [×]`	$\div 1.852$
Nautical Miles (NM)	Statute Miles (mi)	$\approx 1.150779$	`[NM] [ENTER] 1.1508 [×]`	$\div 1.1508$
Meters (m)	Centimeters (cm)	$100.0$	`[m] [ENTER] 100 [×]`	$\div 100$
Centimeters (cm)	Millimeters (mm)	$10.0$	`[cm] [ENTER] 10 [×]`	$\div 10$
Kilometers (km)	Meters (m)	$1000.0$	`[km] [ENTER] 1000 [×]`	$\div 1000$

For example, to convert a $100\text{-yard}$ football field into meters, key in 100 [ENTER] 0.9144 [×] to obtain $91.44\text{ meters}$—and dividing right back with [.] 9144 [÷] restores the original $100.00\text{ yards}$.

C.2.2 Area Conversions

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse
Square Feet (sq ft)	Square Inches (sq in)	$144.0$	`[sq ft] [ENTER] 144 [×]`	$\div 144$
Square Yards (sq yd)	Square Feet (sq ft)	$9.0$	`[sq yd] [ENTER] 9 [×]`	$\div 9$
Acres (ac)	Square Feet (sq ft)	$43,560.0$	`[ac] [ENTER] 43560 [×]`	$\div 43560$
Acres (ac)	Hectares (ha)	$\approx 0.4046856$	`[ac] [ENTER] [.] 4047 [×]`	$\div 0.4047$
Hectares (ha)	Square Meters ($\text{m}^2$)	$10,000.0$	`[ha] [ENTER] 10000 [×]`	$\div 10000$
Square Miles ($\text{mi}^2$)	Acres (ac)	$640.0$	`[sq mi] [ENTER] 640 [×]`	$\div 640$
Square Meters ($\text{m}^2$)	Square Feet (sq ft)	$\approx 10.76391$	`[m²] [ENTER] 10.764 [×]`	$\div 10.764$

For instance, to find how many square feet lie inside a $2.5\text{-acre}$ neighborhood park, enter 2.5 [ENTER] 43560 [×], which reveals $108,900\text{ sq ft}$.

C.2.3 Weight & Mass Conversions

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse
Ounces (oz)	Grams (g)	$28.349523$	`[oz] [ENTER] 28.35 [×]`	$\div 28.3495$
Pounds (lb)	Ounces (oz)	$16.0$	`[lb] [ENTER] 16 [×]`	$\div 16$
Pounds (lb)	Kilograms (kg)	$0.45359237$	`[lb] [ENTER] [.] 4536 [×]`	$\div 0.4536$
Kilograms (kg)	Pounds (lb)	$\approx 2.2046226$	`[kg] [ENTER] 2.2046 [×]`	$\div 2.2046$
Tons (US Short)	Pounds (lb)	$2000.0$	`[ton] [ENTER] 2000 [×]`	$\div 2000$
Metric Tons (t)	Kilograms (kg)	$1000.0$	`[t] [ENTER] 1000 [×]`	$\div 1000$
Kilograms (kg)	Grams (g)	$1000.0$	`[kg] [ENTER] 1000 [×]`	$\div 1000$
Grams (g)	Milligrams (mg)	$1000.0$	`[ENTER] 1000 [×]`	$\div 1000$

To convert a $65\text{ lb}$ bicycle shipping crate into kilograms, press 65 [ENTER] 0.453592 [×] to read $29.483\text{ kg}$ on the display.

C.2.4 Capacity & Liquid Volume Conversions

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse
Fluid Ounces (fl oz)	Milliliters (mL)	$29.57352956$	`[fl oz] [ENTER] 29.57 [×]`	$\div 29.5735$
Cups (c)	Fluid Ounces (fl oz)	$8.0$	`[c] [ENTER] 8 [×]`	$\div 8$
Pints (pt)	Cups (c)	$2.0$	`[pt] [ENTER] 2 [×]`	$\div 2$
Quarts (qt)	Pints (pt)	$2.0$	`[qt] [ENTER] 2 [×]`	$\div 2$
Gallons (gal US)	Quarts (qt)	$4.0$	`[gal] [ENTER] 4 [×]`	$\div 4$
Gallons (gal US)	Liters (L)	$3.785411784$	`[gal] [ENTER] 3.7854 [×]`	$\div 3.7854$
Gallons (Imperial)	Liters (L)	$4.54609$	`[gal imp] [ENTER] 4.546 [×]`	$\div 4.5461$
Liters (L)	Milliliters (mL)	$1000.0$	`[L] [ENTER] 1000 [×]`	$\div 1000$
Cubic Meters ($\text{m}^3$)	Liters (L)	$1000.0$	`[m³] [ENTER] 1000 [×]`	$\div 1000$

To convert a $15\text{-gallon}$ camp water tank into liters, run 15 [ENTER] 3.7854 [×] to find $56.781\text{ liters}$.

C.2.5 Speed Conversions

From Unit	To Unit	Conversion Number	SC-6 Keystrokes
Miles per Hour (mph)	Kilometers per Hour (km/h)	$1.609344$	`[mph] [ENTER] 1.609 [×]`
Miles per Hour (mph)	Feet per Second (ft/s)	$\approx 1.466667$	`[mph] [ENTER] 5280 [×] 3600 [÷]`
Kilometers per Hour (km/h)	Meters per Second (m/s)	$\approx 0.277778$	`[km/h] [ENTER] 1000 [×] 3600 [÷]`
Knots (kt)	Miles per Hour (mph)	$\approx 1.150779$	`[kt] [ENTER] 1.151 [×]`
Knots (kt)	Kilometers per Hour (km/h)	$1.852$	`[kt] [ENTER] 1.852 [×]`

When a quadcopter drone flies at $45\text{ mph}$, you can find how many feet it covers each second by keying in 45 [ENTER] 5280 [×] 3600 [÷], which lands on $66.0\text{ ft/s}$.

C.2.6 Tire & Air Pressure Conversions

From Unit	To Unit	Conversion Number	SC-6 Keystrokes
Atmospheres (atm)	Pounds per Sq Inch (psi)	$14.6959488$	`[atm] [ENTER] 14.696 [×]`
Atmospheres (atm)	Kilopascals (kPa)	$101.325$	`[atm] [ENTER] 101.325 [×]`
Bar (bar)	Kilopascals (kPa)	$100.0$	`[bar] [ENTER] 100 [×]`
Pounds per Sq Inch (psi)	Kilopascals (kPa)	$6.894757$	`[psi] [ENTER] 6.895 [×]`

If a road bicycle tire is inflated to $85\text{ psi}$, keying in 85 [ENTER] 6.89476 [×] converts that pressure to $586.05\text{ kPa}$.

C.2.7 Time Conversions

From Unit	To Unit	Conversion Number	SC-6 Keystroke Sequence
Minutes (min)	Seconds (s)	$60.0$	`[min] [ENTER] 60 [×]`
Hours (hr)	Minutes (min)	$60.0$	`[hr] [ENTER] 60 [×]`
Hours (hr)	Seconds (s)	$3600.0$	`[hr] [ENTER] 3600 [×]`
Days (d)	Hours (hr)	$24.0$	`[days] [ENTER] 24 [×]`
Weeks (wk)	Days (d)	$7.0$	`[wk] [ENTER] 7 [×]`
Years (yr standard)	Days (d)	$365.0$	`[yr] [ENTER] 365 [×]`
Years (yr leap)	Days (d)	$366.0$	`[yr] [ENTER] 366 [×]`

To count how many seconds tick by in a full 24-hour day, enter 24 [ENTER] 3600 [×] to reach $86,400\text{ seconds}$.

C.2.8 Temperature Conversions (Fahrenheit and Celsius)

Because water freezes at $32^\circ\text{F}$ ($0^\circ\text{C}$) and boils at $212^\circ\text{F}$ ($100^\circ\text{C}$), converting between Fahrenheit and Celsius uses two simple word rules:

\[\text{Celsius} = (\text{Fahrenheit} - 32) \times 5 \div 9 \qquad \text{Fahrenheit} = (\text{Celsius} \times 9 \div 5) + 32\]

Conversion	Word Rule	SC-6 Step-by-Step Keystrokes (or One-Tap Key)
Fahrenheit to Celsius	$(\text{Fahrenheit} - 32) \times 5 \div 9$	`[Temp F] [ENTER] 32 [-] 5 [×] 9 [÷]` (or `[Temp F] [f] [▸°C]`)
Celsius to Fahrenheit	$(\text{Celsius} \times 9 \div 5) + 32$	`[Temp C] [ENTER] 9 [×] 5 [÷] 32 [+]` (or `[Temp C] [g] [▸°F]`)

Here is how normal human body temperature ($98.6^\circ\text{F}$) converts to Celsius step by step on the stack:

Keystroke	Register Y	Register X	What Happens on the Stack
`98.6`	0	98.6	Enter normal body temperature ($98.6^\circ\text{F}$)
`[ENTER]`	98.6	98.6	Copy $98.6$ up onto Shelf $Y$
`32`	98.6	32	Enter freezing-point offset ($32^\circ\text{F}$)
`[-]`	0	66.6	Subtract 32 ($98.6 - 32 = 66.6$)
`5`	66.6	5	Enter numerator 5 of the $5/9$ scale factor
`[×]`	0	333.0	Multiply $66.6 \times 5 = 333.0$
`9`	333.0	9	Enter denominator 9 of the $5/9$ scale factor
`[÷]`	0	37.0	Divide $333.0 \div 9 =$ 37.0°C

Normal human body temperature is $37.0^\circ\text{C}$ (or in one tap on the SC-6 keypad, 98.6 [f] [▸°C] $\to$ 37.00).

C.2.9 Chaining Multi-Step Unit Conversions on the Stack

When a real-world measurement needs two conversion steps in a row, the stack lets you chain them without parentheses or scratch paper. For a marathon footrace ($26.2\text{ miles} \to \text{kilometers} \to \text{meters}$), since $1\text{ mile} = 1.609344\text{ kilometers}$ and $1\text{ kilometer} = 1,000\text{ meters}$, multiplying $26.2 \times 1.609344$ first gives $42.164813\text{ km}$ and multiplying by $1,000$ converts to meters:

Keystroke	Register Y	Register X	What Happens on the Stack
`26.2`	0	26.2	Enter marathon distance ($26.2\text{ miles}$)
`[ENTER]`	26.2	26.2	Copy $26.2$ up onto Shelf $Y$
`1.609344`	26.2	1.609344	Enter kilometers per mile ($1.609344$)
`[×]`	0	42.164813	Multiply for marathon in kilometers ($42.165\text{ km}$)
`1000`	42.164813	1000	Enter $1,000\text{ meters per kilometer}$
`[×]`	0	42164.81280	Multiply for marathon in meters: 42,164.81 m

Ground Shelf $X$ shows the full Olympic marathon distance in meters: $42,164.81\text{ meters}$.

