We got to this quest for clarity of a few basic mathematical concepts as
a consequence of thinking about the issues that we talk about in our post, ‘So Simple, Yet So Hard: The Square Root Drama’ (Pinheiro, 2014).
As a reply to our post, Master Brian Treadway (Treadway, 2014) wrote (Treadway, 2014b):
“And yet (36) ^
(1/2) = ((-6) ^ 2) ^ (1/2).”
Notice that what is now under
discussion is the transitivity of the equalities plus the replacement of
numbers that are seen as parts of a computation.
Sadly, also Dr. Maths (The Math Forum, 1991-2013) goes wrong on this one. We
have made the important parts of their discourse appear in yellow here:
“The two are very closely related; the difference lies in their generality and their role in logic. Often they are interchangeable, but not always.
Substitution is a "common-sense" concept: if two things are equal, then one can be put in place of the other and nothing will change. Essentially, it is part of the definition of "equal": two things are equal if and only if they can be substituted for one another. It can be used to explain why, for example, a = b + 5 and b = c implies that a = c + 5.
The first statement here could be replaced by ANY statement about b; it is very general.
Transitivity is a little more formal; it is one of a set of properties (relexivity, symmetry, and transitivity) used to define the concept of "equivalence relation" (of which equality is one example). It also has a more specific definition than substitution; it only applies when we have two equalities: a = b and b = c implies that a = c.
This can be considered a special case of substitution, replacing b with c in the equation a = b. So we could always use the term "substitution" if we wished; but we could not use the term "transitivity" in place of "substitution" in cases where the same quantity (b above) is not found alone on one side of each equation.
You can see why we call transitivity a "property of equality" (or, more generally, of an equivalence relation), but do not call substitution a "property" of anything in particular. It is more general than that.
Here is one place where I commented on the relationship of these concepts: Isosceles Trapezoid Proof
http://mathforum.org/library/drmath/view/55425.html"
On (The
Math Forum, 1991-2013b),
Dr. Maths says (again we have used the color yellow to
mark the most important parts of their discourse):
“The transitive property looks a lot like substitution, and I wouldn't really count it wrong to say the latter when the former is true. In a sense the transitive property is a special case of substitution; it says that if a = b and b = c, then a = c and you could express this in terms of substitution as by substituting c for b in "a = b", we find that a = c.
But since the transitive property is stronger (more clearly defined, perhaps), it is better to use that whenever you can, and to reserve "substitution" for more complex cases where you substitute within an expression, rather than replacing a whole side of an equality. But again, that's far from a major error. The basic idea is right either way.”
We believe that we may have been a bit lazy and superficial when teaching Mathematics to people this far.
It is clearly missing either reinforcing boundaries or mentioning them.
Why can we not replace the 6 in (62)1/2
with a -6?
Basically, that would be the same as saying that, because 62=36
and 36=(-6)2, we can replace 36 with B, 62 with A, and (-6)2
with C to then have that A = B and B = C and then utter that A = C
and therefore that 62=(-6)2.
Philosophically, what we are saying is that ‘the how does
not matter’ or ‘the way to get to somewhere does not matter, that is, all paths
that lead to place X are equivalent’.
If we go to Philosophy, we immediately understand that that
cannot be true in Mathematics, since all that matters in Mathematics is
obviously how we get to somewhere.
How we get to the 36 is obviously the only thing that
matters.
The fact that both A and C lead to 36 cannot make of A
something immediately equivalent to C: Restrictions would have to be imposed,
as a minimum thing, for this equivalency to hold.
Notice that if we had only one number to each side, the
transitivity of the equality would be true.
For instance, consider ½ = 0.5 and 0.5 = 2/4. We know that,
in this case, ½ = 2/4.
One would then say that division is an operation and we
actually have more than one number playing the role of A and C.
That is true, but, as long as the rules of Mathematics are
obeyed by all equalities involved, the transitivity will always be verified for
this sort of ‘operation’ then.
It will also be verified when we have a simple
multiplication to both sides.
Consider, for instance, 2 x 3 = 6 and 6 = 6 x 1. We know
that, in this case, 2 x 3 = 6 x 1.
Perhaps because both exponential and logarithm are what
could be called ‘compound operations’ (formed from multiple operations: multiplication
and sum) but both multiplication and division are what could be called ‘simple
operations’ (formed from one operation: sum in one case and distribution in
another), we have restrictions to be imposed only when we have exponentials and
logarithms in what regards transitivity and replacement.
We can always write operations that are not possible in
Mathematics, or equalities that do not hold, and ask for the person to ‘solve our
problem’, is it not?
We could then have 2/0=1 and 1=3/2, for instance. We
obviously would know that we could never apply transitivity and replacement
here because the original problem could not have existed.
If we say that log 100 = 2 and 100 = (-200)/(-2), we are not
copying the model we had when we spoke about transitivity and replacement. Is that clear?
We now have log B = A and B = C, not A =B and
B = C, as it was before.
We must here remember that, at least in the reals, we do not allow our log to have a negative
base or a negative 'logarithmand' (number), even though we allow it to have a negative value.
If we actually replaced our first 100, which is our B, with C, that is, with (-200)/(-2), we could think of splitting what we get into two other logs using the property of the log
of the division, for instance, what will then clearly show to us that we cannot
do it.
In our case, which is very similar to the original problem, things are
subtler, for we have an exponential instead.
Notice, however, that for us to be able to apply the
property that allows us to multiply exponents, we must have a ‘true
multiplication’.
The fact that we cannot apply commutativity to the exponents
tells us that we do not have a true multiplication.
Notice that there is no square root of negative numbers. We
at most write the negative part in terms of the complex numbers.
We therefore cannot swap ½ with 2 in [(-6)2]1/2
. We can do that in (62)1/2 however.
Now, what looked equivalent to us in the beginning of this
text cannot look that way anymore: They are definitely different items.
The reason why we cannot replace 6 with -6 in this
particular situation then is that we would stop having an allowed operation in
Mathematics to have one that is impossible to be solved, that has result ‘infinity’,
or that cannot exist, or that will be at most coded by means of 'complex numbers'.
We can then say that we should be adding more details
when teaching Mathematics to others: When we teach transitivity, we should speak
about simple and compound operations and say that transitivity with replacement can only happen when we have simple operations.
References:
Pinheiro, M. R. (2014). http://herewedoteachmaths.blogspot.com.au/2014/01/so-simple-yet-so-hard-square-root-new.html
Treadway, B. (2013). Brian Treadway. Retrieved January 11 2014 from ca.linkedin.com/pub/brian-treadway/2b/46/33
Treadway, B. (2013b). Retrieved January 11 2014 from http://www.linkedin.com/groupAnswers?trk=view_disc&gid=33207&commentID=5827521797708353536&ut=3RiAvd0ioDm641&viewQuestionAndAnswers=&fromEmail=&discussionID=5827238737582514179
The Math Forum. (1991-2013). Comparing Transitivity and
Substitution. Retrieved January 11 2014 from http://mathforum.org/library/drmath/view/64446.html
