MAGICAL MATH RESULTS
…and Their Explanations
Robert “Dr. Bob” Gardner
September 22, 2017
The purpose of mechanics is to describe how bodies change their position in space with “time.” I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations.
FROM: Albert Einstein, Relativity (first published in 1916), Chapter 3, “Space and Time in Classical Mechanics.”
Sloppy Algebra
Pre-Calculus Algebra (MATH 1710)
So what’s up?
DIVISION BY 0!!!
EXPLANATION
You can’t divide by zero!!!
Note. You can’t divide by zero in algebra, calculus, or analysis. There isn’t an advanced class somewhere out there (in mathematics) where people are dividing by zero.
So what’s up?
EXTRANEOUS ROOTS!!!
EXPLANATION
SQUARING BOTH SIDES OF AN EQUATION INTRODUCES EXTRANEOUS ROOTS!!!
Note. That is, in an equation with an unknown, the operation of squaring both sides introduces a new equation with values satisfying the new equation which do not satisfy the original equation. This is illustrated for the real numbers, but holds for other algebraic structures as well.
Calculus
Calculus 2 (MATH 1920)
Finite Paint for an Infinite Wall
1
Note. The resulting object is called “Gabriel’s trumpet.” We will show that it has finite volume and infinite surface area.
Finite Paint for an Infinite Wall
Finite Paint for an Infinite Wall
Note. So the area of the surface of revolution is:
EXPLANATION
Note. I explain this to Calculus 2 students as saying that it isn’t “fair” to compare two different dimensional quantities, say surface area (2 dimensional) and volume (3 dimensional); a physical argument could be based on the units by which these are measured. I pose the question: “How much paint does it take to paint an infinite surface?” Well, it depends on how thick you paint the surface… if you paint it 0 thick, then it will take 0 (volume) of paint; not to say that it will take no paint, but that I could start with 1 gallon of paint, paint an infinite surface with a 0-thick coat, and have 1 gallon of paint left over. This foreshadows measure theory and the fact that a measure 0 set can still have a lot of stuff in it!
EXPLANATION
1
Nobody objects to this, but it is the same story as above, just dropped down one dimension (from finite volume to finite area, and from infinite surface are to infinite perimeter). It is the introduction of the paint terminology that makes things suspect in the discussion of Gabriel’s trumpet.
Everything Equals Everything Else
Everything Equals Everything Else
Note. Let’s rearrange the alternating harmonic series to get something different… say 1.
Note. We take positive terms until the partial sum is greater than or equal to 1, then take negative terms until the partial sum is less than 1, then take positive terms… This gives process yields:
Everything Equals Everything Else
Note. The choice of 1 in the previous argument is arbitrary and could be replaced with any real number. So we conclude that any two real numbers are equal…
EXPLANATION
YOU CAN ONLY REARRANGE ABSOLUTEY CONVERGENT SERIES, WITHOUT AFFECTING THE SUM!!!
Note. This is expressed as “The Rearrangement Theorem for Absolutely Convergent Series”:
EXPLANATION
Note. It is not that The Rearrangement Theorem for Absolutely Convergent Series is surprising; one would probably expect any rearrangement of a series to have no effect on the sum of the series. The surprise is how conditionally convergent series behave:
Set Theory
Analysis 1 (MATH 4217/5217)
Bigger Than Infinity
Bigger Than Infinity
Note. This is called the “Cantor diagonalization argument.”
Bigger Than Infinity
Note. It is easy to say when two finite sets are the same size; it’s when they both have the same number of elements. But for infinite sets, it is more complicated.
EXPLANATION
Note. These ideas about the cardinalities of sets were introduced by Georg Cantor in 1873.
1845-1918
Neither True Nor False
Question: Is The Continuum Hypothesis true or false?
Neither True Nor False
Note. The truth value of The Continuum Hypothesis is a complicated story.
Note. In 1939 Kurt Gödel proved that the Continuum Hypothesis is consistent with the (ZFC) axioms of set theory.
1906-1978
Neither True Nor False
Note. In 1964 Paul Cohen proved that the Continuum Hypothesis is does not follow from the (ZFC) axioms of set theory (it is independent of them).
1934-2007
EXPLANATION
1906-1978
Note. In fact, Kurt Gödel studied the consistency and completeness of axiomatic systems. He showed that in certain axiomatic systems there are meaningful claims
which are neither true nor false; they are “undecidable.”
The Continuum Hypothesis is an example of an undecidable in the ZFC axiomatic system of set theory.
A Set or Not a Set?
1872-1970
Note. In 1903 Bertrand Russell, while working on Principia Mathematica, discovered what became known as Russell’s Paradox.
