Jim Loy's Mathematics Page

NOTE:  The original site "jimloy.com" is no longer existing, but all the content can be found in the wayback archive:

http://web.archive.org/web/20140105155353/http://www.jimloy.com/math/math.htm

http://www.jimloy.com/math/math.htm

Go to my home page

---

Jim Loy's Mathematics Page

Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com.

"He must be a 'practical' man who can see no poetry in mathematics." - W. F. White.

Dedicated to the memory of Isaac Asimov. See the top of my Science pages for comments on Dr. Asimov.

My Mathematics Pages were described briefly in the Math Forum Internet News No. 5.48 (27 November 2000).

My Mathematics Pages were listed on ENC Online's Digital Dozen for Sep. 2003, as one of the most educational sites on the WWW. ENC is the Eisenhower National Clearinghouse, and is concerned with science and mathematics education.

---

My theorem: There are no uninteresting numbers. Assume that there are. Then there is a lowest uninteresting number. That would make that number very interesting. Which is a contradiction.

A number of readers have objected that "numbers" in the above theorem should be "natural numbers" (non-negative integers). My reply to one reader was this:

Yes, but I wanted to keep it simple and quotable. And the proof that all numbers are interesting should not be boring. From natural numbers, it can be generalized to rationals, as fractions with interesting numerators and denominators are obviously interesting. And what could be more interesting than an irrational that cannot be formed from any finite combination of rationals? I see that David Wells' book Curious and Interesting Numbers has something similar: "39 ... seems to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number having the property of being uninteresting."

Also "uninteresting" is not well-defined. Several mathematics teachers have asked permission to quote my theorem.

---

Contents:

• Algebra
• Geometry
• Calculus and Pre-Calculus
• Arithmetic, Roman Numerals ...
• Calendars
• Number Theory
• Fractals & Chaos
• Other Topics

• Book Reviews
• Other Links

---

Algebra:

• Intro:
  • Before You Took Algebra (includes the most basic laws of algebra)
  • The Essence of Algebra (equations and variables)
  • Word Problems
  • Distributive Law
  • Some Rules and Equations NEW
  • Intro to Matrices
  • Determinants
  • Fahrenheit to Celsius
  • Two Equals One?
  • Division by 1/2 NEW
  • Square Both Sides?
  • (x+y)² or (x+y)^2
  • Quadratic Formula
  • Cubic Equations
  • Irrational Numbers
  • e (also under Calculus and Pre-Calculus)
  • Logarithms (also under Calculus and Pre-Calculus)
  • The Slide Rule
  • The Words of Mathematics (basic glossary)
  • Practice Quizzes (require JavaScript-enabled browser)
    • Algebra Practice 001 - Solve for x NEW
    • Math Practice 002 - Reduce the fraction NEW
    • Algebra Practice 003 - Factor the polynomial NEW
    • Math Practice 004 - Improper fractions NEW
    • Algebra Practice 005 - Parentheses NEW
    • Math Practice 006 - Adding fractions NEW
    • Math Practice 007 - Multiplication NEW
• Sequences and Series:
  • Infinite Series (Identify your series here)
  • Geometric Series
  • Fibonacci Numbers
  • Harmonic Series
  • Two Infinite Series
  • How To Be A Little Gauss
  • Triangular Numbers
  • .999999...
  • Two Series for Taking Square Roots
  • A Leaning Tower (also under Geometry)
  • Zeno's Paradoxes in my Physics pages
• Complex (and Imaginary) Numbers:
  • Imaginary Numbers (and complex numbers)
  • Roots of Unity
  • Gaussian Primes
• Others:
  • Surprise?
  • Modular Arithmetic (under Number Theory)
  • Synthetic Division
  • 1/37 and 1/47
  • Rationalizing the Denominator
  • Finite Difference
  • The Yanghui Triangle
  • My Equation
  • 0!
  • Irrational Accounting in my Humor pages
  • The Fixed Point Theorem (under Geometry)

---

Geometry:

Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com.

