In the days before calculators and personal computers
an engineer always had a slide rule nearby. These days it
is difficult to locate a slide rule outside of a museum.
I don't know how many of those who have used a slide rule
ever thought of it as an analog computer, but that is
really what it is. As such, I think that it is an ideal
tool for teaching the mathematical concept of
transformations. With the proper scales, a slide rule can
be used not only to multiply but to find the third side
of a right triangle and add velocities relativistically.
In order to introduce the concepts that will be used, we
will start with the simplest type of slide rule - one
that is made up of two ordinary rulers.
The addition slide rule was compared above to a mirror. Perhaps a more apt metaphor would be a translation.
Suppose you traveled back in time to the days of the Roman Empire. You notice someone doing arithmetic using Roman numerals and you want to verify your understanding of this numeric representation. Let T be the transformation from Arabic numerals to Roman numerals. According to what you were taught T(5) = V, T(III) = 3 and T(8) = VIII.
You observe that this particular person seems to be using ".." to stand for "+". You hand over a sheet of paper with V .. III written on it. When the person writes VIII on the paper this increases your confidence that the above transformation is correct. You have established a degree of internal consistency. We can think of addition as taking two input values to produce an output value. Since the output is determined by the inputs, for one process to be a translation of the other it is sufficient for equivalent inputs to result in equivalent outputs. (5+3) is the output of the Arabic numeral arithmetic. T(5)..T(3) is the output of the Roman numeral arithmetic. For the outputs to be equivalent we must have T(5)..T(3) = T(5+3) . If we could prove that in general T(X)..T(Y) = T(X+Y)then we could say that the one process is a translation of the other.
For the addition slide rule we have D(X) // D(Y) = D(X+Y) . The combining of distances is a translation of the addition of numbers. It does not matter that numbers and distances are different entities. Adding numbers is structually equivalent to combining distances. The next section states this in more formal terms.
The logarithmic function satisfies the relationship
log (x * y) = log(x) + log(y).
The log function is an isomorphism. It transforms a
multiplication problem into an addition problem. Since
addition of numbers is isomorphic to addition of
distances, consider the effect of applying the distance
function D to both sides of the equation.
D(log(x*y)) = D( log(x) + log(y) ) = D( log(x)) // D(log(y) )
D(log(x*y)) = D(log(x)) // D(log(y))
It follows that the transform (D log) formed by composing
the D and log transforms is an isomorphism. The same
process could be used to show that in general the
composition of two isomorphic transforms is an isomorphic transform -
the mirroring of the mirroring of a domain is itself a
mirroring of the domain;
the translation of a translation is a translation of the original.
The construction of a slide rule for multiplication
follows from the above equation. If the distance
that a number is placed is equal to the log of the
number then the result is the standard slide rule.
Figure 3 shows how the slide rule is used to
multiply 10 and 1000.
Shortly after their discovery it was realized that
logarithm functions loga(x) were the inverse
of exponential functions ax. For convenience,
let us write the exponential function as EXP(X) and the
logaritm function as LOG(X) for some common base a.
Since LOG(X) is an isomorphism and EXP(X) is its inverse,
then by what was shown above we get:
EXP(X+Y) = EXP(X) * EXP(Y),
which is an expression of the law of exponents.
The final slide rule will compute relativistic velocity.
As a quick explanation of what this is, consider a
person walking forward with velocity u in a train
traveling with velocity v. Then the exact velocity w of
the person relative to the ground can be expressed
using the isomorphic relationship
(c + w)/(c - w) = (c + u)/(c - u) * (c + v)/(c - v), where c is
the speed of light (about 186,000 miles per second).
To avoid having to use the speed of light in our calculations
we can express all the velocities as fractions of the
speed of light. Dividing the numerator and denominator
of all three terms by c gives
(1 + p)/(1 - p) = (1 + q)/(1 - q) * (1 + r)/(1 - r), where
p = w/c, q=u/c and r = v/c.