A Set or Not a Set?
Note. Russell’s Paradox can be informally described in the following story.
Imagine a town with a barber. The barber cuts the hair of all of those who do not cut their own hair (and only those). Who cuts the barber’s hair?
If he does not cut his own hair then he must cut his own hair (since that is his job). If he does cut his own hair, then he cannot cut his own hair since his job is to cut the hair of those who do not cut their own hair.
A Set or Not a Set?
EXPLANATION
Note. This revision of the axioms of set theory in the very early 1900s set the stage for the later work by Kurt Gödel in the 1920s.
Weird Functions
Introduction to Topology (MATH 4357/5357)
Painting a Plane with a Line
Question. Can we map the interval one to one, onto, and continuously to the square?
Answer. NO! [Netto, 1879]
Question. Can we map the interval continuously onto the square?
Painting a Plane with a Line
Guiseppe Peano
1858-1932
Peano’s paper: “Sur une courbe, qui remplit toute une aire plane,” Mathematische Annalen, 36(1), 157–160 (1890).
Painting a Plane with a Line
Note. Additional examples were given by David Hilbert (1862–1943), Eliakim H. Moore (1862–1932), Henri Lebesgue (1875–1941),
Wacław Sierpiński (1882–1969), George Pólya (1887–1985) and others.
David Hilbert
Henri Lebesgue
George Pólya
Painting a Plane with a Line
Note. The desired function is given as a limit of a sequence of functions. There is a “fractal nature” to this sequence of functions
Painting a Plane with a Line
Painting a Plane with a Line
Painting a Plane with a Line
Painting a Plane with a Line
EXPLANATION
A Very Sharp Function
Note. So a continuous function can be non-differentiable at one point. How badly non-differentiable can a continuous function be?
A Very Sharp Function
Note. Weierstrass’ example was the first of many continuous but nowhere differentiable functions. In fact, there are entire books devoted to the topic.
EXPLANATION
Karl Weierstrass, 1815-1897
Note. The graph of Weierstrass’ function has a fractal nature. The animated picture show the first few partial sums of Weierstrass’ Fourier series.
EXPLANATION
(The notes for this topic and these two images are from Wikipedia.)
Measure Theory
Real Analysis 1 (MATH 5210)
Note. The Banach-Tarski Theorem states: “A solid ball may be separated into a finite number of pieces and reassembled in such a way as to create two solid balls, each identical in shape and volume to the original.”
“The Banach-Tarski Paradox does not hold in the plane; a space of three or more dimensions is required.” We have similar paradoxes for an interval and a disk, but these examples required an infinite number of pieces. [L.M. Wapner, The Pea and the Sun—A Mathematical Paradox. A.K. Peters, Ltd., 2005.]
Volume from Nowhere!
Note. Polish mathematician Stefan Banach (1892–1945), of “Banach space” fame, and Alfred Tarski (1902–1983) published “On the Decomposition of Sets of Points in Respectively Congruent Parts” (in French in Fundamenta Mathematicae 6) in 1924. Their work was heavily dependent on earlier work of Vitali and Hausdorff.
Volume from Nowhere!
From: https://warosu.org/sci/thread/6690359
Note. In 1947, R.M. Robinson (1911–1995) showed that the construction could be accomplished using only five pieces (and could not be accomplished using fewer pieces). Two of the pieces form one new sphere and the other three create a second sphere (“On the Decomposition of Spheres,” Fundamenta Mathematicae, 34 (1947), 246–260).
Volume from Nowhere!
Axiom of Choice
EXPLANATION
Note. The Axiom of Choice allows us to show the existence of sets so exotic that we cannot assign to them a volume in a meaningful way. The problem is that the pieces of the sphere are not measurable. So the sum of the measures of the disjoint pieces does not equal the measure of the union of the pieces.
Note. This is addressed in Real Analysis 1 (MATH 5210) when constructing a nonmeasurable subset of the set [0,1]. The Axiom of Choice is used to partition [0,1] into a countable number of disjoint pieces which union to give [0,1]. Each of these pieces has the same measure (if this makes sense), since any one can be rigidly transformed to any other.
EXPLANATION
EXPLANATION
or
EXPLANATION
Note. So it is not that volume can be created out of nothing. It is simply that the Axiom of Choice allows us to create sets which have such bizarre shapes that it is meaningless to assign to them a measure (of length, area, volume,…).
Note. Nonmeasurable sets are a standard part of analysis these days. The story about creating two spheres from one is not meant to be taken as a physical possibility, simply that nonmeasurable sets may violate additivity.
The Sacred Spirit of Lucidity!
Thank you!
Questions?