Next, consider an airliner cruising at $36,000\text{ ft}$ ($36,000\text{ ft} \to \text{meters} \to \text{kilometers}$). Using $1\text{ foot} = 0.3048\text{ meters}$, we multiply $36,000 \times 0.3048$ to get meters and immediately divide by $1,000$ for kilometers:

Keystroke	Register Y	Register X	What Happens on the Stack
`36000`	0	36000	Enter cruising altitude ($36,000\text{ ft}$)
`[ENTER]`	36000	36000	Copy $36,000$ up onto Shelf $Y$
`.3048`	36000	0.3048	Enter meters per foot ($0.3048\text{ m/ft}$)
`[×]`	0	10972.8	Multiply to get altitude in meters ($10,972.8\text{ m}$)
`1000`	10972.8	1000	Enter $1,000\text{ meters per kilometer}$
`[÷]`	0	10.9728	Divide by 1,000 for kilometers: 10.97 km

Cruising height is $10,972.8\text{ meters}$, or $10.97\text{ km}$ above sea level.

What about comparing family car fuel mileage ($35.0\text{ Miles per Gallon} \to \text{Liters per 100 km}$)? Dividing the conversion constant $235.215$ by the Miles per Gallon ($\text{Liters per 100 km} = 235.215 \div \text{MPG}$) flips American mileage into metric fuel consumption:

Keystroke	Register Y	Register X	What Happens on the Stack
`235.215`	0	235.215	Enter the fuel-economy constant ($235.215$)
`[ENTER]`	235.215	235.215	Copy $235.215$ up onto Shelf $Y$
`35`	235.215	35	Enter fuel rating ($35\text{ MPG}$) onto Shelf $X$
`[÷]`	0	6.720429	Divide $235.215 \div 35 =$ 6.72 L/100 km

A car rated at $35\text{ MPG}$ burns $6.72\text{ liters of fuel per 100 kilometers}$.

Finally, we can test a snowy winter morning temperature ($14^\circ\text{F} \to -10^\circ\text{C} \to 14^\circ\text{F}$) by converting $14^\circ\text{F}$ into Celsius and then running the inverse steps right on the stack to return to $14^\circ\text{F}$:

Keystroke	Register Y	Register X	What Happens on the Stack
`14`	0	14	Enter winter morning temperature ($14^\circ\text{F}$)
`[ENTER]`	14	14	Copy $14$ up onto Shelf $Y$
`32`	14	32	Enter freezing-point offset ($32^\circ\text{F}$)
`[-]`	0	-18	Subtract 32 ($14 - 32 = -18^\circ\text{F}$ below freezing)
`5`	-18	5	Enter numerator 5
`[×]`	0	-90	Multiply $-18 \times 5 = -90$
`9`	-90	9	Enter denominator 9
`[÷]`	0	-10	Divide $-90 \div 9 =$ -10°C
`9 [×] 5 [÷]`	0	-18	Reverse the scale factor ($-10 \times 9 \div 5 = -18$)
`32 [+]`	0	14	Add 32 to return to 14°F

Here $14^\circ\text{F}$ equals $-10^\circ\text{C}$ (ten degrees below freezing), and reversing the steps returns $14^\circ\text{F}$.

C.3 Prime Number Catalog & Factorization Directory

A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Every composite number greater than 1 can be built by multiplying a unique team of prime numbers together.

C.3.1 Complete Catalog of All 46 Prime Numbers Under 200

There are 46 prime numbers less than 200. The table below lists each prime number, its Twin Prime partner (a prime just 2 steps away), and a fun number pattern:

Prime #	Prime Number	Twin Prime Partner	Fun Number Pattern
#1	2	Only Even Prime	Only even prime; smallest prime number ($1 + 1 = 2$).
#2	3	Twin with 5	Smallest odd prime ($2^2 - 1 = 3$).
#3	5	Twin with 3 and 7	Only prime in two twin pairs ($3,5$ and $5,7$); sum of two squares ($1^2 + 2^2 = 5$).
#4	7	Twin with 5	Largest one-digit prime ($2^3 - 1 = 7$).
#5	11	Twin with 13	Smallest two-digit prime; reads the same forward and backward.
#6	13	Twin with 11	Reversing its digits makes $31$ (also prime!); sum of two squares ($2^2 + 3^2 = 13$).
#7	17	Twin with 19	Sum of the first four primes ($2 + 3 + 5 + 7 = 17$); $4^2 + 1 = 17$.
#8	19	Twin with 17	Twin prime partner of $17$ ($1 + 6 + 12 = 19$).
#9	23	Solo Prime	Sum of three consecutive primes ($5 + 7 + 11 = 23$).
#10	29	Twin with 31	Sum of three consecutive squares ($2^2 + 3^2 + 4^2 = 29$); also $2^2 + 5^2 = 29$.
#11	31	Twin with 29	One less than $32$ ($2^5 - 1 = 31$); digit-flip partner of $13$.
#12	37	Solo Prime	Sum of two squares ($1^2 + 6^2 = 37$); digit-flip partner of $73$.
#13	41	Twin with 43	Sum of the first six primes ($2 + 3 + 5 + 7 + 11 + 13 = 41$); $4^2 + 5^2 = 41$.
#14	43	Twin with 41	Twin prime partner of $41$ ($6^2 + 7 = 43$).
#15	47	Solo Prime	Two less than a square ($7^2 - 2 = 47$).
#16	53	Solo Prime	Sum of two squares ($2^2 + 7^2 = 53$); sum of five primes ($5 + 7 + 11 + 13 + 17 = 53$).
#17	59	Twin with 61	Sum of three consecutive primes ($17 + 19 + 23 = 59$).
#18	61	Twin with 59	Sum of two squares ($5^2 + 6^2 = 61$); difference of two cubes ($5^3 - 4^3 = 61$).
#19	67	Solo Prime	Sum of five consecutive primes ($7 + 11 + 13 + 17 + 19 = 67$).
#20	71	Twin with 73	Sum of three consecutive primes ($19 + 23 + 29 = 71$); digit-flip partner of $17$.
#21	73	Twin with 71	Sum of two squares ($3^2 + 8^2 = 73$); digit-flip partner of $37$.
#22	79	Solo Prime	Digit-flip partner of $97$ ($79$ and $97$ are both prime).
#23	83	Solo Prime	Sum of three consecutive primes ($23 + 29 + 31 = 83$); $9^2 + 2 = 83$.
#24	89	Solo Prime	Fibonacci prime ($34 + 55 = 89$); sum of two squares ($5^2 + 8^2 = 89$).
#25	97	Solo Prime	Largest two-digit prime; sum of two squares ($4^2 + 9^2 = 97$).
#26	101	Twin with 103	Smallest three-digit prime; reads the same forward and backward ($10^2 + 1 = 101$).
#27	103	Twin with 101	Twin prime partner of $101$.
#28	107	Twin with 109	Twin prime partner of $109$.
#29	109	Twin with 107	Sum of two squares ($3^2 + 10^2 = 109$); twin prime partner of $107$.
#30	113	Solo Prime	Sum of two squares ($7^2 + 8^2 = 113$); every rearrangement of its digits ($113, 131, 311$) is prime!
#31	127	Solo Prime	One less than $128$ ($2^7 - 1 = 127$).
#32	131	Solo Prime	Reads the same forward and backward; digit partner of $113$ and $311$.
#33	137	Twin with 139	Sum of two squares ($4^2 + 11^2 = 137$); twin prime partner of $139$.
#34	139	Twin with 137	Twin prime partner of $137$; sum of five primes ($19 + 23 + 29 + 31 + 37 = 139$).
#35	149	Twin with 151	Sum of two squares ($7^2 + 10^2 = 149$).
#36	151	Twin with 149	Reads the same forward and backward; twin prime partner of $149$.
#37	157	Solo Prime	Sum of two squares ($6^2 + 11^2 = 157$); halfway between $151$ and $163$.
#38	163	Solo Prime	Solo prime between $157$ and $167$ ($9^2 + 9^2 + 1 = 163$).
#39	167	Solo Prime	Two less than $169$ ($13^2 - 2 = 167$).
#40	173	Solo Prime	Sum of two squares ($2^2 + 13^2 = 173$); sum of three primes ($53 + 59 + 61 = 173$).
#41	179	Twin with 181	Twin prime partner of $181$.
#42	181	Twin with 179	Reads the same forward and backward; sum of two squares ($9^2 + 10^2 = 181$).
#43	191	Twin with 193	Reads the same forward and backward; twin prime partner of $193$.
#44	193	Twin with 191	Sum of two squares ($7^2 + 12^2 = 193$); twin prime partner of $191$.
#45	197	Twin with 199	Sum of the first twelve primes ($2 + 3 + \dots + 37 = 197$); $14^2 + 1 = 197$.
#46	199	Twin with 197	Largest prime under $200$; sum of three consecutive primes ($61 + 67 + 71 = 199$).

C.3.2 The $\sqrt{N}$ Sieve Stop Rule & Eratosthenes' Shadow Stick

In Chapter 7, you used the Sieve of Eratosthenes to cross out multiples of $2, 3, 5,$ and $7$ on a $1\text{--}100$ grid. Notice the remarkable shortcut that happens when you extend Eratosthenes' sieve all the way to $200$ in the 46-prime reference table above: once you finish crossing out multiples of $2, 3, 5, 7, 11,$ and $13$, every single remaining uncrossed number up to $200$ is guaranteed to be prime!

Why don't you have to test $17, 19, 23,$ or any larger prime? Because whenever a composite number $N$ factors into two numbers $A \times B = N$, the two factors cannot both be larger than $\sqrt{N}$—if both were larger than $\sqrt{N}$, their product would already exceed $N$! That means every composite number up to $200$ must have at least one prime factor less than or equal to $\sqrt{200}$. On your SC-6, tapping 200 [f] [√x] displays 14.142136, and the largest prime below $14.142136$ is $13$ (since the smallest composite number built purely from $17$ or larger primes is $17 \times 17 = 289$, well past $200$). That exact $\sqrt{N}$ Stop Rule is how the SC-6 [PRIME] and [g] [FACT] keys test even a four-digit number like $1,999$ in a tiny fraction of a second: since 1999 [f] [√x] is $44.71$, the processor checks only the 14 primes up to $43$ and immediately flashes PRIME!

