Download 9.1

QUANTIFICATION
One source of the great explanatory and predictive power contemporary science is
its quantitative character. A very large number of our most interesting questions
begin with "how much" or "how often." Less obviously, even outwardly
qualitative questions often yield answers only in response to investigations that are
quantitative at every stage. We're all taught to use numbers, although with some
curious practical omissions; but the skills involved even in simple mathematics
corrode swiftly. Biomechanics may not necessarily be highly mathematical, but it’s
inevitably quantitative; if you don't commonly work with numbers or if you are, say,
an accountant, and use them in quite a different way, the following quick brush-up
and commentary might prove useful.
I. ARITHMETIC
Numbers can be divided into "positive," "negative," and "zero." If positive
numbers are envisioned as a scale running from left to right, then negative numbers
mark a continuation of the same scale from right to left beginning just to the left of
zero. Numbers are assumed to be positive unless preceded by a minus sign.
Equipped with negative numbers, it is easy to see that the operation called
"subtraction" is a special case of a more general operation called "addition"—
subtraction is merely the addition of a negative number...
5 - 3 = 2 will henceforth mean "add -3 to 5";
-3 + 5 = 2 is the same as 5 - 3 = 2.
So subtraction needn't concern us further.
Similarly it is easy to see that the operation called "division" is a special case of a
more general operation called "multiplication"—division is merely multiplication by
a fraction...
6 divided by 3 = 2 is the same as
which is, as mentioned, the same as
6 x 1/3 = 2
6 x 0.333 = 2.
So we don't really need division either. My word processor lacks the division
symbol; but until now, I never even noticed! A horizontal bar is all we need to
indicate division—in the example above (second line), one can multiply the 6 by the
1 and then use the bar for the subsequent division:
6 x 1/3 = 6/3 = 2
The appropriate signs to use in multiplication (and division) aren't instantly
obvious. Just remember that the product of two negative or two positive numbers is
positive, while the product of a positive and a negative number is negative*
-4 x -2.5 = +10
-4 x +2.5 = -10
Two more operations need mention at this point, and again the second is just a
special case of the first—"exponentiation" and "extraction of roots". (The former
sounds hortative, the latter dentistical.) Linguistic conventions vary—"three to the
second power," "three squared," or "three exponent two" all mean "three multiplied
by three" or the product of two threes, and they correspond to...
32 = 9
The same notation works equally well for roots; thus we can write the "cube root
of 8" as "8 to the power 1/3" and, if desired, put the latter in decimal notation (the
actual operation is not readily done without a calculator)...
81/3 = 2
or
80.333 = 2
Dividing by a number that has an exponent is equivalent to multiplying by that
number with the same exponent times -1...
3
-3
1/2 = 2 = 1/8
So these exponents can be integers, fractions, or combinations; they can be
positive or negative as well; and they can even take a value of zero. The latter is a
rather peculiar case—anything with an exponent of zero is equal to one...
170 = 1
*
Certain of the statements here quite shamelessly lack even the merest pretense of honest
explanations. While they aren't really just conventions, for present purposes they're most
easily treated as Revealed Truth....
Multiplying a set of identical numbers with exponents is the same as adding their
exponents. It's obvious in the first example, less so in the subsequent ones...
1
1
3 x 3 x 31 = 33 = 27
31 x 32 = 33 = 27
34.2 x 3–1.2 = 33 = 27
Taking a number to a non-integer power such as 3 in 34.2 is easily done with a
hand calculator that has a key labelled "yx". Otherwise it's awkward.
In order to avoid ambiguity, it's necessary to establish rules for precedence of
operations. Proceeding from initial operations to later ones, the following
conventions are used...
a. exponentiation and extraction of roots
b. multiplication and division
c. addition and subtraction
As an example, the following expressions yield 8, not 12...
2+2x3
and
3x2+2
The way to circumvent these conventions is with another convention, the latter
involving parentheses (in pairs, always). The transcendent rule is always to perform
the operation in the innermost parentheses first, proceeding to outer parentheses.