Applying in succession the log and distance
functions to both sides gives:
D(log ((1 + p)/(1 - p) )) =
D(log( (1 + q)/(1 - q) )) // D(log( (1 + r)/(1 - r))).
To construct a slide rule to add velocities, set the
distance of a fraction x equal to
log( (1+x) / (1-x) ).
Figure 5 shows a slide rule for adding velocities.
Consider again the original formula for velocity (c + w) / (c - w) = (c + u)/(c - u) * (c + v)/(c -v). Let u & v be the sum of velocities u and v. Let T(x) = (c + x)/(c - x). Then T(u & v) = T(u)*T(v) How do we add three velocities? In the example of the person walking with velocity u in a train traveling with velocity v, let s be the velocity of the earth relative to the sun. What is the velocity of the person relative to the sun? Velocities can only be added two at a time. There are two ways of doing this and we would hope that they come out the same. We could first find the velocity of the person relative to the earth and then add this velocity to the velocity of the earth relative to the sun. This would give T(s & (v & u)) = T(s) * T(v & u) = T(s) * (T(v) * T(u)) On the other hand we could first find the velocity of the train relative to the sun and then add the person's velocity. We then have T((s & v) & u) = T(s & v) * T(u) = (T(s) * T(v)) * T(u). The two values are of course the same. The reason for this is that multiplication is associative, i. e., (a * b) * c = a * (b * c). We see that this causes addition of velocities to be associative - s & (u &v) = (s & u) &v. The above argument could have been used for any isomorphism. Thus we have the property that isomorphisms preserve the associative property just we showed earlier that they preserve the commutative property. We could have used this property to state immediately that it does not matter which of the two ways the velocities are added because multiplication is associative. It could also have been argued that using the velocity slide rule, it is obvious that it does not make any difference in which order the velocities are added. This is because the addition of distances is both commutative and associative and any calculation for which we can construct a slide rule must therefore also be both commutative and associative. In the above formula for velocity addition it is possible to solve for w to get u & v = (u + v)/ (1 + u*v/c2). To show that addition of velocities is associative we could have then solved explicitly for both (s & (v & u)) and ((s & v) & u), but this involves a great deal more effort. We can also apply the above results to the parallel resistor and right triangle examples. Using the notation in the section on right triangles, the distance (x & y) that results from East and North displacements of x and y is given by s(x & y) = s(x) + s(y). To generalize to three dimensions we get s(s & y & z) = s(x) + s(y) + s(z). For combining several parallel resistors we get: 1/r = 1/r1 + 1/r2 + ... + 1/rn.
Although only a small portion of high school students is
likely to become computer programmers, most of them will
probably be using computers in one way or another. The
distinction between program developers and program users
has been blurred by such software as spreadsheets and
database query programs. There are thus good practical
reasons for introducing Boolean algebra in high school.
The DeMorgan rules reveal a fundamental isomorphism that
facilitates teaching this subject and makes it more
interesting.
The variables in Boolean algebra can take on only two
values - TRUE or FALSE and the principal operations are
AND, OR and NOT. The variables can stand for any
statement; we are concerned with whether the combination
of such statements is either true or false.
The AND, OR and NOT operators agree with what common
sense would say they should be. NOT is the simplest
operator. It takes only one variable.
NOT X = FALSE if X is TRUE and TRUE if X is FALSE.
Because Boolean variables, unlike numerical values, take
on only two values we can completely specify the AND and
OR operations with tables.
| Y | ||||
|---|---|---|---|---|
| TRUE | FALSE | |||
| X | ||||
| TRUE | TRUE | FALSE | ||
| FALSE | FALSE | FALSE | ||
Whenever possible, check to see if a result "makes
sense". The above table says that X AND Y is TRUE
only if both X and Y are both TRUE, in agreement with
how the term AND is used in everyday use.
| Y | ||||
|---|---|---|---|---|
| TRUE | FALSE | |||
| X | ||||
| TRUE | TRUE | TRUE | ||
| FALSE | TRUE | FALSE | ||
The table for X OR Y says that X OR Y is FALSE only if
both X and Y are both FALSE.
