Mike DeHaan

According to Hall’s Marriage Theorem, a computer algorithm can correctly assign brides to grooms for optimum happiness. Dating services use computer programs to match prospective mates, but group matching by list is pure math theory.

Tweet

Alan Turing (1912-1954) “invented” his Turing Machine to represent the process of making mathematical inferences. The penultimate goal was to determine whether the “Halting Problem” could be solved. So, what’s the “Halting Problem?

Tweet

Can one Turing machine emulate another? Are Turing machines guaranteed to finish a task? As Tolkien said about the advice that elves provide, the answer is “both yes and no.” Essentially, the Universal Turing machine represents the ability for a “computer” to manipulate a program just as it deals with data. Review of a Standard Turing Machine A standard Turing machine, or ‘TM’, is able to solve a great variety of problems in mathematics, pattern recognition, and pattern creation, although it is only a theoretical construct rather than a physical computer. The Turing machine’s program consists of a set of states. Such a program can process any number of different patterns on its input tape, with the option to accept or reject the input or to update cells on the tape. Variations on Turing machines include having more than one tape and a read/write head for each tape. This may make it easier to describe a program, but does not add any greater “power” to the class of problems it can solve. One tape can always be segregated by special symbols, and any length of symbol can be shifted onto blank areas of the tape. Using multiple tapes allows the [...]

Tweet

The ACM Turing Award winner for 2011 is Judea Pearl. This computing award has been presented annually since 1966, with multiple recipients in some years. This year’s winner has advanced artificial intelligence by improving the way in which AI programs acquire additional information, among other things. The first winner of the Turing award, Alan Jay Perlis, was well-known for his advances in the area of programming languages. Judea Pearl won the Turing Award for 2011 in Artificial Intelligence The most recent ACM Turing Award winner, announced in March 2012, is Judea Pearl. He has “transformed artificial intelligence (AI)…for the processing of information under uncertainty.” Pearl was born in Tel Aviv. He earned his bachelor’s degree in Israel, his Master’s at Rutgers and his PhD from the Brooklyn Polytechnic Institute. He devotes considerable time to the Daniel Pearl Foundation, which promotes cross-cultural understanding in memory of Judea’s son Daniel. One early approach to artificial intelligence created AI systems by asking experts to write rules; the AI then required precise information about a specific situation in order to apply and follow those rules. Pearl recognized that experts must also deal with uncertainty. He implemented the mathematics of probability theory as algorithms, or [...]

Tweet

Alan Turing (1912-1954) “invented” the Turing machine (TM) as a powerful theoretical model for mathematicians exploring rules-based mathematics. The  Non-deterministic Turing machine, or NTM, extends the basic concept by permitting multiple instructions for one state-input combination. The Deterministic Turing Machine A Turing machine has a finite number of states, symbols and instructions. A pattern of symbols are presented on an infinitely long “tape”. The TM reads or rewrites one symbol in one cell of the tape, then moves right or left by one cell. A TM is programmed with instructions. A deterministic Turing machine is limited to one instruction for each state to deal with each symbol, plus optional states to “halt and accept” or “halt and reject” the tape’s initial pattern. A TM either halts or continues processing without halting. What is a Non-Deterministic Turing Machine? A non-deterministic Turing machine allows multiple instructions for any one state as it reads any one input symbol. Mathematically, the mapping from the state plus symbol is a relation, rather than a function. Conceptually, a non-deterministic TM runs many isolated computations in parallel. Another useful concept is a decision tree structure. A node with branches is created each time an NTM begins a parallel computation. Each “leaf” of [...]

Tweet

A Turing machine, or TM, is a theoretical model devised by Alan Turing to explore the limits of rule-based math. The model has a finite number of rules, states and symbols, and an infinite tape with cells, each of which can contain a single symbol. The TM can either read the current cell, rewrite it, move to the next cell on the left or right, or transition to a new state. The description of any Turing Machine consists of a list or table of 4-tuples, including the state number, one input symbol that it can read and accept while in this state, the next state to which it should transition, and the action it should take: either the symbol it must rewrite into the current cell, or the direction in which to move. The following are examples of simple Turing machines: A Turing Machine that Either Reads to the End, or Loops Forever This Turing machine just reads and goes in the indicated direction until it encounters an ‘H’ to direct it to halt. It has two states and four instructions. The tape has three values: {‘L’, ‘R’ and ‘H’} which stand for left, right, and halt. *State #0 is [...]

