## Dr. Kromm's Patented Math and Physics Tutorial

In the course of editing and proofing several rather technical GURPS books, I was amazed to discover that not all GURPS authors and editors are experts at mathematics and the physical sciences! To this end, I have drafted this "cheat sheet" to help those with deficient quadrivium.

## Numbers

Let's start out simple. Everyone knows how to write numbers, right? Wrong! There are strict rules governing the proper treatment of numerical quantities, and I'm going to fill you in on them.

### Fractions

Let's start with fractions. The following conventions hold when expressing fractional quantities:

#### Exact Fractions

When a fraction is exact (see Precision, below) do not express it in decimal notation, but as one number over the other. For example, when you are writing a rule and mean precisely "two-thirds," write it as 2/3 (e.g., "Your parrying ability is 2/3 your Fencing skill."). The number on the top (the "2" in 2/3) is called the numerator. The number on the bottom (the "3" in 2/3) is called the denominator. Remember, "2/3" is far more precise than "0.67" if what you really mean is "two-thirds."

Fractions are most intuitive when they have been reduced by dividing both the numerator and denominator by the largest whole number that can divide into both and still give a whole number. For instance, never write 8/12 if you can write 2/3. It looks bad, takes up more space and is utterly nonintuitive for most readers.

#### Decimal Fractions

Reserve decimal notation for when you are speaking of physical quantities and the results of formulae; otherwise, use an exact fraction. For instance, if you really mean HT/2, then write "HT/2" and not 0.5×HT. This is more intuitive and takes up less space.

Do not add unnecessary trailing zeroes. If you mean "precisely one half," then "0.50" is no closer than "0.5." All the extra zeroes do is take up space. On the other hand, try not to mix precision within a table. If you write 0.25, then write 0.50 as well. Aside from being correct, this makes the table easier to typeset. Of course, if you really mean 1/4 and 1/2, then write that!

Do not omit the leading zero, either. The decimal system is a place value system, which means that the numbers only have meaning because of their position with respect to other numbers. If you leave those numbers out, then the notation is meaningless. For instance, "0.5" is correct, but ".5" is not. Aside from being incorrect, a number like ".5" can become ambiguous if preceded by an ellipsis or leader.

### Scientific Notation

This is an extension of the decimal system for really large or small numbers. The number is expressed as a decimal fraction multiplied by an appropriate power of 10. This is only really useful for numbers with lots of zeroes in them, in which case it can be a major space-saver. For instance, 1,200,000,000 is really just 1.2 times 1,000,000,000, and 1,000,000,000 is just 109, so one could write this as 1.2×109 in scientific notation. If you have to write this more than once, then you will save a significant amount of space. Keep in mind that not everyone knows what this means, though; if you plan to use this notation, it's a good idea to explain it to the reader in the introduction. See Prefixes (below) for another way to handle this.

If you don't want to get into scientific notation in a book, another alternative is to simply spell out the number; e.g., "1 trillion" rather than "1,000,000,000,000." For very large numbers, this is often both more intuitive and more space-efficient. This is done most often with dollar values; e.g., "\$1 billion" means more to most people than "\$1,000,000,000" and is also shorter. Note that in American usage, a million has six zeroes, a billion has nine, a trillion has 12 and a quadrillion has 15.

### Precision

Precision is the degree of certainty with which you know a number.

An exact quantity is just that – there is no doubt as to what the quantity is, you are completely certain of it. Exact quantities are understood to come in units. Exact quantities in GURPS include things like skill modifiers, a character's ST score or the Shots stat of a gun. Such a number is never expressed as a decimal; e.g., you would never say "-1.0 to Guns skill." The same holds for exact fractions ("Fencing Parry is 2/3 Skill," "Speed = (DX+HT)/4").