Eratosthenes of Cyrene (chief librarian of the great Library of Alexandria around 240 BCE) used that same eye for clean mathematical bounds to pull off one of the greatest measurements in science history: measuring the distance around the entire planet Earth using nothing but a stick's shadow! Travelers told him that at noon on the summer solstice in the southern city of Syene (modern Aswan), the sun shone straight down to the bottom of a deep water well with zero shadow ($0^\circ$). On that exact same day in Alexandria—$800\text{ kilometers}$ due north—Eratosthenes stood a vertical measuring stick in the ground and used a protractor wheel like your $360^\circ$ Unit Circle Volvelle to measure the sun's shadow angle at $7.2^\circ$.

Because a full circle around the curved Earth is $360^\circ$, Eratosthenes divided the full circle by his shadow angle (360 [ENTER] 7.2 [÷] $\to \mathbf{50}$) to discover that the $800\text{ km}$ road between Syene and Alexandria was exactly $\frac{1}{50}$ of the way around the world! Multiplying that fraction by $800\text{ km}$ right on the stack (360 [ENTER] 7.2 [÷] 800 [×]) gives $40,000\text{ kilometers}$—matching modern satellite measurements of Earth's circumference ($40,008\text{ km}$ around the poles) to within two-tenths of one percent!

C.3.3 Quick Divisibility Rules with SC-6 Checks

Before testing a big number with [PRIME], you can spot small factors right away using these mental math rules:

Divide By	Quick Mental Rule	How to Check on Your SC-6 (`[g] [÷R]`)	Example
2	Last digit is even ($0, 2, 4, 6, 8$).	`[Number] [ENTER] 2 [g] [÷R]` (Remainder in $Y$ is `0`)	$84 \div 2 = 42$ with Remainder $0$
3	Add the digits; the sum divides evenly by 3.	Add the digits on the stack, then press `3 [g] [÷R]`	$153 \to 1+5+3=9$, and $9 \div 3 = 3$
4	Last two digits divide evenly by 4.	`[Last 2 Digits] [ENTER] 4 [g] [÷R]`	$3,728 \to 28 \div 4 = 7$ (Remainder $0$)
5	Last digit is 0 or 5.	Look at the ones digit on the screen.	$95$ ends in $5$ ($95 \div 5 = 19$)
6	Passes both the 2-test (even) and the 3-test.	Check that it is even and its digits add to a multiple of 3.	$78$ is even, and $7+8=15$ ($78 \div 6 = 13$)
8	Last three digits divide evenly by 8.	`[Last 3 Digits] [ENTER] 8 [g] [÷R]`	$5,128 \to 128 \div 8 = 16$ (Remainder $0$)
9	Add the digits; the sum divides evenly by 9.	Add the digits on the stack, then press `9 [g] [÷R]`	$423 \to 4+2+3=9$, and $9 \div 9 = 1$
10	Last digit is 0.	Look at the ones digit on the screen.	$340$ ends in $0$ ($340 \div 10 = 34$)

C.3.4 Ten Classic Prime Factorizations

Every composite whole number greater than 1 breaks down into a unique product of prime numbers—just like a tower built from your Prime Factor Studio blocks. The table below shows 10 classic composite numbers, their prime factor blocks, and how to multiply the prime blocks back together on your SC-6:

Composite Number	Prime Factor Blocks	SC-6 `[g] [FACT]` Screen Output	RPN Keystrokes to Multiply Them Back
72	$2^3 \times 3^2$	`2³ × 3²`	`2 [ENTER] 3 [g] [yˣ] 3 [x²] [×]` $\to 72$
84	$2^2 \times 3 \times 7$	`2² × 3 × 7`	`2 [x²] 3 [×] 7 [×]` $\to 84$
180	$2^2 \times 3^2 \times 5$	`2² × 3² × 5`	`2 [x²] 3 [x²] [×] 5 [×]` $\to 180$
252	$2^2 \times 3^2 \times 7$	`2² × 3² × 7`	`2 [x²] 3 [x²] [×] 7 [×]` $\to 252$
360	$2^3 \times 3^2 \times 5$	`2³ × 3² × 5`	`2 [ENTER] 3 [g] [yˣ] 3 [x²] [×] 5 [×]` $\to 360$
504	$2^3 \times 3^2 \times 7$	`2³ × 3² × 7`	`2 [ENTER] 3 [g] [yˣ] 3 [x²] [×] 7 [×]` $\to 504$
720	$2^4 \times 3^2 \times 5$	`2⁴ × 3² × 5`	`2 [ENTER] 4 [g] [yˣ] 3 [x²] [×] 5 [×]` $\to 720$
1,000	$2^3 \times 5^3$	`2³ × 5³`	`2 [ENTER] 3 [g] [yˣ] 5 [ENTER] 3 [g] [yˣ] [×]` $\to 1000$
1,024	$2^{10}$	`2¹⁰`	`2 [ENTER] 10 [g] [yˣ]` $\to 1024$
1,729	$7 \times 13 \times 19$	`7 × 13 × 19`	`7 [ENTER] 13 [×] 19 [×]` $\to 1729$

Why Master Builders & Astronomers Love "Highly Composite" Numbers (72, 84, 180, 360, 720): Look closely at the prime factor blocks of the first seven numbers in the table above. In 1915, Srinivasa Ramanujan (whose $1,729 = 7 \times 13 \times 19$ Taxicab Number from Chapter 7 caps the bottom row) published a landmark study on what he named Highly Composite Numbers—numbers packed with small prime factors ($2, 3, 5,$ and $7$) so they can be divided evenly in more ways than any smaller number! You can actually calculate how many whole-number divisors any number has straight from the exponents in its [g] [FACT] output: add $1$ to each prime's exponent and multiply them together.

For example, your 1-Whole Fraction Tower stands $84\text{ mm}$ tall ($2^2 \times 3^1 \times 7^1$). Adding $1$ to each exponent gives $(2+1) \times (1+1) \times (1+1) = 3 \times 2 \times 2 = \mathbf{12\text{ divisors}}$ ($1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$), allowing halves, thirds, fourths, sixths, sevenths, and twelfths to land on exact whole-millimeter heights. A $360^\circ$ circle ($2^3 \times 3^2 \times 5^1$) is even richer: adding $1$ to each exponent (3 [ENTER] 1 [+] 2 [ENTER] 1 [+] [×] 1 [ENTER] 1 [+] [×] $\to 4 \times 3 \times 2$) reveals $24$ exact divisors, so a $360^\circ$ circle splits into clean whole degrees whether you divide it into $2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120,$ or $180$ equal wedges! Doubling $360$ to $720$ ($2^4 \times 3^2 \times 5^1$) yields $(4+1) \times (2+1) \times (1+1) = 5 \times 3 \times 2 = \mathbf{30\text{ divisors}}$ (which is why two full turns of a $360^\circ$ wheel or the total interior angles of a 6-sided hexagon equal $720^\circ$). And look at 1,000 ($2^3 \times 5^3$) beside 1,024 ($2^{10}$): metric decimal blocks (1,000) use three $2$-blocks and three $5$-blocks ($(3+1) \times (3+1) = 16$ divisors), whereas computer memory chips (1,024 bytes in a traditional kilobyte) stack ten pure $2$-blocks (2 [ENTER] 10 [g] [yˣ] $\to 1,024$, with $10 + 1 = 11$ power-of-two divisors).

C.3.5 GCF and LCM Reference Table for Common Number Pairs

The table below shows 9 popular number pairs, the prime blocks they share ([GCF]), the product of the two numbers, and their Least Common Multiple ([f] [LCM]):

Number Pair	Shared Prime Blocks	Greatest Common Factor (`GCF`)	First $\times$ Second	Least Common Multiple (`LCM`)
$24\text{ and }18$	$2 \times 3$	6	432	72
$36\text{ and }24$	$2 \times 2 \times 3$	12	864	72
$60\text{ and }42$	$2 \times 3$	6	2,520	420
$84\text{ and }36$	$2 \times 2 \times 3$	12	3,024	252
$105\text{ and }45$	$3 \times 5$	15	4,725	315
$120\text{ and }75$	$3 \times 5$	15	9,000	600
$144\text{ and }96$	$2^4 \times 3$	48	13,824	288
$210\text{ and }165$	$3 \times 5$	15	34,650	2,310
$252\text{ and }105$	$3 \times 7$	21	26,460	1,260

C.3.6 Why "Goldbach's Comet" Has High Ridges and Low Valleys

In Chapter 7, you met Christian Goldbach's famous 1742 letter to Leonhard Euler—observing that every even number greater than $2$ can be written as the sum of two prime numbers ($4 = 2 + 2$, $6 = 3 + 3$, $8 = 3 + 5$, $10 = 3 + 7 = 5 + 5$). When mathematicians plot how many different prime pairs add up to each even number, the dots streak across the page like a glowing comet tail (Goldbach's Comet) split into low valleys and high ridges.

Why do powers of $2$ like $8$ ($1$ pair: $3 + 5$), $16$ ($2$ pairs: $3 + 13, 5 + 11$), $32$ ($2$ pairs: $3 + 29, 13 + 19$), and $64$ ($5$ pairs) sit in the lowest valleys? Because powers of $2$ have no odd prime factors at all (they are not divisible by $3, 5,$ or $7$), nearly half of the odd primes below them are blocked from pairing up when their partners land on multiples of $3$ or $5$! Conversely, multiples of $6$ ($2 \times 3$) and $30$ ($2 \times 3 \times 5$) like $18$ ($2$ pairs), $30$ ($3$ pairs), $42$ ($4$ pairs), $60$ ($6$ pairs), and $90$ ($9$ pairs) ride the top ridge of the comet: because multiples of $6$ already divide evenly by both $2$ and $3$, subtracting any prime greater than $3$ from a multiple of $6$ is guaranteed never to land on a multiple of $2$ or $3$, giving every prime a much higher chance of meeting a prime partner!