• Euclidean Geometry:
  • Geometric Constructions
    • Trisection of an Angle
    • Collapsible Compasses
    • The 17-gon
    • Golden Rectangle and Golden Ratio (also listed below)
    • The Regular Pentagon (also listed below)
    • Squaring a Pentagon
    • Spherical Construction Problem
    • Inscribing a Square in a Quadrilateral
    • More Geometric Constructions
    • Trisecting a Line Segment NEW
  • Congruence Of Triangles, Part I
    • Congruence Of Triangles, Part II
  • Isosceles Triangles
    • The Equal Angle Bisector Theorem (under construction)
  • The Pythagorean Theorem
    • Pythagorean Triples (also under Number Theory)
    • The 3-4-5 Right Triangle In Ancient Egypt in my Egyptology pages
    • My Three Triangle Puzzle in my Puzzle pages
  • The Centers of a Triangle
  • The Nine Point Circle
  • Morley's Theorem
  • Some Triangle Formulas
  • Desargues' Theorem
  • Pappus' Theorem
  • Hero's Formula
  • Napoleon's Theorem NEW
  • My Theorem? (the theorem was known before)
  • False Proofs:
    • A Right Angle Is Equal To an Obtuse Angle?
    • Every Triangle is Isosceles?
    • An Angle Cuts Two Parallel Lines Into Equal Segments?
    • A Rectangle Inscribed in a Square is a Square NEW
    • A Point Inside a Circle is On the Circle NEW
    • A Parallelogram?
    • Zeno's Paradoxes in my Physics pages
    • Aristotle's Wheel Paradox
  • Pi
  • The Area of a Crescent in my puzzle pages
  • Regular Solids
  • Archimedian Solids NEW
  • Tesselations NEW
  • Golden Rectangle and Golden Ratio
  • The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com
  • Areas and Perimeters
  • Quadrilateral With Two Nonadjacent Right Angles
  • Parallelogram?
  • The Regular Pentagon
    • Home Plate
    • Five Tetrahedra in my Puzzle pages
  • Shoot Hard? in my Billiards/Pool pages
  • Euclid III-31 and Pool in my Billiards/Pool pages
  • A Leaning Tower in my Puzzle pages
  • Kicks and Banks in my Billiards pages (which may load slowly)
  • Mirror Image
  • Volumes Of Cones and Pyramids
    • Cone-Shaped Milkshake Glasses
  • Conic Sections
    • The Ellipse
    • The Hyperbola NEW
  • The Astroid
  • The Cycloid
  • Figures of Constant Width
  • Malfatti's Conjecture NEW
  • Cinderella Interactive Geometry Program (Requires Java 1.1, loads slowly)
  • The Sextant in my Astronomy pages
  • Non-Euclidean Geometries (about Euclid's Fifth Postulate)
  • The Fixed Point Theorem NEW
• Non-Euclidean Geometry:
  • Non-Euclidean Geometries
  • Saccheri
• Other Geometries (Topology, Analytic Geometry, Trigonometry, graph theory, ...):
  • Left-Handed Dice
  • How Tall Is That Tree? (a little trig)
  • Mobius Strip
  • Klein Bottle
  • Tesseract
  • Analytic Geometry
  • Dividing The Plane (mathematical induction)
  • Roots of Unity (also under Algebra, above)
  • The Bridges of Königsberg in my Puzzle pages
  • Four Color Theorem Intro
  • The Travelling Salesman Problem in my Computer/Internet pages
  • Sin(x+y) and Cos(x+y)
  • Some Curves NEW
  • Time, The Fourth Dimension in my Physics pages
  • Life (has some gif animations)
  • The Fixed Point Theorem (also listed above)
• Also see Fractals and Chaos below

---

Calculus and Pre-Calculus

• Two Equals One? (also under Algebra)
• Limits
• Analytic Geometry (also under Geometry)
• Derivatives
• Infinity
• Integrals NEW
• The Hotel Infinity
• e
• Logarithms
• Computers and Derivatives
• Zeno's Paradoxes in my Physics pages
• Prof. E. McSquared's Calculus Primer - by Howard Swann and John Johnson (a book review)
• Newton vs. Leibniz
• Some Curves NEW
• A Maximum Area in my Puzzle pages

For the definition of the word function, see Analytic Geometry

---

Arithmetic, Roman Numerals ...