Virtually all other quantities are expressed as decimals, and therefore have a finite precision. Generally speaking, the more decimals places you show, the more precise the number. For instance, "0.12000" implies more precision than "0.12." However, only indicate precision that is really there. If you round 0.11999 to 0.12 (see Rounding Conventions, below), then you have to write "0.12," because "0.12000" would imply the number is known to be 0.12000 to five decimal places, and since it isn't (it's 0.11999 to five places), this would be incorrect. Likewise, if you round to 0.120, then you have to write "0.120" and if you round to 0.1200, then you must write "0.1200." In no case would "0.12000" be correct for 0.11999.

#### Significant Figures and Doubtful Digits

Measured physical quantities are always given to a certain number of decimal places; the number of places is indicative of how precise the measurement is. For instance, an automobile manufacturer might give the top speed of a car as "127 mph," meaning the speed is known to a precision of three places. On the other hand, a scientist might give the diameter of the Earth as "12,756.3 km," meaning the diameter is known to a precision of six places. The digits which occupy these decimal places are known as significant figures. For example, the speed of the car above is known to three significant figures, the diameter of the Earth to six.

When measuring a quantity, one always reads the instrument to the limit of its capabilities and estimates the final digit. So with any measured physical quantity, it is implicit that the last significant digit is also the least well known. This last digit is called the doubtful digit. Remember: whenever you state a physical quantity using decimal notation, especially one you have taken from a scientific handbook, the last digit is implicitly doubtful.

When quantities of finite precision are used in an arithmetic equation, the precision of the result can never be higher than that of the lowest-precision quantity in the equation. Thus, the result should always be rounded off to that number of places (see Rounding Conventions, below). For example, if a rule requires the player to multiply two numbers together, and one is 0.12, the other 96.87, then the result is 12, not 11.6244. Since 0.12 has only two significant figures, so must the result.

### Rounding Conventions

When doing calculations, one often ends up with long decimal numbers. As per the guidelines above, these must be rounded off to the appropriate number of significant figures. Note that in a long, complex calculation, you only do this once, at the end of the calculation, not at each step along the way.

The rules for rounding, called rounding conventions, are as follows: When rounding off a number, look at the digit one place beyond the place you are rounding off to. If this digit is 0, then simply drop the 0; e.g., 56.70 becomes 56.7. If this digit is between 1 and 4, round the number down; e.g., 56.73 becomes 56.7 (and not 56.8). If this digit is between 6 and 9, then round the number up; e.g., 56.77 becomes 56.8 (and not 56.7). If this digit is 5, then round an odd number up and an even number down; e.g., 56.75 becomes 56.8, but 56.65 becomes 56.6. This may seem arbitrary, but it all comes out in the wash. Note that other conventions exist and are equally valid; the important thing is to stick to just one, such as the one above.

Instead of using the conventions above, one can also say "round any fraction up the next whole number" or "drop all fractions" (this is done for a character's Move score, for instance). However, you must always state such things explicitly – they are never the norm.

## Mathematical Notation

Numbers are to equations what letters are to words and sentences. In the same way that we have grammar and spelling for text, there are rules for writing equations – there's a Right and a Wrong way to do these things; use the Wrong way and you risk confusing anyone who happens to know the Right way.

### Mathematical Operations

The basic arithmetic operations are addition, subtraction, multiplication and division. Addition is always indicated with a "plus" (+). Subtraction is indicated with a "minus" (-), equivalent to an ordinary hyphen, not a dash (--). Multiplication is indicated using a cross (×), not an asterisk (*) or a Roman "x." Note that the "×" is often omitted when a number and an algebraic symbol or two algebraic symbols multiply one another; for instance, "2 × x" can be written "2x." Division is normally indicated with a forward slash (/), except when used in the sense of a ratio, in which case a colon (:) is appropriate. For the sake of clarity, always insert a space before and after any of these symbols.

More complicated operations tend to confuse the uninitiated. To facilitate understanding, indicate exponents by spelling them out or by using multiplication; e.g., write ST2 as "ST squared" or "ST×ST." Square and cube roots should be spelled-out as well, rather than attempting to use the radical ("square root") sign. Note that when spelling out an operation, place parentheses or brackets around everything it applies to; e.g., "Square root of (motive thrust/loaded weight)." Other functions (trigonometric functions, logarithms, etc.) should rarely be used in a game; there are usually simpler rules of thumb that serve the purpose just as well.