You can test any even number on Goldbach's Comet right on your SC-6 by picking a prime candidate with [PRIME], subtracting it from your even number, and tapping [PRIME] on the partner:

Even Number	How Many Prime Pairs?	All Prime Pairs That Add to This Even Number	Why It Sits Low or High on Goldbach's Comet
4	1 pair	$2 + 2$	Only even number that uses the even prime $2$
8	1 pair	$3 + 5$	Power of $2$ ($2^3$): only one odd prime pair works
10	2 pairs	$3 + 7,\; 5 + 5$	Includes the twin-center pair $5 + 5$
16	2 pairs	$3 + 13,\; 5 + 11$	Power of $2$ ($2^4$): sits on the bottom edge of the comet
24	3 pairs	$5 + 19,\; 7 + 17,\; 11 + 13$	Multiple of $6$: three symmetric prime pairs around $12$
30	3 pairs	$7 + 23,\; 11 + 19,\; 13 + 17$	Primorial ($2 \times 3 \times 5$): every non-3, non-5 prime pairs up!
32	2 pairs	$3 + 29,\; 13 + 19$	Power of $2$ ($2^5$): drops back down to just $2$ pairs
36	4 pairs	$5 + 31,\; 7 + 29,\; 13 + 23,\; 17 + 19$	Square multiple of $6$: four distinct prime pairs
48	5 pairs	$5 + 43,\; 7 + 41,\; 11 + 37,\; 17 + 31,\; 19 + 29$	Multiple of $6$: climbs to five prime pairs
60	6 pairs	$7 + 53,\; 13 + 47,\; 17 + 43,\; 19 + 41,\; 23 + 37,\; 29 + 31$	Champion under $60$ ($2^2 \times 3 \times 5$): six distinct prime pairs!
64	5 pairs	$3 + 61,\; 5 + 59,\; 11 + 53,\; 17 + 47,\; 23 + 41$	Power of $2$ ($2^6$): fewer pairs than $60$ even though $64 > 60$
100	6 pairs	$3 + 97,\; 11 + 89,\; 17 + 83,\; 29 + 71,\; 41 + 59,\; 47 + 53$	Six pairs of two-digit primes add to an exact $100$

C.3.7 Perfect Numbers & Giant Factor Challenges

A Perfect Number is a special whole number whose smaller factors add up to the number itself. Look at the first four perfect numbers:

Perfect Number	Prime Recipe	Smaller Factors Added Together
6	$2 \times 3$	$1 + 2 + 3 = \mathbf{6}$
28	$4 \times 7$	$1 + 2 + 4 + 7 + 14 = \mathbf{28}$
496	$16 \times 31$	$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = \mathbf{496}$
8,128	$64 \times 127$	$1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + \dots = \mathbf{8,128}$

You can build the third perfect number ($496$) right on the stack by following Euclid's ancient recipe: multiply $2^4$ ($16$) by the prime number $2^5 - 1$ ($31$) to reach $16 \times 31 = 496$:

Keystroke	Register Z	Register Y	Register X	What Happens on the Stack
`2 [ENTER]`	0	2	2	Enter base 2 and copy onto Shelf $Y$
`5 [g] [yˣ]`	0	0	32	Raise $2^5 = 32$
`1 [-]`	0	0	31	Subtract 1 for Mersenne candidate $2^5 - 1 = 31$
`[PRIME]`	0	0	31	Screen flashes `PRIME` to confirm that $31$ is a prime number
`2 [ENTER]`	31	2	2	Enter base 2; $31$ lifts to Shelf $Z$
`4 [g] [yˣ]`	0	31	16	Raise $2^4 = 16$; $31$ drops to Shelf $Y$
`[×]`	0	0	496	Multiply $31 \times 16 =$ 496 (the third perfect number)

The display lands on $496$, the third perfect number. That same curiosity about giant powers of two led to one of history's most famous division checks: in 1640, Pierre de Fermat guessed that $2^{32} + 1 = 4,294,967,297$ was a prime number, and for nearly a hundred years no one could factor it—until Leonhard Euler proved in 1732 that $641$ divides into it evenly with zero remainder: $$4,294,967,297 = 641 \times 6,700,417$$ You can test Euler's discovery right on your SC-6 by keying in 4294967297 [ENTER] 641 [÷], which lands cleanly on $6,700,417.00$ with no decimal remainder.

C.4 Everyday Rates, Money & Shopping Word Rules

Whether you are comparing unit prices in a grocery store aisle, computing average speed on a family road trip, or adding sales tax at a register, the SC-6 stack keeps your starting numbers visible at every step:

Everyday Math Task	What You Know	Word Rule	SC-6 Keystroke Pattern
Speed (Unit Rate)	Distance, Time	$\text{Distance} \div \text{Time}$	`[Distance] [ENTER] [Time] [÷]`
Travel Distance	Speed, Time	$\text{Speed} \times \text{Time}$	`[Speed] [ENTER] [Time] [×]`
Travel Time	Distance, Speed	$\text{Distance} \div \text{Speed}$	`[Distance] [ENTER] [Speed] [÷]`
Unit Price (Best Buy)	Total Cost, Count	$\text{Total Cost} \div \text{Count}$	`[Total Cost] [ENTER] [Count] [÷] [$]`
Density of an Object	Mass (g), Volume ($\text{cm}^3$)	$\text{Mass} \div \text{Volume}$	`[Mass] [ENTER] [Volume] [÷]`
Percent of a Number	Whole, Percent	$\text{Whole} \times (\text{Percent} \div 100)$	`[Whole] [ENTER] [Percent] [%]`
Percent Change	Starting, Ending	$(\text{Ending} - \text{Starting}) \div \text{Starting} \times 100$	`[Starting] [ENTER] [Ending] [g] [%CHG]`

Look closely at the first five rows of that table and you will spot a wonderful shortcut: Speed ($\text{miles} \div \text{hours}$), Unit Price ($\text{dollars} \div \text{ounces}$), and Density ($\text{grams} \div \text{cm}^3$) are all the exact same mathematical idea! Every time you see the word "per" in everyday life—miles per hour, dollars per pound, heartbeats per minute, or grams per cubic centimeter—the word "per" simply means "divide the total on Shelf $Y$ by the number of units on Shelf $X$" ([÷]):

Finding the Best Buy in the Grocery Store Aisle: Suppose a $12\text{-ounce}$ box of cereal costs $\$4.56$, while a giant $18\text{-ounce}$ family box of the same cereal costs $\$6.30$. Which box gives you more cereal for each dollar?
Key in the first box's price and ounces (4.56 [ENTER] 12 [÷]): Shelf $X$ shows 0.38 ($38\text{ cents per ounce}$).
Now key in the family box right on top (6.30 [ENTER] 18 [÷]): Shelf $X$ shows 0.35 ($35\text{ cents per ounce}$), while the first box's 0.38 waits right above it on Shelf $Y$ so you can compare both unit prices side by side! Subtracting [-] confirms that the family box saves $3\text{ cents per ounce}$ ($\$0.54$ saved across the $18\text{-ounce}$ box).
Combining Road-Trip Speed and Clock Time ([g] [▸HMS]): Suppose your family is driving $210\text{ miles}$ to a state park at a steady highway speed of $60\text{ miles per hour}$. Dividing 210 [ENTER] 60 [÷] gives 3.5 hours in decimal time—and one tap of [g] [▸HMS] ([g] then [🕒]) translates 3.5 decimal hours directly into 3.3000 ($3\text{ hours, } 30\text{ minutes}$)!

C.4.2 Adding Tax, Tips & Discounts in One Tap

Because [%] on the SC-6 computes the percentage on Ground Shelf $X$ while keeping your original dollar amount untouched on Shelf $Y$, you never have to retype a price to add sales tax or subtract a sale discount:

Money Task	Word Rule	SC-6 Keystroke Pattern
Simple Interest Earned	$\text{Starting Money} \times \text{Yearly Rate \%} \times \text{Years}$	`[Money] [ENTER] [Rate] [%] [Years] [×] [$]`
Sales Tax Amount	$\text{Price} \times \text{Tax Rate \%}$	`[Price] [ENTER] [Tax Rate] [%] [$]`
Total Price with Tax	$\text{Price} + \text{Sales Tax}$	`[Price] [ENTER] [Tax Rate] [%] [+] [$]`
Discount Saved	$\text{Regular Price} \times \text{Discount \%}$	`[Price] [ENTER] [Discount] [%] [$]`
Sale Price After Discount	$\text{Regular Price} - \text{Discount}$	`[Price] [ENTER] [Discount] [%] [-] [$]`
Restaurant Tip ($18\%$)	$\text{Bill} \times 18\%$	`[Bill] [ENTER] 18 [%] [$]`
Splitting a Bill Equally	$(\text{Bill} + \text{Tip}) \div \text{Diners}$	`[Bill] [ENTER] 18 [%] [+] [Diners] [÷] [$]`

Even two-step store sales—like taking a $20\%$ coupon off a $\$45.00$ backpack and then adding $8\%$ sales tax at the register—flow in one unbroken chain without writing down a single intermediate number: 1. Key in 45 [ENTER] 20 [%] to see the 9.00 discount on Shelf $X$ while 45 waits on Shelf $Y$, then tap [-] to land on the 36.00 sale price. 2. Now type 8 [%] (2.88 tax on Shelf $X$, 36.00 sale price on Shelf $Y$), tap [+], and tap [$] to round cleanly to 38.88 at the checkout counter!

When money stays in a savings account for several years, the interest itself starts earning interest! For example, to watch a $\$500$ savings account grow at $4\%$ interest compounded yearly for $3$ years, multiply $\$500$ by $1.04$ three times ($1.04^3 \times 500 = 562.43$ dollars):

Keystroke	Register Y	Register X	What Happens on the Stack
`1.04`	0	1.04	Enter yearly growth multiplier ($1 + 0.04 = 1.04$)
`[ENTER]`	1.04	1.04	Copy $1.04$ up onto Shelf $Y$
`3`	1.04	3	Enter 3 years of compounding
`[g] [yˣ]`	0	1.124864	Raise $1.04^3 = 1.124864$
`500`	1.124864	500	Enter starting savings ($\$500$); multiplier lifts to $Y$
`[×]`	0	562.43	Multiply by $\$500$ for $562.43 total savings

After $3$ years, the account holds $\$562.43$ total savings ($\$62.43$ in interest earned).

C.5 Annotated Bibliography & Primary Mathematical Sources

Every numbered reference across this volume links to a classic book, historical source, or mathematical study in the Annotated Bibliography below. If a story or puzzle in this book sparked your curiosity, several unforgettable books from our bibliography are waiting for you at your school or public library. For puzzle lovers and aspiring math magicians, Martin Gardner's The Colossal Book of Short Puzzles and Problems (Gardner, 1959) and George Pólya's How to Solve It (Pólya, 1945) show how drawing a picture, testing a smaller case, or working backward can crack seemingly impossible riddles. For young engineers and machine builders, David Macaulay's The Way Things Work Now (Macaulay, 2016) and Randall Munroe's What If? (Munroe, 2014) open up gears, levers, pendulums, and wild physics questions with playful illustrations that pair naturally with Chapter 6 and Chapter 9, while Simon Singh's The Code Book (Singh, 1997) tells the true history of secret ciphers from ancient Rome to modern RSA prime-number internet security.

