Monthly Archives: November 2008

36 Methods of Mathematical Proof

image If you ask an average person on the street what is the highest level of mathematics, the most common answer would probably be Calculus.  There might even be a few throwing College Algebra out there as fairly advance.  However, if you ask a math major or engineering student the question of what is the lowest level of mathematics, the foundation of the mathematics they use, the most common answer would likely be Calculus.   Why the disparity? 

The mathematics that is taught from kindergarten through secondary is often limited to procedural techniques to solve specific problem types without recognizing that advanced mathematics is all about recognizing patterns and using axioms, definitions, and theorems to formalize those patterns thereby leading to new patterns.

So, of course, in upper level mathematics we spend a great deal of time moving from procedural mathematics, to proving theorems, to developing new theorems.  Unfortunately, the rigor is sometimes lost in the classroom for many reasons.  I’m sure I’ve used just about every one of the following invalid proof techniques.

  • Proof by obviousness: "The proof is so clear that it need not be mentioned."
  • Proof by general agreement: "All in favor?…"
  • Proof by imagination: "Well, we’ll pretend it’s true…"
  • Proof by convenience: "It would be very nice if it were true, so…"
  • Proof by necessity: "It had better be true, or the entire structure of mathematics would crumble to the ground."
  • Proof by plausibility: "It sounds good, so it must be true."
  • Proof by intimidation: "Don’t be stupid; of course it’s true!"
  • Proof by lack of sufficient time: "Because of the time constraint, I’ll leave the proof to you."
  • Proof by postponement: "The proof for this is long and arduous, so it is given to you in the appendix."
  • Proof by accident: "Hey, what have we here?!"
  • Proof by insignificance: "Who really cares anyway?"
  • Proof by mumbo-jumbo: Mumbo-Jumbo
  • Proof by profanity: (example omitted)
  • Proof by definition: "We define it to be true."
  • Proof by tautology: "It’s true because it’s true."
  • Proof by plagiarism: "As we see on page 289,…"
  • Proof by lost reference: "I know I saw it somewhere…."
  • Proof by calculus: "This proof requires calculus, so we’ll skip it."
  • Proof by terror: When intimidation fails…
  • Proof by lack of interest: "Does anyone really want to see this?"
  • Proof by illegibility: Illegibility
  • Proof by logic: "If it is on the problem sheet, it must be true!"
  • Proof by majority rule: Only to be used if general agreement is impossible.
  • Proof by clever variable choice: "Let A be the number such that this proof works…"
  • Proof by tessellation: "This proof is the same as the last."
  • Proof by divine word: "…And the Lord said, ‘Let it be true,’ and it was true."
  • Proof by stubbornness: "I don’t care what you say- it is true."
  • Proof by simplification: "This proof reduced to the statement 1 + 1 = 2."
  • Proof by hasty generalization: "Well, it works for 17, so it works for all reals."
  • Proof by deception: "Now everyone turn their backs…"
  • Proof by supplication: "Oh please, let it be true."
  • Proof by poor analogy: "Well, it’s just like…"
  • Proof by avoidance: Limit of proof by postponement as it approaches infinity
  • Proof by design: If it’s not true in today’s math, invent a new system in which it is.
  • Proof by authority: "Well, Gauss says it’s true, so it must be!"
  • Proof by intuition: "I have this gut feeling."

  • Finished Reading – Ender’s Game

    image I just finished reading Orson Scott Card’s Ender’s Game.  I had a great time reading it and will definitely continue on in the series.  I’ll not spoil anything but the ending wasn’t too much of a surprise until I recognized the connection between Ender and the buggers.  I’ll say nothing more than that.  If you haven’t read it and you’re into science fiction, it’s definitely worth your time.  There’s a reason it is considered one of the modern greats.

    4 1/2 stars out of 5

    Next Read: The Children of Dune by Frank Herbert

    Conducting an online mathematics course

    I gave a presentation today to my colleagues at WBU in order to train them on teaching an online mathematics course.  We have offered College Algebra online for the last year and, so far, I have been the sole instructor.  I developed a series of lecture videos and notes for the students.  We are now ready to have additional instructors leading the course that was developed.

    For your own edification, I have posted the presentation to AuthorStream.  I’m certainly open to any questions or critiques you may have regarding our system.

    http://www.authorstream.com/player.swf?p=112729_633631661611168656
    Uploaded on authorSTREAM by splineguy

    Why didn’t I think of that?

    image Problem: Which is larger, [tex]2.2^{3.3}[/tex] or [tex]3.3^{2.2}[/tex]?

    Sure, there’s the easy way: stick it in your calculator.  So, make it interesting.  Answer the question without using the calculator.