Beyond that, the previous precedential rules apply. Parentheses also replace the sign
"x" for multiplication...
3(2 + 2) = 12
3x2+2=8
(3 x 2)2 = 36
3 x 22 = 12
Note that with enough parentheses the "x" need never be used at all. In fact, it
becomes handy to avoid the "x" designation for multiplication in algebra since "x" is
commonly used to represent an unknown quantity, and using the same symbol for
two different items is a Bad Thing.
II. SCIENTIFIC NOTATION
Which gets around the inconvenience of writing very large and very small
numbers. The essential feature consists of expressing a number in two parts—the
significant figures and the number of places by which the decimal point has been
moved in the shift from ordinary notation. Each place, of course, represents a power
of ten in our counting system; and the ten is given explicitly in the second part.
Thus...
126,000,000 can be written as (1.26 x 108)
0.000,000,126 can be written as (1.26 x 10-7)
Note that the exponent is obtained just by counting the number of places by
which the decimal point is shifted, with shifts to the left being positive and shifts to
the right negative. The notation simplifies calculations—for instance, in multiplying
two such numbers, the significant figures (on the left) are first multiplied and then the
exponents (on the right) are added...
(3.4 x 106)(2.3 x 10-3) = 7.82 x 103 = 7820
Division is as easy—one just switches the sign of the divisor's exponent (what
something is being "divided by") before adding the exponents.
III. MEASUREMENT
Numbers are used in "scales of measurement". It's useful to distinguish among
four basic sorts of scales and to ask which arithmetic operations are permissible with
each (Stevens, 1946). It’s also useful to imagine each kind of scale displayed as the
horizontal axis of a graph (more below), asking what such a graph might mean.
a. Nominal Scale: numbers are used only as labels—examples are the numbers
on football jerseys and serial or model numbers. The only permissible operation is
the determination of equality (is this model a 610 or not?).
Graphs should be used sparingly—they’re really just tables—since the order of the
items on the horizontal axis has no certain meaning. Only bar graphs are even
acceptable.
b. Ordinal Scale: numbers are used to rank order things—examples are
acquisition numbers in a library, lumber grades, and Mohs' scale of hardness of
minerals. In the latter, rank is based on who scratches whom—a kind of geological
pecking order. Additional permissible operations are determinations of
inequalities: "greater (or harder) than" (>) and "smaller (or less hard) than" (<). One
can't assert that something numbered 10 is twice as hard or twice as recent as
something numbered 5, or that a 7 is two standard units of hardness or time less than
a 9.
Bar graphs work nicely, but points connected by lines should be avoided—since
the spacing of points across the graph is arbitrary, the slopes of the lines lack real
meaning.
c. Interval Scale: equal differences between pairs of numbers imply equal
intervals between them, so the scale is truly quantitative. But the zero point is a
matter of convention or convenience. The commonest cases are our temperature
scales, Fahrenheit and Celsius, and calendar dates. Addition and subtraction work
nicely—40 degrees is as much hotter than 30 as 25 degrees is hotter than 15. But 40
is not twice as hot as 20 degrees in any sense at all—multiplication and division are
precluded.
Conventional connect-the-dots graphs are swell, with only the caveat that
extrapolation to zero means little or nothing—there’s no natural starting point on the
horizontal axis. Slopes, though, convey meaning.
d. Ratio scale: numbers imply equal intervals, now with an absolute zero as
well. So multiplication and division are possible, along with the other operations
above. Most physical scales are of this type, as is the scale of numbers with which
we count things. Five apples are indeed half as many as are ten. One set of values
(say, in feet) can be converted to another (in meters) with a simple multiplication by a
constant conversion factor. The absolute temperature scale (Kelvin degrees) is of
this type—zero is the temperature at which molecular motion ceases, and the scale is
a measure of the intensity of such motion.