### Comparing Quantities

When two quantities are mathematically identical, indicate this using an "equal sign" (=). An equation is just two quantities separated by an equal sign.

When comparing two unequal quantities or defining a range of quantities, use the "less than" sign (<) when the quantity on the left is smaller than the one on the right, and the "greater than" sign (>) when it is larger. Note that these signs are exclusive, not inclusive; e.g., for an integer-valued quantity such as ST, saying "ST<10" means a ST of 9 or less; ST 10 is excluded. If you mean 10 or less, then write "ST<11." The mathematical symbols for "greater than or equal to" and "less than or equal to" should be avoided, since many readers will be unfamiliar with them. Instead, spell out what you mean; e.g., say "10 or less" if you mean less than or equal to 10, "10 or more" if you mean greater than or equal to 10. Note that in both cases, 10 is included.

Finally, always insert a space before and after any of these symbols (<, =, >), for clarity's sake.

### Writing Equations and Formulae

An equation consists of two quantities or algebraic expressions with an equal sign in between. The implication is that whatever is to the left of the equal sign is the same as what is to the right. If this is not true, then don't write it as an equation; simply state that the two quantities are approximately equivalent.

A formula is an equation which has an algebraic expression on one side of the equal sign and the quantity to be determined on the other. All the quantities in the algebraic expression should be defined either beforehand or beneath the formula. Letters, symbols and spelled-out words can all be used, but make sure that the reader knows what they refer to.

#### The Order of Operations

When writing an equation or formula, there are some rules of "mathematical grammar" to be obeyed. These rules are called the order of operations, and indicate in what order mathematical operations take place. Limiting ourselves to operations that are likely to occur in the rules of a RPG, this order is as follows: raising a quantity to a power or taking a root comes first, then multiplication and division, then addition and subtraction. Expressions grouped together inside parentheses (), brackets [] and braces {} are always evaluated first, using the above rules. Parentheses go inside brackets; brackets go inside braces.

It is important to realize that equations are not simply read from left to right. The above order always holds, regardless of what is to the left of what. For instance, the formula

a = b + c × d

means, "Multiply c by d first and then add the product to b," and not, "Add b to c first and then multiply the sum by d." If this latter reading is what is intended, then use parentheses to indicate that the addition comes first; i.e.,

a = (b + c) × d.

The most commonly confused formulae are those involving the division of one complicated expression by another. In such a case, it is always a good idea to group things in parentheses. For instance, if you want to add x to y and z, then divide by a times b, then write

(x + y + z) / (a × b)

to avoid ambiguities.

## Measurements and Units

A measurement is a numerical rating of a property of a physical system: the height and weight of a character, the speed and acceleration of a vehicle, the energy contained in an ultra-tech power cell, etc. A measurement always consists of two parts: a magnitude and its units. The magnitude indicates "how much," while the units indicate what property is being measured and which standard of measure is being used to rate it; i.e., "How much of what?" For instance, "7 lbs." indicates that weight is being measured in pounds, and that it has been found to be 7 pounds. A measurement that is missing either units or a magnitude is meaningless.

When writing equations that contain quantities that have units, it always a good idea to include the units. For instance, if a character can jump three times ST minus 10 inches, this should be written as

(3×ST - 10) inches.

### Basic Quantities and Their Units

Below is a small table of physical quantities, along with the units and abbreviations used for each in GURPS. Note that many of these units and abbreviations are not standard; they have become "standard" in GURPS through repeated use.