    This raises a couple of interesting questions:

    1. What do we mean by raising a number to a (terminating) decimal or fractional power?

    2. Can I do that without a calculator?

    The answer to the first question is, of course, roots.  The exponents above can be written as fractions, and we learn in algebra that

    [tex]displaystyle a^{frac{m}{n}} = sqrt[m]{a^n}[/tex]

    So, we interpret [tex]2.2^{3.3}[/tex] as

    [tex]2.2^{displaystyle frac{33}{10}} = sqrt[10]{2.2^{33}}[/tex]

    Similary, we interpret [tex]3.3^{2.2}[/tex] as

    [tex]3.3^{displaystyle frac{11}{5}} = sqrt[5]{3.3^{11}}[/tex]

    What about question 2, then?  The answer is, I certainly can’t.  There is a fairly easy-to-use algorithm (step-by-step) procedure for calculating square roots by hand. A simple Google search reveals many places to find the algorithm explained. However, the only method I would try for calculating a 10th root or, even, a 5th root by hand might be a series approximation of each root function.  You will quickly decide that…

    There must be an easier way:

    Continue reading Why didn’t I think of that?

    A math puzzle to start the day

    I’m on a math puzzle and modeling kick.  I’ve gotten back into writing code (in MATLAB, at least) and am having fun with a few blogs that post problems regularly.  For example

    Below is a problem that I thought was fun and interesting to solve.  I found it interesting because I could write a very short program that would execute quickly to do a brute force search for the solution.  However, a quick algebraic approach yields the answer just as quickly.

    Let [tex]x[/tex] and [tex]y[/tex] be two-digit numbers such that [tex]y[/tex] is formed by reversing the digits of [tex]x[/tex].  The integers [tex]x[/tex] and [tex]y[/tex] satisfy the equation [tex]x^2 – y^2 = m^2[/tex] for some positive integer [tex]m[/tex].  Find [tex]x + y + m[/tex].

    My solution can be found below.  (Spoiler alert!! – Try it yourself before you peek)

    Continue reading A math puzzle to start the day

    Forest Fire Simulation in MATLAB

    In my Fall course of Math Models, I have three groups working on projects to finish up the semester.  One of the groups have an assignment to explore a model of the spread of a forest fire.  The assumptions are that the trees are on a rectangular grid, or a lattice.  The time is a discrete variable and at each time step the probability that the fire spreads from one point in the lattice to an adjacent point (up, down, left or right) is given by p.  For simplicity, the event that the fire spreads to each point is assumed to be independent of any other point.

    Part of their project is to implement a numerical simulation of their forest fire.  I couldn’t let them have all the fun, so below is an example of my version of the simulation in MATLAB.  I have to hold off on posting the code until after they have handed in their project.

    In the graphical representation of my simulation, green represents an unburnt tree, black is burnt and red is currently on fire.  The fire lasts for exactly one time step.  I also implemented a 3-D version, where a height of 1 is unburnt, 2 is on fire, and 0 is burnt.  I’ll confess to having way too much fun with this.

    I have used a 200×200 lattice with p = 0.5.

    Update (3/4/16)
    I promised a LONG time ago that I would post the code.  I finally got around to it.  Here are links to the m files that were used to generate the graphical simulations above:

    I am definitely a math freak

    Who else could have so much fun answering emails full of algebra questions from my online students?  Really, what I am enjoying most is the using the new equation editor in word and “spicing” up my responses with some highlighting and striking out.

    Here is a sampling of what is tickling my fancy this morning (Did I actually just say that?)

    A student clicked a link on their homework labeled, “Ask My Instructor a question”.  I am sent a link to their problem and they have a chance to type a message to me regarding the problem.

    This problem was sent to me,

    image

    with the message:

    “Help having trouble with this one I don’t understand how to simplify”

    So I walk them through the problem in the following way:

    We begin by inverting the second fraction and changing the division to multiplication (using the rule that to divide fractions we “invert and multiply”):

    clip_image002

    Now, we factor each numerator and denominator. We begin by pulling out all the GCFs (greatest common factors):

    clip_image004

    I can factor the trinomial in the first denominator further,

    clip_image006

    We now cancel all the common factors:

    clip_image008

    Multiplying straight across, this leaves

    clip_image010

    Let me know if this makes sense.

    I have say to that I am really enjoying the new equation editor in Word 2007 because it allows me to do just about everything from the keyboard.  For example, I can insert the multiplication “dot” by simply typing the LaTEX form “cdot” and Word 2007 immediately replaces my text.

    The other surprising feature I discovered today was the ability to use standard font formatting such as highlighting and striking out.

    Now, off to class…