On graphs, both slopes and zero points have real meaning.
Strictly speaking, no measurement is of any value unless information is available
about its accuracy. No measurement is absolutely accurate; and, indeed, the
accuracy required depends on the use to which the measurement will be put. It's no
trick to obtain a balance which will weigh five pounds to the nearest thousandth of a
pound, but would you insist that your bag of apples be so accurately determined? It
would require either an expensive selection process or the use of fractional apples,
certainly a rotten idea.
Accuracy, though, reflects the effects (or lack of effects) of two quite distinct
components. First, "imprecision," a term used here for the scatter among a series of
separate determinations. Put a bunch of thermometers in boiling water—their
readings may vary from 97 to 101 degrees Celsius. The commonest measure of this
scatter is something called the "standard deviation"—one figures the average of the
measurements, then the average of the squares of the deviations of the individual
measurements from the first average, and then takes the square root (1/2 power,
remember) of the latter average. Often the standard deviation is given following a
datum and preceded by a plus/minus: 5.67 ± 0.03 kilograms for items made to a very
close tolerance. Obviously it would be a joke to report the mass above as 5.668734
kilograms even if the balance could be read this closely! One should bear this in
mind when reporting digital read-outs, which commonly display more figures than
the underlying measurement justifies.
The other thing limiting accuracy is "systematic error." It amounts to a measure
of the accuracy of the calibration of the measuring device. Its estimation, though,
requires some knowledge of an absolute (or much more accurate) standard. Weigh
yourself on a non-digital bathroom scale ten times—you can, with care, read it to the
nearest half-pound, and the standard deviation ought to be somewhere around such a
value. But there may be a concealed error of a pound or two, observable only if you
have access to a standard weight whose value is trustworthy to perhaps a few ounces.
No measurement is better than the calibration of the measuring device, no matter how
impressive the flashing digital read-out!
In the primary scientific literature, it’s important that sufficient indication be
given of both the imprecision and the systematic error of every item of data, either
with the items of data or in the section presenting the methods used. In a general
discussion (such as in a textbook) one can get away with looser practices; still, it's a
good idea to provide, in context, sufficient distinction among numbers meaningful to
an order of magnitude (a ten-fold factor), to about 50% either way, to about 10%, and
to about 1%. Saying something is "of the order of 100,000", something else is "very
roughly 100,000", another number is "about 100,000" and still another is "101,000"
makes about these distinctions.
IV. ALGEBRA—EQUATIONS
Algebraic statements, which we've been tacitly using, are essentially simple
sentences, with the ordinary sequence, subject-verb-object. Subject and object are
mathematical expressions. The commonest verb is "equals"; a statement using
"equals" is called an "equation"...
Two plus four equals six.
2 + 4 = 6.
With ordinary algebra, an equation is equally valid read either from subject end
or from object end. In an algebraic statement, one or more numerals are replaced by
letters. A letter (Roman, Greek, or other) stands for a number which is, at the
moment of writing, unspecified (almost always—, the ratio of the diameter to the
circumference of a circle, with a value of 3.1416, is an exception). Queerly enough,
as we'll see, it proves very useful to symbolize something unknown. The choice of
letter is fairly arbitrary—it may be a de facto abbreviation (r for radius, V for volume)
or a convention (S for area). Beyond that, letters early in the alphabet (a, b, c) most
commonly refer to constants; letters late in the alphabet (x, y, z) most commonly refer
to variables...
2+a=6
(a is obviously 4).
Two (or more) letters run together are (with very few exceptions) treated as if
separated by a multiplication sign; thus the expression, "F = ma", implies that m is
multiplied by a.
An equation might describe how to figure the surface area of a rectangular
block...
S = 2 lw + 2 lh + 2 wh,
where S is area, l is length, w is width, and h is height. Notice how much more
economical of space and easily perused is the equation compared to a verbal
statement such as the one earlier for standard deviation.