### Unit Table

Quantity GURPS Units Abbreviation
Acceleration yards per second per second yps/s
miles per hour per second mph/s
Area square feet sf
square yards sy
Energy kilojoules kJ
kilowatt-seconds kWs
Frequency hertz Hz
Length inches in. or "
feet ft. or '
yards yd.
miles mi.
Power kilowatts kW
Speed yards per second yps
miles per hour mph
Temperature degrees Farenheit °F
kelvins (for stars only) K
Time seconds sec.
minutes min.
hours hr.
Volume gallons gal.
cubic feet cf
cubic yards cy
Weight pounds lb.
tons ton

### Prefixes

When very large or small quantities are being expressed, it is both more efficient and more intuitive to use units of the appropriate size. Like scientific notation (see above), this can often reduce the length of text. This is accomplished by preceding the unit name with a prefix that indicates multiplication by the appropriate power of ten; e.g., "1 megaton" instead of "1,000,000 tons," or "9 mm" (9 millimetres) instead of "0.009 m" (0.009 metres). Note that this is generally only done with metric units, not American or Imperial ones. The following "metric prefixes" are standard:

### Prefix Table

Multiplier Power of 10 Prefix Symbol
0.000000000000000001 10-18 atto- a
0.000000000000001 10-15 femto- f
0.000000000001 10-12 pico- p
0.000000001 10-9 nano- n
0.000001 10-6 micro- µ
0.001 10-3 milli- m
0.01 10-2 centi- c
0.1 10-1 deci- d
10 101 deka- da
100 102 hecto- h
1,000 103 kilo- k
1,000,000 106 mega- M
1,000,000,000 109 giga- G
1,000,000,000,000 1012 tera- T
1,000,000,000,000,000 1015 peta- P
1,000,000,000,000,000,000 1018 exa- E

### Exceptions and Special Cases

Computer Storage: Many GURPS books discuss computers, and list storage capabilities in "gigs" (gigabytes). A gigabyte is actually 1,073,741,824 "bytes" or characters. The prefix "giga-" is used approximately here.

Obscure Prefixes: The prefixes hecto-, deka-, deci-, and centi- are rare. "Hectare" and "centimeter" are commonly used in metric, but in general these prefixes should be avoided. For instance, you would not likely be understood by most readers if you said "1 dekasecond"; just say "10 seconds."

Temperature: The kelvin (not "degrees Kelvin") is preferred by scientists and is used in GURPS for the temperature of stars; however, the day-to-day unit of temperature is "degrees Farenheit" (°F). The degree symbol must appear to avoid confusion with other units (F by itself means "farad," a unit of capacitance). Unlike the kelvin, the degree Farenheit is a relative unit and not an absolute one; therefore, one cannot apply prefixes to it. For instance, there is no such thing as a "kilodegree Farenheit."

### Units in Equations and Formulae

As mentioned above, units are very important, and without them, a measurement is meaningless. A similar statement can be made about the units appearing in an equation or formula: The units of quantities on either side of an equal sign must always be identical. You cannot equate a quantity measured in pounds to one measured in hours! If you encounter a formula that does do this, then there's probably an error in it somewhere; go back and check it. (Scientists often call checking one's math this way "dimensional analysis.")

Likewise, you cannot add or subtract two quantities with unlike units within a formula (e.g., adding distance in yards to speed in yards/second); always make sure that the units of all quantities being added together (or subtracted) are the same. This rule applies even when you have different units that measure the same thing; e.g., you cannot add ounces to pounds, even though both measure weight. Note that people add unlike units all the time in day-to-day life; e.g., 1' + 2" = 1'2", 1 lb. + 8 oz. = 1 lb., 8 oz. and so on. This is fine at the market, but it's not terribly intuitive when one has to use those numbers in a calculation. To this end, always convert things to the same units; e.g., in the examples above, it would be much better to say 12" + 2" = 14" or 1 lb. + 0.5 lb. = 1.5 lb.

#### Empirical Formulae

Some formulae disobey the rules above, but make sense in a certain context. For example, people often equate every five seconds between a flash of lightning and clap of thunder to a distance of one mile between the observer and a thunderstorm, even though "miles" and "seconds" are not identical units! "Rules of thumb" like this are called empirical formulae. In this case, it is acceptable to break the rules provided that this is clearly indicated. This is normally done by putting the units in parentheses and prefixing them with "in." For instance

Distance (in miles) = Time (in seconds)/5.

An example in GURPS is the way Speed/Range penalties are calculated in the advanced combat system. There, one adds speed (in yards/second) to range (in yards) before looking up the S/R modifier. No one is claiming that this is a scientifically exact way of doing things – it's just a gameable rule of thumb.