To explore the history and beauty of everyday math, look for Steven Strogatz's The Joy of x (Strogatz, 2012), Jordan Ellenberg's How Not to Be Wrong (Ellenberg, 2014), and Georges Ifrah's The Universal History of Numbers (Ifrah, 2000), which trace how human beings use place value, fractions, geometry, and probability to understand the natural world. And for parents, teachers, and math coaches, Jerome Bruner's Toward a Theory of Instruction (Bruner, 1966), Liping Ma's Knowing and Teaching Elementary Mathematics (Ma, 1999), and Zoltan Dienes's Building Up Mathematics (Dienes, 1960) explain why building with physical blocks before moving to visual diagrams and symbols builds deep, lasting mathematical confidence.

C.5.1 History of Stack Notation & Pocket Calculators

[lukasiewicz1924] Jan Łukasiewicz (1929) — Elementy logiki matematycznej (Elements of Mathematical Logic; parenthesis-free notation devised 1924). Komisja Wydawnicza Koła Matematyczno-Fizycznego Słuchaczów Uniwersytetu Warszawskiego (Warsaw; English trans. Pergamon Press, 1963).

Annotation: Introduces parenthesis-free prefix and postfix notation (devised in 1924 and published in his 1929 Warsaw logic treatise), demonstrating that operator position alone uniquely determines evaluation order without grouping symbols.
Cited in Chapters: App C, Ch 02

[burks1954] Arthur W. Burks, Don W. Warren & Jesse B. Wright (1954) — An Analysis of a Logical Machine Using Parenthesis-Free Notation. Mathematical Tables and Other Aids to Computation, Vol. 8, No. 46, pp. 53–57.

Annotation: Formalizes Polish parenthesis-free postfix notation for electronic digital computers, laying the theoretical foundation for hardware-evaluated operational stacks.
Cited in Chapters: App B

[hamrick1980] Kathy B. Hamrick (1996) — The History of the Hand-Held Electronic Calculator. The American Mathematical Monthly, Vol. 103, No. 8, pp. 633–639.

Annotation: Traces the historical evolution of Polish notation from abstract Warsaw mathematical logic to Corvallis handheld scientific computing.
Cited in Chapters: App B, App C

C.5.2 Classic Discoveries in Mathematics & Physics

[napier1614] John Napier (1614) — Mirifici Logarithmorum Canonis Descriptio. Andrew Hart, Edinburgh.

Annotation: The invention of logarithms, transforming laborious multi-digit multiplication into addition and doubling the working lifespan of astronomers.
Cited in Chapters: Ch 05

[galilei1638] Galileo Galilei (1638) — Discorsi e Dimostrazioni Matematiche, intorno à due nuoue scienze (Dialogues Concerning Two New Sciences). Louis Elsevier, Leiden.

Annotation: Establishes the geometric Square-Cube Law governing surface-area-to-volume scaling in structural mechanics and biological thermoregulation, as well as the isochronous law of the pendulum.
Cited in Chapters: Ch 06, Ch 09

[gauss1809] Carl Friedrich Gauss (1809) — Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. Perthes and Besser, Hamburg.

Annotation: Introduces the Method of Least Squares and the Gaussian Normal error distribution, recovering the lost celestial orbit of Ceres.
Cited in Chapters: Ch 04, Ch 07

[kahan1979] William M. Kahan (1979) — Personal Calculator Has Key to Solve Any Equation f(x) = 0. Hewlett-Packard Journal, Vol. 30, No. 12, pp. 20–26.

Annotation: Establishes the non-destructive base-preserving percentage invariant and foundational floating-point error bounds for handheld solvers.
Cited in Chapters: App B

C.5.3 Mathematics, Nature & Popular Nonfiction

[sartorius1856] Wolfgang Sartorius von Waltershausen (1856) — Gauss zum Gedächtniss. S. Hirzel, pp. 12–13, Leipzig.

Annotation: Primary biographical source recording young Carl Friedrich Gauss's arithmetic series solution in J.G. Büttner's Brunswick classroom and his Low German exclamation 'Ligget se!' ('There it lies!').
Cited in Chapters: Ch 04

[gardner1959] Martin Gardner (1959) — The Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster, New York.

Annotation: Pioneered popular mathematical recreations, flexagons, hexaflexagons, and spatial puzzles that celebrate mathematical play.
Cited in Chapters: App C, Ch 07

[gould1977] Stephen Jay Gould (1977) — Ever Since Darwin: Reflections in Natural History. W. W. Norton & Company, New York.

Annotation: Analyzes the evolutionary biology of 13-year and 17-year periodical cicadas (Magicicada) and prime-number predator avoidance cycles.
Cited in Chapters: Ch 07

[singh1997] Simon Singh (1997) — Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. Walker & Company, New York.

Annotation: Captures the three-century human quest to prove Fermat's Last Theorem, conveying the passion and stakes of mathematical discovery.
Cited in Chapters: App C, Ch 07

[ifrah2000] Georges Ifrah (2000) — The Universal History of Numbers: From Prehistory to the Invention of the Computer. John Wiley & Sons, New York.

Annotation: Comprehensive historical account of tally stones, Mesopotamian sexagesimal accounting, Egyptian hieroglyphic numerals, and the independent invention of zero and positional place value in India and Mesoamerica.
Cited in Chapters: App C, Ch 03

[webb2001] Glenn F. Webb (2001) — The Prime Number Periodical Cicada Problem. Discrete and Continuous Dynamical Systems - Series B, Vol. 1, No. 3, pp. 387–399.

Annotation: Mathematical population model proving that 13-year and 17-year prime emergence periods minimize resonance with multi-year predator-prey cycles.
Cited in Chapters: Ch 07

[gilbert2006] Caroline Gilbert, Graham Robertson, Yvon Le Maho, Yasuhiko Naito & André Ancel (2006) — Huddling behavior in emperor penguins: Dynamics of huddling. Physiology & Behavior, Vol. 88, No. 4-5, pp. 479–488.

Annotation: Quantifies how Emperor penguins (Aptenodytes forsteri) reduce exposed windward surface area per bird to survive Antarctic winter blizzards.
Cited in Chapters: Ch 06

[strogatz2012] Steven Strogatz (2012) — The Joy of x: A Guided Tour of Math, from One to Infinity. Eamon Dolan / Houghton Mifflin Harcourt, Boston.

Annotation: Guides the reader through arithmetic, algebra, geometry, and calculus using everyday curiosity and intuitive geometric insight.
Cited in Chapters: App C, Ch 06, Foreword

[ellenberg2014] Jordan Ellenberg (2014) — How Not to Be Wrong: The Power of Mathematical Thinking. Penguin Press, New York.

Annotation: Exposes common cognitive biases, non-linear scaling traps, and statistical survivorship fallacies with conversational wit.
Cited in Chapters: App C

[munroe2014] Randall Munroe (2014) — What If?: Serious Scientific Answers to Absurd Hypothetical Questions. Houghton Mifflin Harcourt, Boston.

Annotation: Applies order-of-magnitude dimensional analysis, physics Fermi approximations, and creative mathematical modeling to wild scenarios.
Cited in Chapters: App C

[macaulay2016] David Macaulay (2016) — The Way Things Work Now. HMH Books for Young Readers, Boston.

Annotation: Visualizes mechanical physical mechanisms and gear systems through step-by-step illustrations, inspiring the SC-6 tactile manipulative studio design.
Cited in Chapters: App A, App C, Ch 01, Ch 05, Foreword

[parker2019] Matt Parker (2019) — Humble Pi: A Comedy of Maths Errors. Riverhead Books, New York.

Annotation: Explores famous engineering catastrophes and numerical bugs caused by round-off errors, integer overflows, and unit conversion failures.
Cited in Chapters: Ch 06

C.5.4 Hands-On Mathematics Education & Curriculum Standards

[polya1945] George Pólya (1945) — How to Solve It: A New Aspect of Mathematical Method. Princeton University Press, Princeton, NJ.

Annotation: Defines the heuristic problem-solving framework (Understand, Plan, Carry Out, Look Back), aligned with step-by-step RPN stack inspection.
Cited in Chapters: App B, App C

[miller1956] George A. Miller (1956) — The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information. Psychological Review, Vol. 63, No. 2, pp. 81–97.

Annotation: Identifies the 7 ± 2 item capacity limit of human short-term working memory, validating the 4-register stack as the optimal cognitive span.
Cited in Chapters: App B

[dienes1960] Zoltan P. Dienes (1960) — Building Up Mathematics. Hutchinson Educational, London.

Annotation: Introduces Multibase Arithmetic Blocks and the Dynamic Principle of mathematical learning, grounding the SC-6 Place Value Studio.
Cited in Chapters: App A, App B, App C

[bruner1966] Jerome S. Bruner (1966) — Toward a Theory of Instruction. Harvard University Press, Cambridge, MA.

Annotation: Establishes the developmental progression from physical action to visual imagery and symbolic notation underpinning the StackCalc manipulative studios and display modes.
Cited in Chapters: App A, App B, App C

[sweller1988] John Sweller (1988) — Cognitive Load During Problem Solving: Effects on Learning. Cognitive Science, Vol. 12, No. 2, pp. 257–285.

Annotation: Establishes Cognitive Load Theory, demonstrating that parentheses and operator precedence buffering in algebraic calculators impose extraneous cognitive load.
Cited in Chapters: App B

[falkner1999] Karen P. Falkner, Thomas P. Carpenter & Elizabeth Fennema (1999) — Children's Understanding of Equality: A Foundation for Algebra. Teaching Children Mathematics, Vol. 6, No. 4, pp. 232–236.

Annotation: Landmark empirical study demonstrating that elementary students misinterpret the equals sign (=) as a command to execute an operation rather than relational equivalence.
Cited in Chapters: App B

[ma1999] Liping Ma (1999) — Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Lawrence Erlbaum Associates, Mahwah, NJ.

Annotation: Demonstrates Profound Understanding of Fundamental Mathematics (PUFM) through transparent conceptual operations and division with remainders.
Cited in Chapters: App B, App C

[ccssm2010] National Governors Association Center for Best Practices & Council of Chief State School Officers (2010) — Common Core State Standards for Mathematics. National Governors Association & CCSSO, Washington, DC.