Letters or numerals below the level of the others ("subscripts") are just labels put
there for convenience. Thus one way of stating that the net pressure across a barrier
is the difference between outside and inside pressures is the following...
p = po - pi .
(, the Greek "delta" is thus a shorthand for "difference between values of".)
By far the most important rule for working with algebraic statements is the
following. You can always do the same thing to both sides of an equation without
changing the essential meaning or truth of the statement. Put in terms of equations,
it says that you can do just about anything as long as you do it simultaneously to the
expressions on both sides of the equal sign.
The commonest thing one has to do with an equation is to "solve" it—to find the
value of a particular unknown. It's usually done by combining things on each side of
the equation wherever possible and by applying the rule above.
2a + 4 = 5 + 5
2a + 4 = 10
2a = 6
and the equation is solved!
combine 5's and you get...
then subtract 4 from both sides to get...
then divide both sides by 2 and you have a = 3
One can get quite accustomed to "reading" equations for some idea of how one
quantity changes in response to the change in one or more others. It may be useful
to rearrange an equation (again applying the rule above) to assist the process. We
can consider a few cases as examples.
a. A hypothetical relationship between the number of organisms which can feed
in a given area and the mass of each organism...
C = n(Mb)
(where C is a constant, n is the number of creatures, and Mb is the body mass of
each.) Clearly a doubling of numbers would necessitate a halving of mass; a tripling
of mass would require reducing numbers by a third, etc.
b. A fast fox catches a slower rabbit. How is the time it takes related to the
distance between them when pursuit starts and to the difference in their speeds? For
steady speeds, distance is the product of speed and time.
Pursuit takes a time t,
the rabbit runs at speed vr and the fox at vf, and they're initially a distance do apart. A
little thought should convince you that
do = vft – vrt .
A little additional messing around gives
do = t(vf – vr) .
And one might then divide both sides by (vf - vr):
t = do / (vf - vr) .
All of which says that the time it takes is equal to the initial distance between
them divided by the difference in their speeds. Or that the initial distance can be no
greater than the time available for pursuit times the difference in their speeds. And
all of which tells us when stalking can stop and pursuit begin. It also tells us how far
away the prey must detect the predator for a certain distance between prey and refuge
(rabbit and hole, here), if the prey is to live to tell of the encounter.
c. The surface area S is being compared to the volume V for a set of cubeshaped objects of different sizes. The following equation is obtained...
S = 6V2/3
What the equation says is that for any change (increase or decrease) in volume,
surface area changes in the same direction (the exponent is positive) but by a smaller
factor (the exponent is less than one). (It's true, in fact, for any set of similarly
shaped solids—only the 6 is specific for cubes.) If you triple the volume you
(roughly) double the area. If you double the length of each side, you increase the
volume by a factor of 8 but
increase the surface by only a factor of four. The
biological implications are truly profound, believe it or not.
V. ALGEBRA—PROPORTIONALITIES
Consider an equation relating two variables and containing a constant, k, of the
form,
y = kx .
What this says is that any multiplicative operation done to x implies the same
operation done to y—triple x and you must thereby triple y, for instance. To avoid
the distraction of that mild red herring, k, we can state the relationship between x and
y as a so-called "proportionality:" "y is proportional to x,", and write it using a
symbol for "is proportional to" as
y  x.
For this particular proportionality we can make the more specific statement that
"y is directly proportional to x," distinguishing it from
y
 1/x
for which we say that "y is inversely proportional to x," or, for that matter, to
distinguish it from any other proportionality such as
y  x3
or "y is proportional to the cube of x." For this latter, doubling x implies an
eight-fold increase in y.
Note that you can multiply or divide both sides of a proportionality by the same
number or you can raise both sides to the same power, but you violate the
relationship if you add the same thing to both sides.