## Science!

This section discusses a few basic scientific principles that are often confused or overlooked in games with a science-fiction or technological theme.

### Physical Quantities

Certain physical quantities are commonly confused; the five most common errors are discussed below.

#### Energy and Power

In simple terms, energy is the ability to do work (e.g., move an object against friction or gravity, heat it up, etc.). It can be neither created nor destroyed, but it can be transformed (see Conservation of Energy, below). For example, a car turns the chemical energy of gasoline into the mechanical energy of the moving car, and uses an alternator to turn some of this mechanical energy into the electrical energy that powers the car's lights and radio. In GURPS, energy is measured in kilowatt-hours (kWh) or in kilowatt-seconds (kWs), which are also known as kilojoules (kJ).

Power is simply the rate at which energy is generated (i.e., gathered or converted into a useful form by a power plant, reactor or engine) or consumed. The car mentioned above might have an engine that generates a certain amount of power, meaning it provides a certain quantity of energy each second. It may also have a radio that requires a certain amount of power, meaning it consumes a certain quantity of energy every second. In GURPS, power is measured in kilowatts (kW), equal to one kilowatt-second of energy being produced or consumed per second.

Energy can be stored in batteries and power cells, and it is proper to speak of these things as "containing" and "storing" energy. Likewise, each "shot" from an energy weapon and each "use" of a gadget requires a certain amount of energy. Power, on the other hand, cannot be stored; power is only really meaningful when speaking of the output of a power plant of some kind (engine, nuclear reactor, etc.), or when referring to the rate of energy consumption of a gadget that operates continuously.

#### Mass and Weight

Mass (m) is a fundamental quantity, and is essentially a measure of how much matter you have present. It is measured in grams in the metric system, or in slugs in Imperial units. Weight (w) is the force acting upon a mass in a given gravity field (g). It is given in Newtons in the metric system, or in pounds in the Imperial system. Weight is related to mass as follows

w = mg.

Obviously, in zero gravity (g = 0), objects have no weight; however, they always have mass. In the Imperial system, it has become traditional to assume a gravity field of 1 g (i.e., Earth-normal gravity) and equate mass with weight. This has led to the widespread use of pounds as a unit of mass, even though this is technically incorrect. GURPS is no exception to this. Remember, though, that mass is not weight; when an object is weightless, it is not massless – even if its mass is being measured in pounds. Physical calculations that depend on mass (like the top speed and acceleration of a vehicle) will not change when gravity does; those that depend on weight (like the stall speed of an aircraft) will.

#### Momentum and Kinetic Energy of a Moving Object

An object with mass m and speed v has a momentum (abbreviated as p by physicists) given by

p = mv

and a kinetic energy (abbreviated as T by physicists) given by

T = (1/2)mv2.

Momentum and kinetic energy are not the same thing. When determining the damage from a collision, or a speeding bullet, the appropriate quantity to gauge it by is its kinetic energy. This means that a bullet moving twice as fast is four times as damaging! On the other hand, when determining knockback and things like skill or DX penalties for collisions, momentum is what matters.

#### Speed and Acceleration

Speed is a measure of the distance covered by a moving object in a given amount of time. In GURPS, it is usually measured in yards per second (as per the Move score of characters, for instance) or in miles per hour (for vehicles and so on). Acceleration is the rate of change of speed – how much the speed increases or decreases in a given amount of time. In GURPS, this is generally measured in yards per second per second or in miles per hour per second. The most common error is to give acceleration in "yards per second" or "miles per hour" – don't!

#### Speed and Velocity

As mentioned above, speed is simply a measure of "how fast." Velocity, on the other hand, includes both the speed and the direction that something is moving in (it is what scientists and mathematicians call a vector). Strictly speaking, it is incorrect to use the term "velocity" for speed – the two are not synonymous.

### Basic Kinematics

Without getting into complexities, there are a few equations that govern the motion of objects when they are accelerating at a constant rate. (I.e., none of these apply when acceleration is changing.) These equations are especially useful when writing supplements that have to do with space travel!