Annotation: The foundational curriculum standards framework governing K-12 operations, algebraic thinking, number sense, and geometry in US classrooms.
Cited in Chapters: App B

[singapore2014] Ministry of Education Singapore (2014) — Primary Mathematics Teaching and Learning Syllabus. Curriculum Planning and Development Division, Singapore.

Annotation: The Singapore Math framework emphasizing bar modeling, hands-on block-to-symbol transitions, and relational understanding in fractions and ratios.
Cited in Chapters: App B

C.5.5 Complete BibTeX Catalog (StackCalc SC-6 Elementary)

@book{lukasiewicz1924,
  author    = {Łukasiewicz, Jan},
  title     = {Elementy logiki matematycznej (Elements of Mathematical Logic; parenthesis-free notation devised 1924)},
  publisher = {Komisja Wydawnicza Koła Matematyczno-Fizycznego Słuchaczów Uniwersytetu Warszawskiego (Warsaw; English trans. Pergamon Press, 1963)},
  year      = {1929},
  note      = {Introduces parenthesis-free prefix and postfix notation (devised in 1924 and published in his 1929 Warsaw logic treatise), demonstrating that operator position alone uniquely determines evaluation order without grouping symbols.}
}

@article{burks1954,
  author    = {Burks, Arthur W. and Warren, Don W. and Wright, Jesse B.},
  title     = {An Analysis of a Logical Machine Using Parenthesis-Free Notation},
  journal   = {Mathematical Tables and Other Aids to Computation},
  volume    = {8},
  number    = {46},
  pages     = {53--57},
  year      = {1954},
  note      = {Formalizes Polish parenthesis-free postfix notation for electronic digital computers, laying the theoretical foundation for hardware-evaluated operational stacks.}
}

@article{hamrick1980,
  author    = {Hamrick, Kathy B.},
  title     = {The History of the Hand-Held Electronic Calculator},
  journal   = {The American Mathematical Monthly},
  volume    = {103},
  number    = {8},
  pages     = {633--639},
  year      = {1996},
  note      = {Traces the historical evolution of Polish notation from abstract Warsaw mathematical logic to Corvallis handheld scientific computing.}
}

@book{napier1614,
  author    = {Napier, John},
  title     = {Mirifici Logarithmorum Canonis Descriptio},
  publisher = {Andrew Hart},
  address   = {Edinburgh},
  year      = {1614},
  note      = {The invention of logarithms, transforming laborious multi-digit multiplication into addition and doubling the working lifespan of astronomers.}
}

@book{gauss1809,
  author    = {Gauss, Carl Friedrich},
  title     = {Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium},
  publisher = {Perthes and Besser},
  address   = {Hamburg},
  year      = {1809},
  note      = {Introduces the Method of Least Squares and the Gaussian Normal error distribution, recovering the lost celestial orbit of Ceres.}
}

@article{kahan1979,
  author    = {Kahan, William M.},
  title     = {Personal Calculator Has Key to Solve Any Equation f(x) = 0},
  journal   = {Hewlett-Packard Journal},
  volume    = {30},
  number    = {12},
  pages     = {20--26},
  year      = {1979},
  note      = {Establishes the non-destructive base-preserving percentage invariant and foundational floating-point error bounds for handheld solvers.}
}

@book{strogatz2012,
  author    = {Strogatz, Steven},
  title     = {The Joy of x: A Guided Tour of Math, from One to Infinity},
  publisher = {Eamon Dolan / Houghton Mifflin Harcourt},
  address   = {Boston},
  year      = {2012},
  isbn      = {978-0547517650},
  note      = {Opens every mathematical section with everyday curiosity and intuitive aha-moments, showing how mathematical beauty connects to human experience.}
}

@book{ellenberg2014,
  author    = {Ellenberg, Jordan},
  title     = {How Not to Be Wrong: The Power of Mathematical Thinking},
  publisher = {Penguin Press},
  address   = {New York},
  year      = {2014},
  isbn      = {978-1594205224},
  note      = {Exposes common cognitive biases, non-linear scaling traps, and statistical survivorship fallacies with conversational wit.}
}

@book{parker2019,
  author    = {Parker, Matt},
  title     = {Humble Pi: A Comedy of Maths Errors},
  publisher = {Riverhead Books},
  address   = {New York},
  year      = {2019},
  isbn      = {978-0593084694},
  note      = {Explores famous engineering catastrophes and numerical bugs caused by round-off errors, integer overflows, and unit conversion failures.}
}

@book{munroe2014,
  author    = {Munroe, Randall},
  title     = {What If?: Serious Scientific Answers to Absurd Hypothetical Questions},
  publisher = {Houghton Mifflin Harcourt},
  address   = {Boston},
  year      = {2014},
  isbn      = {978-0544272996},
  note      = {Applies order-of-magnitude dimensional analysis, physics Fermi approximations, and creative mathematical modeling to wild scenarios.}
}

@book{macaulay2016,
  author    = {Macaulay, David},
  title     = {The Way Things Work Now},
  publisher = {HMH Books for Young Readers},
  address   = {Boston},
  year      = {2016},
  isbn      = {978-0544824386},
  note      = {Visualizes mechanical physical mechanisms and gear systems through step-by-step illustrations, inspiring the SC-6 tactile manipulative studio design.}
}

@book{singh1997,
  author    = {Singh, Simon},
  title     = {Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem},
  publisher = {Walker & Company},
  address   = {New York},
  year      = {1997},
  isbn      = {978-0385493628},
  note      = {Captures the three-century human quest to prove Fermat's Last Theorem, conveying the passion and stakes of mathematical discovery.}
}

@book{gardner1959,
  author    = {Gardner, Martin},
  title     = {The Scientific American Book of Mathematical Puzzles and Diversions},
  publisher = {Simon & Schuster},
  address   = {New York},
  year      = {1959},
  note      = {Pioneered popular mathematical recreations, flexagons, hexaflexagons, and spatial puzzles that celebrate mathematical play.}
}

@article{miller1956,
  author    = {Miller, George A.},
  title     = {The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information},
  journal   = {Psychological Review},
  volume    = {63},
  number    = {2},
  pages     = {81--97},
  year      = {1956},
  note      = {Identifies the 7 ± 2 item capacity limit of human short-term working memory, validating the 4-register stack as the optimal cognitive span.}
}

@book{bruner1966,
  author    = {Bruner, Jerome S.},
  title     = {Toward a Theory of Instruction},
  publisher = {Harvard University Press},
  address   = {Cambridge, MA},
  year      = {1966},
  note      = {Establishes the developmental progression from physical action to visual imagery and symbolic notation underpinning the StackCalc manipulative studios and display modes.}
}

@book{dienes1960,
  author    = {Dienes, Zoltan P.},
  title     = {Building Up Mathematics},
  publisher = {Hutchinson Educational},
  address   = {London},
  year      = {1960},
  note      = {Introduces Multibase Arithmetic Blocks and the Dynamic Principle of mathematical learning, grounding the SC-6 Place Value Studio.}
}

@book{polya1945,
  author    = {Pólya, George},
  title     = {How to Solve It: A New Aspect of Mathematical Method},
  publisher = {Princeton University Press},
  address   = {Princeton, NJ},
  year      = {1945},
  note      = {Defines the heuristic problem-solving framework (Understand, Plan, Carry Out, Look Back), aligned with step-by-step RPN stack inspection.}
}

@article{sweller1988,
  author    = {Sweller, John},
  title     = {Cognitive Load During Problem Solving: Effects on Learning},
  journal   = {Cognitive Science},
  volume    = {12},
  number    = {2},
  pages     = {257--285},
  year      = {1988},
  note      = {Establishes Cognitive Load Theory, demonstrating that parentheses and operator precedence buffering in algebraic calculators impose extraneous cognitive load.}
}

@book{ma1999,
  author    = {Ma, Liping},
  title     = {Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States},
  publisher = {Lawrence Erlbaum Associates},
  address   = {Mahwah, NJ},
  year      = {1999},
  note      = {Demonstrates Profound Understanding of Fundamental Mathematics (PUFM) through transparent conceptual operations and division with remainders.}
}

@techreport{singapore2014,
  author      = {{Ministry of Education Singapore}},
  title       = {Primary Mathematics Teaching and Learning Syllabus},
  institution = {Curriculum Planning and Development Division},
  address     = {Singapore},
  year        = {2014},
  note        = {The Singapore Math framework emphasizing bar modeling, Concrete-Pictorial-Abstract transitions, and relational understanding in fractions and ratios.}
}

@techreport{ccssm2010,
  author      = {{National Governors Association Center for Best Practices and Council of Chief State School Officers}},
  title       = {Common Core State Standards for Mathematics},
  institution = {NGA Center for Best Practices and CCSSO},
  address     = {Washington, DC},
  year        = {2010},
  note        = {The foundational curriculum standards framework governing K-12 operations, algebraic thinking, number sense, and geometry in US classrooms.}
}

@article{falkner1999,
  author = {Karen P. Falkner and Thomas P. Carpenter and Elizabeth Fennema},
  title = {Children's Understanding of Equality: A Foundation for Algebra},
  year = {1999},
  journal = {Teaching Children Mathematics},
  volume = {6},
  number = {4},
  pages = {232--236},
}

@book{ifrah2000,
  author = {Georges Ifrah},
  title = {The Universal History of Numbers: From Prehistory to the Invention of the Computer},
  year = {2000},
  publisher = {John Wiley & Sons},
  address = {New York},
  isbn = {978-0471393405},
}

@book{sartorius1856,
  author = {Wolfgang Sartorius von Waltershausen},
  title = {Gauss zum Gedächtniss},
  year = {1856},
  pages = {12--13},
  publisher = {S. Hirzel},
  address = {Leipzig},
}

@book{galilei1638,
  author = {Galileo Galilei},
  title = {Discorsi e Dimostrazioni Matematiche, intorno à due nuoue scienze (Dialogues Concerning Two New Sciences)},
  year = {1638},
  publisher = {Louis Elsevier},
  address = {Leiden},
}

@article{gilbert2006,
  author = {Caroline Gilbert and Graham Robertson and Yvon Le Maho and Yasuhiko Naito and André Ancel},
  title = {Huddling behavior in emperor penguins: Dynamics of huddling},
  year = {2006},
  journal = {Physiology & Behavior},
  volume = {88},
  number = {4-5},
  pages = {479--488},
}

@book{gould1977,
  author = {Stephen Jay Gould},
  title = {Ever Since Darwin: Reflections in Natural History},
  year = {1977},
  publisher = {W. W. Norton & Company},
  address = {New York},
  isbn = {978-0393308181},
}

@article{webb2001,
  author = {Glenn F. Webb},
  title = {The Prime Number Periodical Cicada Problem},
  year = {2001},
  journal = {Discrete and Continuous Dynamical Systems - Series B},
  volume = {1},
  number = {3},
  pages = {387--399},
}

For detailed hands-on lesson plans implementing these rules in classroom workshop stations, see Appendix B: Teacher Lesson Plans & Station Guide. For 3D-printing and cardstock cutouts of the hands-on block kits, see Appendix A: Hands-On Block Studio Guide & How We Built StackCalc.