1999=MCMXCIX (or is it MIM?)
2000=MM.

• Arithmetic (See Speed Arithmetic, below):
  • A Simple Number Puzzle
  • Addition and Subtraction by Counting
  • A New Look At Subtraction (includes comments on calculators in school)
  • A New Look At Long Division
  • Fractions and Ratios (also see the quizzes below)
  • Casting Out Nines
  • Rounding Down?
  • The Abacus
  • The Slide Rule
  • Taking Square Roots
  • The Words of Mathematics
  • Division by 1/2 (also under Algebra above)
  • Practice Quizzes (require JavaScript-enabled browser); these are also listed under Algebra:
    • Math Practice 002 - Reduce the fraction NEW
    • Math Practice 004 - Improper fractions NEW
    • Math Practice 006 - Adding fractions NEW
    • Math Practice 007 - Multiplication NEW
• Speed Arithmetic:
  • Speed Arithmetic
  • The Trachtenberg System of Speed Arithmetic
  • Super Brain
• Egyptian Mathematics:
  • Egyptian Numbers
  • Egyptian Fractions
  • Egyptian Multiplication
• Roman Numerals:
  • Roman Numerals Defined (1999=MCMXCIX, or is it MIM?)
  • Arithmetic with Roman Numerals
  • Division with Roman Numerals
• Others (see Other Topics, below)

---

Calendars and time:

• Friday the 13th
• Daylight Savings Time
• Various Calendars
• When Is Easter?
• Day Of The Week Of Any Date (Some errors have been corrected)
• How Long Is A Year? in my Astronomy pages
• The Seasons in my Astronomy pages
• Let's Predict An Eclipse in my Astronomy pages
• Make Your Own Working Paper Clock - by James Smith Rudolph in my Book Reviews

Also see the Clocks and Time topic in my Astronomy pages.

---

Number Theory:

• Euclid's Algorithm
• Sieve Of Eratosthenes
• The Infinitude Of Primes
• Perfect Numbers
• Twin Primes
• Goldbach's Conjecture
• Pythagorean Triples
• Primes and 6
• Semiprimes
• Repunit Primes
• Modular Arithmetic (also under Algebra)
• The Case of the Missing Divisor in my puzzle pages
• The Hardy-Ramanujan Number
• Casting Out Elevens
• Casting Out Nines (also under Arithmetic)
• Sum of n Consecutive Integers
• Divisibility Tests
• 1999
• Egyptian Fractions
• Irrational Numbers (also under Algebra)

---

Fractals and Chaos:

• Fractals and Self-Similarity
• The Dragon Curve
• The Lorenz Attractor (with an animation)
• The Koch Curve (with an animation)
• The Cesaro Curve NEW
• The Sierpinski Gasket
• The Mandelbrot Set (Intro)
  • A Tour of the Mandelbrot Set (12 graphic images)
  • My Mandelbrot Set Movie (2 min. to load with 14.4K modem)
• Julia Sets
• Fractal Trees
• A Fractal Wave
• Chaos - Sensitivity To Initial Conditions
• The Chaotic Drip in my Physics pages
• Fractint the Fractal Program (5 pages of graphic images)

See ArtsForge's Fractal Canvas Gallery.

---

Other topics:

• Curtains Without Windows in my Science pages
• Transfinite Numbers NEW
• Here Is Your Mathematics Tutor!
• Ace That Test
• The Words of Mathematics (basic glossary)
• Proof in my Logic pages
• Science Fair Projects
• Probability 101
• Discovering Zero
• Scientific Notation
• Interpolation
• John von Neumann in my Biography pages
• All Numbers Are Less Than A Million
• The MathWorld WWW Site!? in my Issues/Opinions pages
• Base 2 (Binary)
• Million, Billion, Trillion...
• Wheat on a Chess Board in my Puzzle pages
• Combination Locks
• Remember The New Math?
• Unfair Teachers
• The Well-Tempered Scale in my Physics pages
• A Trillion Dollars
• The Monty Hall Trap also in my Puzzle pages
• Unbreakable Ciphers
• You Already Use Metric
• Superfluous Accuracy
• Metric Conversion Blamed for Canadian Jet's Forced Landing
• Shuffling Cards