For an object accelerating with acceleration a and starting at speed vi, the final speed vf after accelerating for time t will be:

vf = vi + at.

If it accelerates for a fixed distance x and time t is unknown, use this equation instead:

vf2 = vi2 + 2ax.

Finally, the distance x covered by the object in time t is given by

x = vi t + 1/2 at2.

Note that in the above equations, an object accelerating from a standing start has vi = 0; one that is coming to a stop has vf = 0.

### Fundamental Principles

What follows is a collection of scientific principles to be obeyed in any work that is supposed to be realistic. Magic or extreme ultra-tech may, of course, ignore some or all of these.

#### Absolute Zero

This is the temperature (-459.67°F) at which objects contain no kinetic energy on the molecular scale. It is the absolute bottom of the temperature scale. As far as we know, this temperature cannot be reached, let alone surpassed. Statements like "as cold as absolute zero" or "colder than absolute zero" are incorrect in any universe that resembles our own. (Note: This hasn't stopped some people from creating mathematical models where temperature is defined to be lower, or where molecules have "kinetic energy" at absolute zero, but these are not physical systems.)

#### Conservation of Energy

Although physics goes into much more detail on the subject, it is sufficient to note one point here: In a closed system, energy may change form, but cannot be created or destroyed. This is sometimes called the First Law of Thermodynamics. In game terms, this means that anything that consumes energy should convert that energy to some other form (e.g., the stored energy in a power cell is converted to a laser beam), and that "miracle" power supplies that give large amounts of energy for "free" are not realistic.

Fuel: Fuel is effectively energy (it's stored chemical energy in the case of gasoline and the like, or nuclear energy in the case of antimatter and nuclear fuel rods). If gadget #1 "creates" a certain amount of fuel for energy cost A and gadget #2 can burn that same amount of fuel to produce a quantity of energy B, then at best, B = A. B can never be greater than A!

Real vs. Ideal Systems: Ideally, energy can simply be shifted from form to form at no cost. In reality, though, it is rare for all the energy being converted from one form to another to actually be converted as planned. While energy is never truly lost, some of it is usually wasted in the form of vibrations, heating, noise and so on. In game terms, this means that at the TL when a device is first introduced, it is a good assumption that a significant amount of energy may be lost this way. And some technologies are less efficient than others: Two ray guns may get the same number of shots from a power cell, but it is quite acceptable for one to inflict half the damage of the other!

Always remember that energy is conserved. "Wasted" energy doesn't just vanish – it usually shakes things loose, heats things up or manifests as stray energy fields. Real-life weapon and vehicle engineers dedicate a lot of time to limiting not just the losses but also the undesirable effects caused by them. This is one reason why higher-TL devices are often lighter: fewer losses means fewer measures taken to radiate waste heat, damp out vibrations and shield against stray fields.

#### Newton's Laws

Isaac Newton came up with three basic laws that sum up the way actual physical forces work. Stated simply, these are:

1. An object at rest will stay at rest and an object in motion will stay in motion until acted upon by an outside force. For instance, in the absence of air drag and other forms of friction, moving objects should not just slow down – they'll keep moving forever. This is important to realize in space settings, where there is very often zero gravity and no atmospheric friction to worry about! This tendency is sometimes known as inertia.

2. F = ma. The force (F) acting upon a mass (m) is proportional to the mass and the acceleration (a). Likewise, a = F/m. This means that given a force, the acceleration of an object subjected to it is inversely proportional to the object's mass. Thus, if a character shoves two objects for 1 second, one with mass m and one with mass 2m, then the lighter one will be moving twice as fast at the end of that time. Likewise, a rocket or parachute that accelerates or decelerates a character with mass m by a certain amount will accelerate or decelerate a character with mass 2m only half as much.

3. Action equals reaction. In other words, if a force is exerted in one direction, then there must be an equal force in the opposite direction. A bullet fired from a gun results in recoil, for example, and a character who throws another character away from him in zero gravity will go flying in the opposite direction. Similarly, rockets move by throwing hot gases in the direction opposite the one they are moving in.