Perfect Number	Prime Recipe	Smaller Factors Added Together
6	\(2 \times 3\)	\(1 + 2 + 3 = \mathbf{6}\)
28	\(4 \times 7\)	\(1 + 2 + 4 + 7 + 14 = \mathbf{28}\)
496	\(16 \times 31\)	\(1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = \mathbf{496}\)
8,128	\(64 \times 127\)	\(1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + \dots = \mathbf{8,128}\)

Math Rule	Rule in Words & Numbers	Physical Block Model	Chapter Link	SC-6 Stack Check (`Example`)
Turn-Around Rule (Addition)	\(\text{First} + \text{Second} = \text{Second} + \text{First}\) (\(3 + 5 = 5 + 3\))	Sliding Number Rack beads left vs. right	Chapter 4	`3 [ENTER] 5 [+]` equals `5 [ENTER] 3 [+]`
Turn-Around Rule (Multiplication)	\(\text{Rows} \times \text{Columns} = \text{Columns} \times \text{Rows}\) (\(4 \times 6 = 6 \times 4\))	Turning a 24-tile rectangle \(90^\circ\) on the desk	Chapter 4	`4 [ENTER] 6 [×]` equals `6 [ENTER] 4 [×]`
Grouping Rule (Addition)	\((\text{First} + \text{Second}) + \text{Third} = \text{First} + (\text{Second} + \text{Third})\)	Joining 3 color trains of snap cubes	Chapter 4	`2 [ENTER] 3 [+] 4 [+]`
Grouping Rule (Multiplication)	\((\text{Length} \times \text{Width}) \times \text{Height}\) in any order	Stacking 3D layers of unit cubes	Chapter 4	`2 [ENTER] 3 [×] 4 [×]`
Split-Apart Rule	\(4 \times (10 + 3) = (4 \times 10) + (4 \times 3)\)	Splitting a 4-color tile rectangle along a color seam	Chapter 4	`10 [ENTER] 3 [+] 4 [×]`
Zero and One Rules	Adding \(0\) or multiplying by \(1\) leaves a number unchanged	Adding zero tiles; taking 1 group of tiles	Chapter 4	`7 [ENTER] 0 [+]` \(\to 7\); `7 [ENTER] 1 [×]` \(\to 7\)
Division with Remainder Check	\(\text{Total} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}\)	Sharing counters equally into cups with leftovers	Chapter 4	`29 [ENTER] 4 [g] [÷R]` \(\to\) Remainder `1` in \(Y\), Quotient `7` in \(X\)
GCF and LCM Partner Rule	\(\text{GCF} \times \text{LCM} = \text{First Number} \times \text{Second Number}\)	Comparing Prime Factor Studio towers side by side	Chapter 7	`12 [ENTER] 18 [GCF] 12 [ENTER] 18 [f] [LCM] [×]` \(\to 216\)
Fraction-to-Decimal Bridge	\(\text{Top Number} \div \text{Bottom Number} = \text{Decimal}\)	Comparing Fraction Towers to the \(1\text{-Whole}\) tower	Chapter 5	`3 [ENTER] 4 [÷]` \(\to 0.75\)

Flat Shape	Word Rule	Chapter Link	SC-6 RPN Keystroke Sequence	Stack Shelves Used
Square (Perimeter)	\(\text{Perimeter} = 4 \times \text{Side}\)	Chapter 6	`[Side] [ENTER] 4 [×]`	\(Y, X\)
Square (Area)	\(\text{Area} = \text{Side} \times \text{Side}\)	Chapter 6	`[Side] [x²]`	\(X\)
Rectangle (Perimeter)	\(\text{Perimeter} = 2 \times (\text{Length} + \text{Width})\)	Chapter 6	`[Length] [ENTER] [Width] [+] 2 [×]`	\(Y, X\)
Rectangle (Area)	\(\text{Area} = \text{Length} \times \text{Width}\)	Chapter 6	`[Length] [ENTER] [Width] [f] [RECT]`	\(Y, X\)
Triangle (Perimeter)	\(\text{Perimeter} = \text{Side 1} + \text{Side 2} + \text{Side 3}\)	Chapter 6	`[Side 1] [ENTER] [Side 2] [+] [Side 3] [+]`	\(Z, Y, X\)
Triangle (Area)	\(\text{Area} = (\text{Base} \times \text{Height}) \div 2\)	Chapter 6	`[Base] [ENTER] [Height] [×] 2 [÷]`	\(Y, X\)
Parallelogram (Area)	\(\text{Area} = \text{Base} \times \text{Height}\)	Chapter 6	`[Base] [ENTER] [Height] [×]`	\(Y, X\)
Trapezoid (Area)	\(\text{Area} = (\text{Top} + \text{Bottom}) \div 2 \times \text{Height}\)	Chapter 6	`[Top] [ENTER] [Bottom] [+] 2 [÷] [Height] [×]`	\(Z, Y, X\)
Circle (Circumference)	\(\text{Distance Around} = \pi \times \text{Width (Diameter)}\)	Chapter 6	`[Diameter] [π] [×]`	\(Y, X\)
Circle (Area)	\(\text{Area} = \pi \times \text{Radius} \times \text{Radius}\)	Chapter 6	`[Radius] [x²] [π] [×]`	\(Y, X\)

Keystroke	Register Y	Register X	What Happens on the Stack
`14`	0	14	Enter the \(14\text{ cm}\) triangle base onto Shelf \(X\)
`[ENTER]`	14	14	Copy the \(14\text{ cm}\) base up onto Shelf \(Y\)
`9`	14	9	Enter the \(9\text{ cm}\) height onto Shelf \(X\)
`[×]`	0	126	Multiply base by height (\(14 \times 9 = 126\text{ cm}^2\) rectangle)
`2`	126	2	Type 2 to halve the rectangle; \(126\) lifts to Shelf \(Y\)
`[÷]`	0	63	Divide by 2 for the triangle area: 63 cm²

Keystroke	Register Y	Register X	What Happens on the Stack
`8`	0	8	Enter the \(8\text{ in}\) top width onto Shelf \(X\)
`[ENTER]`	8	8	Copy \(8\text{ in}\) up onto Shelf \(Y\)
`14`	8	14	Enter the \(14\text{ in}\) bottom width onto Shelf \(X\)
`[+]`	0	22	Add top and bottom widths (\(8 + 14 = 22\text{ in}\))
`2`	22	2	Type 2 to average the two widths
`[÷]`	0	11	Divide by 2 for the middle width (\(11\text{ in}\))
`6`	11	6	Enter the \(6\text{ in}\) straight-up height
`[×]`	0	66	Multiply middle width by height: 66 sq in

Keystroke	Register Y	Register X	What Happens on the Stack
`3.5`	0	3.5	Enter the \(3.5\text{ m}\) radius onto Shelf \(X\)
`[x²]`	0	12.25	Square the radius (\(3.5 \times 3.5 = 12.25\text{ m}^2\))
`[π]`	12.25	3.1415926	Enter \(\pi\); \(12.25\) lifts to Shelf \(Y\)
`[×]`	0	38.484510	Multiply by \(\pi\) for circle area: 38.48 m²

Keystroke	Register Y	Register X	What Happens on the Stack
`14`	0	14	Enter the \(14\text{ m}\) garden length onto Shelf \(X\)
`[ENTER]`	14	14	Copy \(14\text{ m}\) up onto Shelf \(Y\)
`9`	14	9	Enter the \(9\text{ m}\) garden width onto Shelf \(X\)
`[+]`	0	23	Add length and width for the \(23\text{ m}\) half-lap
`2`	23	2	Type 2 to double the half-lap
`[×]`	0	46	Multiply by 2 for the full perimeter: 46 m

3D Solid Shape	Volume Word Rule	Total Surface Area Word Rule	Chapter Link	SC-6 Volume Keystrokes
Cube	\(\text{Volume} = \text{Side} \times \text{Side} \times \text{Side}\)	\(\text{Surface Area} = 6 \times (\text{Side} \times \text{Side})\)	Chapter 6	`[Side] [ENTER] 3 [g] [yˣ]`
Rectangular Box (Prism)	\(\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\)	Add all \(6\) rectangular faces (\(2\text{ Top/Bottom} + 2\text{ Front/Back} + 2\text{ Sides}\))	Chapter 6	`[Length] [ENTER] [Width] [×] [Height] [×]`
Triangular Tent (Prism)	\(\text{Volume} = (\text{Base} \times \text{Height} \div 2) \times \text{Length}\)	Add the \(2\) triangle ends plus the \(3\) rectangular side panels	Chapter 6	`[Base] [ENTER] [Height] [×] 2 [÷] [Length] [×]`
Cylinder (Can)	\(\text{Volume} = (\pi \times \text{Radius}^2) \times \text{Height}\)	\(\text{Curved Side} \ (2 \times \pi \times \text{Radius} \times \text{Height}) + 2\text{ Circle Ends} \ (2 \times \pi \times \text{Radius}^2)\)	Chapter 6	`[Radius] [x²] [π] [×] [Height] [×]`

Keystroke	Register Y	Register X	What Happens on the Stack
`4`	0	4	Enter the \(4\text{ m}\) silo radius onto Shelf \(X\)
`[x²]`	0	16	Square the radius (\(4 \times 4 = 16\))
`[π]`	16	3.1415926	Enter \(\pi\); \(16\) lifts to Shelf \(Y\)
`[×]`	0	50.265482	Multiply for the circular floor area (\(50.27\text{ m}^2\))
`10`	50.265482	10	Enter the \(10\text{ m}\) silo height
`[×]`	0	502.654824	Stack the circle floor by height: 502.65 m³