One complication is that the word "or" in
English can have two different meanings; which one is
being used can usually be determined from the context of
the statement. "or" can be defined as above or
it can be the same except that it is defined as FALSE
when X and Y are both TRUE. For example, if I say
"Either candidate A or candidate B will be the next
president", it is understood that this rules out the
possibility of both A and B being the next president.
This type of "or" is referred to in logic as an
exclusive or (XOR) and by way of contrast OR is sometimes
referred to as an inclusive or. In this section I will
only be dealing with the inclusive or.
Notice that the summary of the X OR Y table is the same
as the summary of the X AND Y table with the words TRUE
and FALSE interchanged. This suggests the following
isomorphism, which is one of DeMorgan's two rules:
NOT (X AND Y) = (NOT X) OR (NOT Y)
In plain language this says that if the statement X AND Y
is not TRUE then either X is FALSE or Y is FALSE. We can
formally prove the statement by using the two above
tables to show that both sides of the equation are equal
for all four combinations of values for X and Y.
To get the other of DeMorgan's rules we apply the inverse
of NOT to show the isomorphism in the other direction.
NOT is its own inverse so we get:
NOT (X OR Y) = (NOT X) AND (NOT Y)
AND and OR are isomorphic. We could in principle discard
OR from our vocabulary and just use AND, though I would
not recommend this unless you are planning a career in
politics. "It will rain today or tomorrow"
would become "It is not true that it will not rain
today and it will not rain tomorrow". There have
been times, however, when I have used DeMorgan's rules
when programming to simplify statements.
The isomorphism simplifies proofs. We can use the table
to show that AND is commutative and associative. It
follows immediately by isomorphism that the same is true
of OR.
We can go further. The fact that NOT is the transform
function for both AND and OR means that we can apply both
transforms simultaneously. For example, consider the
following distributive identity between AND and OR which
is analogous to the distributive operation of
multiplication and addition:
X AND (Y OR Z) = (X AND Y) OR (X AND Z).
We can prove this statement by substituting all 8
combinations of X, Y and Z. We should also test the
reasonableness of the statement by using an example.
"I will speak to Sarah and (I will speak to) John or
Raymond." is the same as "I will speak to Sarah
and John or I will speak to Sarah and Raymond."
Apply NOT to both sides of the equation. Using the AND
isomorphism gives the following on the left side.
NOT(X AND (Y OR Z)) = NOT X OR NOT (Y OR Z)
Applying the OR isomorphism gives
NOT X OR NOT (Y OR Z) = NOT X OR ((NOT Y) AND (NOT Z))
Each of the arguments has been negated and the ANDs and
ORs have been interchanged. The same happens on the right
side of the equation. We get
NOT X OR ((NOT Y) AND (NOT Z)) =
((NOT X) OR (NOT Y)) AND ((NOT X) OR NOT Z)
We can get rid of the NOTs by setting X'=NOT X, Y'=NOT Y,
Z'=NOT Z, so that what end up is the same form as we
started except the ANDs and ORs have been interchanged:
X' OR (Y' AND Z') = (X' OR Y') AND (X' OR Z')
For every identity a new one can be created by
interchanging AND and OR.
The relationship between AND and OR is referred to as a
duality. In this case we have a duality between
operators. There are different types of duality but the
general principle is that a dualism exists when true
statements can be generated from other true statements by
interchanging two terms.
Since, in general, xy is not equal to yx,
there can not be slide rule scales for raising a number
to a power if we require both scales to be the same.
However, if we remove the requirement for isomorphism we
can use two different scales to achieve our purpose. We
can create the slide rule scales if we can find two
different functions T and U that satisfy:
T(xy) = T(x) + U(y)
We then create the slide rule having T(x) as the bottom
scale and U(y) as the top scale. Standard slide rules do
this by having T(x) = log(log(x)) and U(y) = log(y).
To see why this works, find log(log(xy)):
log(log(xy)) = log(y*log(x)) = log(y) + log(log(x)).
