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Fun with Next to Nothing

Polynomial Ratios as Hyperreal Numbers:

Infinitesimals without Handwaving,

Limits without all that Tedious Mucking About in Shrinking Deltas and Epsilons

Calculus, the mathematics of change, helped make possible the modern age of science and technology. But it is all too often perceived as a barrier to understanding those subjects. The fear of calculus and the necessity for proficiency in the subject have resulted in several competing approaches to what has been called "Calculus Reform." We've been there before. Remember the New Math?

In the wake of the launch of Sputnik at the dawn of the Space Age, the American public was in a state of panic: How could we have fallen so far behind, and what do we need to do to catch up? The School Mathematics Study Group (SMSG) was one of many educational think tanks tasked with introducing reform in public school curricula to encourage students to become scientists and engineers. SMSG introduced the "New Math," with an emphasis on mathematical structures such as groups, rings, and fields.

I was one of those precocious students often dismissed as "nerds" (or worse!), interested in math and science, and I welcomed the New Math, then being implemented a grade ahead of my own until my senior year, when both the 11th and 12th grades were implemented together. (The pedagogical pendulum has swung a few times between "new math" formal structure and "old math" back-to-basics since then.)

12th grade SMSG math included an introduction to concepts of Calculus, particularly to the derivative of a function as the "slope" of that function (as a "derived" function of the same variable, the slope at x being understood to mean the slope of a tangent to the function at x:

The slope of f(x) was writtenf ′(x), the Lagrange prime notation for the derivative. (In most fonts, f prime comes out f′, with the prime merging into the f, so I usually insert a space for clarity: f ′.) The Leibniz notation, df(x)/dx was seldom used, mostly because probably reflecting the mathematical "party line" of the time, that there was no such animal as an infinitesimal like dx.

Without actually defining "limit" yet, the tangent to f(x) at x is the line through a point (x, f(x)) that is the limit of a line plotted through that point and (x+δ, f(x+δ)) as δ approaches 0, and the slope of the function at that point is the slope of that limit line.

I do not recall the Weierstrass formal "epsilon-delta" definition of a limit being a part of the SMSG program, but it was hinted that we would encounter it, along with the "other half" of calculus, integration, in college. A more informal approach was sufficient for high school.

We will start with a classic example of non-constant motion, that of a falling object, say a stone dropped from a great height.

Phrasing the problem so that the numbers are all positive, the stone falls 16 feet in the first second, and continues to fall a distance of x feet proportional to the elapsed time t seconds, so that x = 16t².

What is the stone's downward velocity v at time t?

Let ∆t be a change in t, and ∆x be the corresponding change in x. (∆, upper case Delta, stands for Difference, and should not be confused with δ, lower case delta, a variable we will use later.)

Then (writing the square of ∆t as (∆t)² rather than the more customary ∆t²)

∆x = 16(t+∆t)² - x = 16(t²+2t∆t+(∆t)²) - 16t² = 16t²+32t∆t+16(∆t)² - 16t² = 32t∆t+16(∆t)²,

and the average velocity v_av = ∆x/∆t = (32t∆t+16(∆t)²)/∆t = 32t+16∆t,

which gets closer to the "instantaneous" velocity v as ∆t (and ∆x) approaches 0.

Letting ∆t become 0, ∆x/∆t = 32t+16∆t = 32t+0 = 32t, the desired velocity v as a function of t.

At this point, you should be frantically waving your hands: "Wait a minute! You've just divided ∆x by ∆t, and now your making ∆t (and ∆x!) zero. Since when does 0/0 mean anything?"

You would be correct, because that final step is just so much handwaving. If v = 0/0 meant anything, it would have to be that v wsas the solution of the equation, 0v = 0. But every number is a solution to that equation, which would make v anything at all! Newton explained that what he meant by an instinaneous velocity was not the ratio of 0 to 0, but the "ultimate ratio" or limit that ∆x/∆t was approaching just as ∆x and ∆t were becoming 0.

In 1734, George Berkeley published his criticism of the "new analysis," The Analyst, the most famous quotation of which (paraphrased in more modern terms) asked the questions: What are these "instantaneous" rates of change? The ratios of vanishing increments? And what are these "vanishing" Increments? They are neither finite quantities nor "infinitesimal" quantities, nor yet nothing. May we not call them the ghosts of departed quantities?

The properties of the "real numbers" were not understood well enough to justify Newton's ultimate ratios until Richard Dedekind proposed an "axiom of continuity" that "completed" the field of rational numbers, filling in all the irrational numbers mathematicians had been discovering sporadically since the days of the ancient Greeks: Every set of real numbers that has an upper bound, a number equal to or greater than every number in the set, has a unique least upper bound, either the greatest number in the set, or the least number greater than all members of the set.

It follows that every set with a lower bound has a greatest lower bound.

If ∆x/∆t is increasing as ∆t approaches 0, Newton's ultimate ratio is the least upper bound of those ratios, greatest lower bound if it is decreasing.

Since a/b is whatever c uniquely satisfies the equation bc = a, and no c satisfies 0c = a except when a is itself 0 and every c works, we still need a definition of limit that gives us a way to check that all our handwaving has not led us astray.

Karl Weierstrass defined the limit f(x) approaches (if it exists) as x approaches a value c as that number l such that, if for every positive number δ (no matter how small), |x - c| < δ, that is, the absolute value of the difference between x and c, is less than δ, there is a positive number ε such that |f(x) - l| < ε, that is, the absolute value of the difference between f(x) and l, is less than ε.

The limit of f(x) as x approaches c is written lim_x→cf(x). Using this notation, the velocity v = lim_Δt→0Δx/Δt.

The "limit at infinity" or "limit as x approaches infinity" of f(x), written lim_x→∞f(x), is just the limit as 1/x approaches 0, since ∞ is not an actual number. We also encounter "infinite limits" of f(x) as x approaches c, written lim_x→cf(x) = ∞, (even though ∞ is not an actual number), when |f(x)| exceeds all positive numbers as x approaches c. An "infinite limit" at c (or at ∞) is just just a way of saying the absolute value of the function grows without a limit as the variable approaches c (or exceeds every real number).

Before the Weierstrass definition, limits were understood only in vague terms of continuous motion, or even vaguer terms of infinitesimals. Continuity could now be made a rigorous concept by defining it in terms of limits: f(x) is continuous at c if and only if the limit of f(x) as x approaches c is f(c). And f(x) is continuous throughout an entire interval s of real numbers between two real endpoints (even the entire set of real numbers R "between minus infinity and plus infinity") if and only if it is continuous at every real value of x in the interval s.

The rigorous foundation of calculus upon the concept of limits within the real number system was a major triumph of Nineteenth Century mathematics. It transformed the vague notion of x and f(x) approaching c and l with the more concrete concept of choosing any positive δ, letting x be within δ of c, and then finding a positive ε for the chosen δ where f(x) is within ε of l. Pick smaller and smaller deltas, and find smaller and smaller epsilons, trapping the limit l within shorter and shorter intervals of length epsilon. Once you prove you can do this for some l, you can discard all those "shrinking" variables, and emerge with the limit l you were seeking.

The handwaving argument above with its division by a number that is not 0, but is allowed to become 0, provides an "educated guess" that still needs to be tested against the Weierstrass definition before it is accepted as the correct answer.

Generations of calculus students, their eyes glazed over with that tongue-twisting Weierstrass definition, have said, "There must be a better way."

And there is. It's called Infinitesimals.

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