---

Book Reviews:

My book reviews of mathematics related books:

• Make Your Own Working Paper Clock - by James Smith Rudolph
• The Fourth Dimension - by Rudy Rucker
• The Man Who Knew Infinity - by Robert Kanigel
• Innumeracy - by John Allen Paulos
• My Brain Is Open - by Bruce Schechter and The Man Who Loved Only Numbers - by Paul Hoffman
• Prof. E. McSquared's Calculus Primer - by Howard Swann and John Johnson
• 200% Of Nothing - by A. K. Dewdney
• aha! Insight by Martin Gardner and aha! Gotcha - by Martin Gardner
• Prisoner's Dilemma - by William Poundstone
• Chaos - by James Gleick
• Gödel, Escher, Bach - by Douglas R. Hofstadter and Forever Undecided - by Raymond Smullyan
• How To Lie With Statistics - by Darrell Huff (The best book ever written)

---

Links to other mathematics pages:

• my Gambling pages
• my Logic pages
• my Puzzle pages
• Science Fair Projects

• MathWorld (huge online encyclopedia)
• S.O.S. Mathematics
• Interactive Mathematics Online
• Go Math Online Tutoring
• SAT Math Online Tutoring
• MathNotes.com
• The MacTutor History of Mathematics archive
• Sci.Math Frequently Asked Questions in Mathematics
• Usenet Mathematics FAQ Archive
• Ask Dr. Math
• tests.researcheasy.com - Math Practice Sat Test
• Super-Index of An Introduction to the Theory of Numbers
• Charles Sturt University mathematics courses
• Perseus Project - Heath's Euclid's Elements (complete)
• D. Joyce - Euclid's Elements (complete)
• Sloane's On-Line Encyclopedia of Integer Sequences
• Robert Nemiroff - Digit Warehouse (millions of digits for free)
• Constructions - by George W. Hart (also see his virtual polyhedra)
• David Eppstein - The Geometry Junkyard
• Rik Littlefield's proof of the Pythagorean Theorem
• Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com
• Dr. Ron Knott's Fibonacci Numbers and the Golden Section
• Integer Jim's Math Squad (another Jim)
• Cinderella (geometry program)
• Geometer's Sketchpad (geometry program)
• Calculators by Ken (download some calculators)
• Math Made Easy (educational videos)
• Molecular Expressions - Powers of Ten (requires Java)
• Zometool, Inc (educational geometric construction toys)
• Kids.net.au's Math Page
• Cut-the-knot.com (interactive mathematics miscellany and puzzles)
• Karl's Calculus Tutor
• Fourth Dimension: Tetraspace - by Garrett Jones
• books and supplies
  • DirectTextBook (better prices)
  • PearsonAtSchool (books and supplies)
  • TextbookLand.com (find the best prices)
  • Chambal.com (book price comparison)
• American Institute of Mathematics
• Word Problems For Kids
• University of St. Andrews, Scotland - Biographies of Mathematicians
• The Numdrum (fun arithmetic for kids)
• Keiichi Suzuki's Kodawari House
• math+blues.com
• Logo Art Gallery
• Hindu Arabic Numeric Medieval Ideograms
• What are Fractals? - by Rupert Russell

---

Incidentally, in the USA, we often call mathematics, "math." In other English speaking countries, they seem to call it "maths."

Many of my mathematics articles contain special symbols, like ² or ² for "squared." Caveman WWW browsers (as I call them) cannot always deal with these symbols. I am currently working on a project to provide an alternative page without special characters (with ^2 for "squared"). Please let me know if this is working or not, by sending me email. Several of my articles depend heavily on graphics. I hope you don't have trouble with those.

---

Send me email
Go to my home page