VI. GRAPHS
A great aid in envisioning the implications of equations is a graph. In the most
ordinary sort, one considers two variables. As one (the "independent variable")
changes, the other (the "dependent variable") changes in consequence according to
the rules embodied by the underlying equation. This is so even if the equation can't
easily be written down or exists only in theory or is just a summary of some
experimental data. It ought to be noted that the distinction between independent and
dependent variables can be somewhat arbitrary at times. But usually the distinction
is meaningful if less than crucial—for the fox mentioned earlier, the distance covered
increases as time advances. Time is the independent variable, distance dependent,
since it seems less sensible to say that elapsed time increases as the distance
increases.
On the ordinary sort of plot, the independent variable is expressed as horizontal
distance and marked on the so-called "x-axis" or "abscissa". The dependent variable
is given as vertical distance and marked on the "y-axis" or "ordinate". A line then
goes through all the points which satisfy the relationship given by the equation or
through a set of experimentally determined points from which an equation might be
concocted.
Frequently the "slope" of a line on a graph is of great interest. The latter is a
measure of how far upward the line goes for a given distance from left to right. In a
graph of the equation
y = 6x + 4
the slope is 6. And the line intersects the vertical (y) axis at a value of 4 (when x =
0, 6x = 0, and thus y = 4). In short, from the straight line of the graph the equation
can be derived. It is common practice to use y for the dependent variable and for it
to be the sole item on the left of an equation and to use x as independent variable,
with other items, on the right (as above).
VII. FORMULAS
A few specific formulas will be useful...
a. The "Pythagorean theorem." Consider a triangle in which one of the angles
is a right angle (90 ). Calling the two sides adjacent to that angle a and b and the
side opposite c, the relationship among the lengths of the three sides is
c2 = a 2 + b 2
b. For a circle of radius r and diameter d:
Circumference:
C = 2r =
Surface area, one side:
S =
d
r2
c. For a sphere of radius r:
Surface area:
Volume:
S = 4 r2
V = 4/3 r3
d. For a cylinder of radius r and height h:
Surface area:
Volume:
S = 2 rh (excluding ends)
V =
r2h
VIII. CONVERTING UNITS
Biomechanics mainly uses SI units (a particular subset of metric units), some of
which are everyday items and others of which are uncommon outside the scientific
literature. A few conversion factors are given below—to convert to the SI unit on
the left, multiply by the indicated factor; to convert from the SI unit, divide by the
factor.
1 meter
=
3.28 feet
39.37 inches
0.000621 miles
1 kilogram
=
35.3 ounces (avoirdupois)
2.20 pounds weight (on earth)
1 cubic meter
=
1,000,000 cubic centimeters
1000 liters
264.2
US gallons
1 meter/second
=
3.28 feet/second
2.24 miles/hour
3.60 kilometers/hour
1 kg/cubic meter
=
0.001 grams/cubic centimeter
0.0624 pounds/cubic foot (earth)
1 newton
=
100,000 dynes
0.225 pounds
0.102 kilogram (on earth)
1 pascal
=
0.000145 pounds/square inch
0.00750 millimeters of mercury
0.00000987 atmospheres
1 joule
=
10,000,000 ergs
0.738 foot-pounds
0.238 calories
kilocalories (Calories)
1 watt
0.000238
0.000948 BTU
=
0.00134 horsepower
3.412 BTU/hour
To convert temperature in degrees Fahrenheit to degrees Celsius, subtract 32 and
then multiply by 5/9.
To convert temperature in degrees Celsius to degrees Fahrenheit, multiply by 9/5
and then add 32.
To convert degrees Celsius to Kelvin, add 273.
The most useful source of conversion factors I've ever seen for the types of
problems raised here is a booklet by Pennycuick (1988). I use its earlier incarnation
and can't imagine doing much at all at a desk that lacks a copy.
VIII. REFERENCES
Pennycuick, C. J. (1988) Conversion Factors: SI Units and Many Others.
Chicago: University of Chicago Press. 47 pp.
Stevens, S. S. (1946) On the theory of scales of measurement. Science 103: 677680.