#### Relativity

Relativity is a mathematically complex theory of physics, developed by Albert Einstein, that is more general than Newton's laws. In fact, it's really two theories: general relativity, which governs gravity, and special relativity, which governs the behavior of objects moving at or near the speed of light. In general, a realistic treatment of relativity is inappropriate for a game book, so it's usually a good idea to avoid dropping the term.

It is interesting to note, however, that a character who travels near the speed of light does not age as quickly as one who stays put, because time appears to pass more slowly to him. For instance, a character who travels at 99.9% of the speed of light for what seems like a year to him will return to a world where over 22 years have passed! This effect is called time dilation, and can be expressed as

Time perceived by stationary character = Time perceived by moving character / Square root of [1 - (fraction of light speed)2].

#### Scaling Laws

As the size (linear scale, length) of an object increases, its other dimensions increase as well. However, not all quantities increase in a linear fashion! For a factor of s increase in size, surface area and cross-sectional area increase as s2, while volume increases as s3.

Structural Load Limits: In the case of inanimate structures, mass increases with volume (s3), but structural strength (DR, hit points and the ST of cranes and robots) only increases with cross-sectional area (s2). This means that if the size of a structure is increased, the forces acting on it (e.g., planetary gravity, maneuvering g-forces in the case of a vehicle) will eventually exceed its structural strength. This limits the size of things like vehicles and robots, and of any structure that must exist in the presence of a gravity field.

Biological Square-Cube Law: In the case of a living organism, oxygen and food demands, as well as heat uptake and disposal needs, increase with volume (s3), while transfer rates of those things into or out of the organism are proportional to surface area (s2); therefore, there are limits on how large a living thing can be before it becomes inviable. This is often called the square-cube law, because the powers of s involved are 2 ("square") and 3 ("cube"). It limits the size of living creatures in any realistic setting, and explains why we don't see eight-foot long wasps and other "monsters" in real life.

#### Speed of Light

The speed of light in a vacuum is about 186,000 miles per second, or about Move 330,000,000! As far as we know, nothing can move faster than this, and no object with mass can ever move at this speed. Material objects that travel at the speed of light, or anything (objects, weapons, data) that moves faster than the speed of light, are pure science fiction.

## Other Trivia

In general, if you don't know how something works, don't make up "technobabble" that uses real scientific terms. This is misleading to those who aren't "in the know" and insulting to those who are. It's better to avoid the issue altogether and carefully describe the game mechanics of the effect. If it really matters how something works, then either do the research or ask your playtesters to help you do it.

Two examples of commonly confused concepts are given below. Other offenders from physics include entropy, the word "quantum" and various quantum physics notions such as the Uncertainty Principle, tunneling, and Bell's Theorem.

### Chaos

Avoid the tendency to use "chaos" and "chaos theory" to justify questionable science. There is nothing mysterious about chaos theory – it's merely a new area of mathematics that deals with the unpredictable outcomes of sensitive processes. When applied to real-life situations, the outcome may be unpredictable, but it is still believable. For instance, chaos theory explains why it's hard to tell if it will rain today or tomorrow, but that's about as exotic as it gets. It never predicts that it will rain frogs!

### Infinity

Infinity is not a number, but a limit or bound. It is an abstract mathematical concept that means "greater than any finite number, however large." In general, avoid using the term "infinite" to refer to a very large number unless you really mean "infinite." Otherwise, just say "very large."

## Bibliography

The following were helpful when preparing this tutorial, and are also useful reference works, should you run into problems.

CRC Handbook of Chemistry and Physics, 67th Edition; Robert C. Weast, editor. CRC Press Inc., Boca Raton, Florida (1987).

CRC Standard Mathematical Tables, 28th Edition; William H. Beyer, editor. CRC Press Inc., Boca Raton, Florida (1987).

The Feynman Lectures on Physics; Richard P. Feynman, Robert B. Leighton and Matthew Sands. Addison-Wesley Publishing Company, Reading, Massachusetts (1965).