The “epsilon-delta” nature of analysis can be daunting and unintuitive to students, as the heavy reliance on inequalities rather than equalities. But it occurred to me recently that one might be able to leverage the intuition one already has from “deals” – of the type one often sees advertised by corporations – to get at least some informal understanding of these concepts.
Take for instance the concept of an upper bound or a lower bound
on some quantity
. From an economic perspective, one could think of the upper bound as an assertion that
can be “bought” for
units of currency, and the lower bound can similarly be viewed as an assertion that
can be “sold” for
units of currency. Thus for instance, a system of inequalities and equations like
Currency | We buy at | We sell at |
| | |
| – | |
| | |
| – | |
Someone with an eye for spotting “deals” might now realize that one can actually buy for
units of currency rather than
, by purchasing one copy each of
and
for
units of currency, then selling off
to recover
units of currency back. In more traditional mathematical language, one can improve the upper bound
to
by taking the appropriate linear combination of the inequalities
,
, and
. More generally, this way of thinking is useful when faced with a linear programming situation (and of course linear programming is a key foundation for operations research), although this analogy begins to break down when one wants to use inequalities in a more non-linear fashion.
Asymptotic estimates such as (also often written
or
) can be viewed as some sort of liquid market in which
can be used to purchase
, though depending on market rates, one may need a large number of units of
in order to buy a single unit of
. An asymptotic estimate like
represents an economic situation in which
is so much more highly desired than
that, if one is a patient enough haggler, one can eventually convince someone to give up a unit of
for even just a tiny amount of
.
When it comes to the basic analysis concepts of convergence and continuity, one can similarly view these concepts as various economic transactions involving the buying and selling of accuracy. One could for instance imagine the following hypothetical range of products in which one would need to spend more money to obtain higher accuracy to measure weight in grams:
Object | Accuracy | Price |
Low-end kitchen scale | | |
High-end bathroom scale | | |
Low-end lab scale | | |
High-end lab scale | | |
The concept of convergence of a sequence
to a limit
could then be viewed as somewhat analogous to a rewards program, of the type offered for instance by airlines, in which various tiers of perks are offered when one hits a certain level of “currency” (e.g., frequent flyer miles). For instance, the convergence of the sequence
to its limit
offers the following accuracy “perks” depending on one’s level
in the sequence:
Status | Accuracy benefit | Eligibility |
Basic status | | |
Bronze status | | |
Silver status | | |
Gold status | | |
| | |
With this conceptual model, convergence means that any status level of accuracy can be unlocked if one’s number of “points earned” is high enough.
In a similar vein, continuity becomes analogous to a conversion program, in which accuracy benefits from one company can be traded in for new accuracy benefits in another company. For instance, the continuity of the function at the point
can be viewed in terms of the following conversion chart:
Accuracy benefit of | Accuracy benefit of |
| |
| |
| |
| |
Again, the point is that one can purchase any desired level of accuracy of provided one trades in a suitably high level of accuracy of
.
At present, the above conversion chart is only available at the single location . The concept of uniform continuity can then be viewed as an advertising copy that “offer prices are valid in all store locations”. In a similar vein, the concept of equicontinuity for a class
of functions is a guarantee that “offer applies to all functions
in the class
, without any price discrimination. The combined notion of uniform equicontinuity is then of course the claim that the offer is valid in all locations and for all functions.
In a similar vein, differentiability can be viewed as a deal in which one can trade in accuracy of the input for approximately linear behavior of the output; to oversimplify slightly, smoothness can similarly be viewed as a deal in which one trades in accuracy of the input for high-accuracy polynomial approximability of the output. Measurability of a set or function can be viewed as a deal in which one trades in a level of resolution for an accurate approximation of that set or function at the given resolution. And so forth.
Perhaps readers can propose some other examples of mathematical concepts being re-interpreted as some sort of economic transaction?
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