Keystroke	Register Y	Register X	What Happens on the Stack
`2`	0	2	Start the \(2\pi r h\) curved side wall
`[π] [×]`	0	6.2831853	Multiply \(2 \times \pi \approx 6.2832\)
`4 [×]`	0	25.132741	Multiply by the \(4\text{ m}\) radius (\(25.13\text{ m}\) rim)
`10 [×]`	0	251.327412	Multiply by \(10\text{ m}\) height (\(251.33\text{ m}^2\) side wall)
`2`	251.327412	2	Start the 2 circular lids; side wall lifts to Shelf \(Y\)
`[π] [×]`	251.327412	6.2831853	Multiply \(2 \times \pi\) on Shelf \(X\)
`4 [x²] [×]`	251.327412	100.530964	Multiply by \(4^2\) for both lids (\(100.53\text{ m}^2\))
`[+]`	0	351.858376	Add side wall and lids: 351.86 m² total surface

Keystroke	Register Y	Register X	What Happens on the Stack
`6 [x²]`	0	36	Square the \(6\text{ cm}\) side for the \(36\text{ cm}^2\) floor
`8`	36	8	Enter the \(8\text{ cm}\) height; \(36\) lifts to Shelf \(Y\)
`[×]`	0	288	Multiply floor area by height: 288 cm³ volume

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse (Back Again)
Inches (in)	Centimeters (cm)	\(2.54\)	`[in] [ENTER] 2.54 [×]`	\(\div 2.54\)
Feet (ft)	Inches (in)	\(12.0\)	`[ft] [ENTER] 12 [×]`	\(\div 12\)
Feet (ft)	Meters (m)	\(0.3048\)	`[ft] [ENTER] [.] 3048 [×]`	\(\div 0.3048\)
Yards (yd)	Feet (ft)	\(3.0\)	`[yd] [ENTER] 3 [×]`	\(\div 3\)
Yards (yd)	Meters (m)	\(0.9144\)	`[yd] [ENTER] [.] 9144 [×]`	\(\div 0.9144\)
Miles (mi)	Feet (ft)	\(5280.0\)	`[mi] [ENTER] 5280 [×]`	\(\div 5280\)
Miles (mi)	Kilometers (km)	\(1.609344\)	`[mi] [ENTER] 1.609344 [×]`	\(\div 1.609344\)
Nautical Miles (NM)	Kilometers (km)	\(1.852\)	`[NM] [ENTER] 1.852 [×]`	\(\div 1.852\)
Nautical Miles (NM)	Statute Miles (mi)	\(\approx 1.150779\)	`[NM] [ENTER] 1.1508 [×]`	\(\div 1.1508\)
Meters (m)	Centimeters (cm)	\(100.0\)	`[m] [ENTER] 100 [×]`	\(\div 100\)
Centimeters (cm)	Millimeters (mm)	\(10.0\)	`[cm] [ENTER] 10 [×]`	\(\div 10\)
Kilometers (km)	Meters (m)	\(1000.0\)	`[km] [ENTER] 1000 [×]`	\(\div 1000\)

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse
Square Feet (sq ft)	Square Inches (sq in)	\(144.0\)	`[sq ft] [ENTER] 144 [×]`	\(\div 144\)
Square Yards (sq yd)	Square Feet (sq ft)	\(9.0\)	`[sq yd] [ENTER] 9 [×]`	\(\div 9\)
Acres (ac)	Square Feet (sq ft)	\(43,560.0\)	`[ac] [ENTER] 43560 [×]`	\(\div 43560\)
Acres (ac)	Hectares (ha)	\(\approx 0.4046856\)	`[ac] [ENTER] [.] 4047 [×]`	\(\div 0.4047\)
Hectares (ha)	Square Meters (\(\text{m}^2\))	\(10,000.0\)	`[ha] [ENTER] 10000 [×]`	\(\div 10000\)
Square Miles (\(\text{mi}^2\))	Acres (ac)	\(640.0\)	`[sq mi] [ENTER] 640 [×]`	\(\div 640\)
Square Meters (\(\text{m}^2\))	Square Feet (sq ft)	\(\approx 10.76391\)	`[m²] [ENTER] 10.764 [×]`	\(\div 10.764\)

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse
Ounces (oz)	Grams (g)	\(28.349523\)	`[oz] [ENTER] 28.35 [×]`	\(\div 28.3495\)
Pounds (lb)	Ounces (oz)	\(16.0\)	`[lb] [ENTER] 16 [×]`	\(\div 16\)
Pounds (lb)	Kilograms (kg)	\(0.45359237\)	`[lb] [ENTER] [.] 4536 [×]`	\(\div 0.4536\)
Kilograms (kg)	Pounds (lb)	\(\approx 2.2046226\)	`[kg] [ENTER] 2.2046 [×]`	\(\div 2.2046\)
Tons (US Short)	Pounds (lb)	\(2000.0\)	`[ton] [ENTER] 2000 [×]`	\(\div 2000\)
Metric Tons (t)	Kilograms (kg)	\(1000.0\)	`[t] [ENTER] 1000 [×]`	\(\div 1000\)
Kilograms (kg)	Grams (g)	\(1000.0\)	`[kg] [ENTER] 1000 [×]`	\(\div 1000\)
Grams (g)	Milligrams (mg)	\(1000.0\)	`[ENTER] 1000 [×]`	\(\div 1000\)

From Unit	To Unit	Conversion Number	SC-6 Forward Keystrokes	Reverse
Fluid Ounces (fl oz)	Milliliters (mL)	\(29.57352956\)	`[fl oz] [ENTER] 29.57 [×]`	\(\div 29.5735\)
Cups (c)	Fluid Ounces (fl oz)	\(8.0\)	`[c] [ENTER] 8 [×]`	\(\div 8\)
Pints (pt)	Cups (c)	\(2.0\)	`[pt] [ENTER] 2 [×]`	\(\div 2\)
Quarts (qt)	Pints (pt)	\(2.0\)	`[qt] [ENTER] 2 [×]`	\(\div 2\)
Gallons (gal US)	Quarts (qt)	\(4.0\)	`[gal] [ENTER] 4 [×]`	\(\div 4\)
Gallons (gal US)	Liters (L)	\(3.785411784\)	`[gal] [ENTER] 3.7854 [×]`	\(\div 3.7854\)
Gallons (Imperial)	Liters (L)	\(4.54609\)	`[gal imp] [ENTER] 4.546 [×]`	\(\div 4.5461\)
Liters (L)	Milliliters (mL)	\(1000.0\)	`[L] [ENTER] 1000 [×]`	\(\div 1000\)
Cubic Meters (\(\text{m}^3\))	Liters (L)	\(1000.0\)	`[m³] [ENTER] 1000 [×]`	\(\div 1000\)

From Unit	To Unit	Conversion Number	SC-6 Keystrokes
Atmospheres (atm)	Pounds per Sq Inch (psi)	\(14.6959488\)	`[atm] [ENTER] 14.696 [×]`
Atmospheres (atm)	Kilopascals (kPa)	\(101.325\)	`[atm] [ENTER] 101.325 [×]`
Bar (bar)	Kilopascals (kPa)	\(100.0\)	`[bar] [ENTER] 100 [×]`
Pounds per Sq Inch (psi)	Kilopascals (kPa)	\(6.894757\)	`[psi] [ENTER] 6.895 [×]`

From Unit	To Unit	Conversion Number	SC-6 Keystroke Sequence
Minutes (min)	Seconds (s)	\(60.0\)	`[min] [ENTER] 60 [×]`
Hours (hr)	Minutes (min)	\(60.0\)	`[hr] [ENTER] 60 [×]`
Hours (hr)	Seconds (s)	\(3600.0\)	`[hr] [ENTER] 3600 [×]`
Days (d)	Hours (hr)	\(24.0\)	`[days] [ENTER] 24 [×]`
Weeks (wk)	Days (d)	\(7.0\)	`[wk] [ENTER] 7 [×]`
Years (yr standard)	Days (d)	\(365.0\)	`[yr] [ENTER] 365 [×]`
Years (yr leap)	Days (d)	\(366.0\)	`[yr] [ENTER] 366 [×]`

Conversion	Word Rule	SC-6 Step-by-Step Keystrokes (or One-Tap Key)
Fahrenheit to Celsius	\((\text{Fahrenheit} - 32) \times 5 \div 9\)	`[Temp F] [ENTER] 32 [-] 5 [×] 9 [÷]` (or `[Temp F] [f] [▸°C]`)
Celsius to Fahrenheit	\((\text{Celsius} \times 9 \div 5) + 32\)	`[Temp C] [ENTER] 9 [×] 5 [÷] 32 [+]` (or `[Temp C] [g] [▸°F]`)

Keystroke	Register Y	Register X	What Happens on the Stack
`98.6`	0	98.6	Enter normal body temperature (\(98.6^\circ\text{F}\))
`[ENTER]`	98.6	98.6	Copy \(98.6\) up onto Shelf \(Y\)
`32`	98.6	32	Enter freezing-point offset (\(32^\circ\text{F}\))
`[-]`	0	66.6	Subtract 32 (\(98.6 - 32 = 66.6\))
`5`	66.6	5	Enter numerator 5 of the \(5/9\) scale factor
`[×]`	0	333.0	Multiply \(66.6 \times 5 = 333.0\)
`9`	333.0	9	Enter denominator 9 of the \(5/9\) scale factor
`[÷]`	0	37.0	Divide \(333.0 \div 9 =\) 37.0°C

Keystroke	Register Y	Register X	What Happens on the Stack
`26.2`	0	26.2	Enter marathon distance (\(26.2\text{ miles}\))
`[ENTER]`	26.2	26.2	Copy \(26.2\) up onto Shelf \(Y\)
`1.609344`	26.2	1.609344	Enter kilometers per mile (\(1.609344\))
`[×]`	0	42.164813	Multiply for marathon in kilometers (\(42.165\text{ km}\))
`1000`	42.164813	1000	Enter \(1,000\text{ meters per kilometer}\)
`[×]`	0	42164.81280	Multiply for marathon in meters: 42,164.81 m

Keystroke	Register Y	Register X	What Happens on the Stack
`36000`	0	36000	Enter cruising altitude (\(36,000\text{ ft}\))
`[ENTER]`	36000	36000	Copy \(36,000\) up onto Shelf \(Y\)
`.3048`	36000	0.3048	Enter meters per foot (\(0.3048\text{ m/ft}\))
`[×]`	0	10972.8	Multiply to get altitude in meters (\(10,972.8\text{ m}\))
`1000`	10972.8	1000	Enter \(1,000\text{ meters per kilometer}\)
`[÷]`	0	10.9728	Divide by 1,000 for kilometers: 10.97 km