Tweet

Alan Turing (1912-1954) “invented” the Turing machine as a theoretical model for exploring the limits of rules-based mathematics. This purely theoretical device became a powerful tool in the minds of mathematicians, and modern computers still follow many of its principles. The Turing machine is even being honored via art at the Intuition and Ingenuity exhibit in the 2012 Kinetica Art Fair. Components of a Turing Machine The two important components of a Turing machine, or TM, are the “read/write head” and the “tape.” The machine deals with one cell of the tape at a time, similar to the way a movie projector works, except that the movie’s tape only moves in one direction, and the “head” only “reads” the film by shining light through one frame at a time. The Turing machine, however, can read and re-write the active cell, or move the tape either right or left to the next cell. The Turing machine’s “head” also has a finite number of internal states. These correspond to a computer’s programmed instructions, along with some of its active memory. The most important states are the initial state and the “halt” state where no subsequent instruction is specified. A Turing machine’s “tape” [...]

Tweet

Is there a new way to calculate a number’s cubed root? Recent news articles from India report that Mr. Nirbhay Singh Nahar has developed an algorithm to calculate the cube root of any number. Given an equation stating “y = x^3″, Nahar’s method would solve for “x = y^(1/3)” without the need to refine repeated approximations. In this article, we assume that Mr. Nahar refers to finding the real cubic root of a real number, rather than also finding complex roots. A Summary of Nahar’s Claim for a Cubic Root Formula Apparently Mr. Nahar has obtained a copyright for his formula, “NAHNO” (“NAHar Number”), and is pursuing a patent for it. According to reports, he wishes to collaborate with one or more well-known mathematicians, presumably to have his work pubished in a respected mathematics journal, and has stated that his overarching aim is for “the credit for my work to go to India, my country“. In the meantime, however, he has not released his formula for scrutiny. Previous Methods of Calculating a Cubic Root Several methods already exist for calculating a cubic root, including Newton’s Method for an ‘N’-th Root, Halley’s Method for an ‘N’-th Root, and a Long Division Method for a [...]

Tweet

Richard’s Paradox demonstrated that a simple rule to define a set of numbers may lead to a paradox. Predecessors to the Richard Paradox In 1905, French mathematician Jules Richard shifted the focus from certain earlier mathematical paradoxes by showing that the definitions themselves might be at fault. In the very early 1900s, paradoxes in the mathematics of set theory had already been found and published. In 1903, for example, Bertrand Russell’s “Principles of Mathematics” discussed paradoxes that dealt with the ordering of infinite ordinal numbers, or with defining a self-containing set with a condition that both includes and excludes itself. An English Version of Richard’s Paradox for Real Numbers Richard built his main paradox in this manner. Create a set of Real numbers, ‘E’, with the following definition. “E = {e}, where 0 < e < 1 in the Real numbers” and each ‘e’ is further defined by a finite number of words in the English language. Order this set by the number of letters in the string and, within each length, alphabetically. To create the paradox, form a new Real number ‘f’ in the range “0 < f < 1″ and based on the existing set ‘E’ as follows. [...]

Tweet

Confusing the “given” event (the event that you assume to have occurred) with the combined event (for which you are calculating probability) is a common pitfall with conditional probability. A Recap of Conditional Probability Recall that “the conditional probability of event ‘A’, given that event ‘B’ has occurred, is calculated as the probability that both ‘A’ and ‘B’ occur, divided by the probability of ‘B’ alone”. The equation is “P(A|B) = P(A∩B) / P(B)”. Just to be clear: “P(A|B)” is “the conditional probability of ‘A’, given ‘B’ occurred”; while “P(A∩B)” is “the probability of the intersection of outcomes for both ‘A’ and ‘B’, ie that both ‘A’ and ‘B’ happened”. Combined and Independent Events Imagine tossing a “fair” (unweighted, unshaved) coin twice. Each toss is independent; either could result in heads or tails, and the first toss does not influence the second. However, even independent events can have a combined outcome. If there were a wager that both independent coin tosses would land heads, then the probability space is {(heads, heads), (heads, tails), (tails, heads), (tails, tails)} and the wager has one chance in four of paying off. Click to Read Page Two: Conditional Probability and Given Outcomes Tweet

Tweet

Despite the value of knowing the probability of an event before it occurs, it can be even more valuable to know how learning part of the outcome changes the conditional probability. The Foundation for Understanding Conditional Probability This article continues a series about probability, by introducing “conditional probability.” If the terms are unfamiliar, consider reviewing “Introducing Probability Theory without Statistics“. Recall that the “probability of an event” is a numeric value that states how likely it is for a specific outcome to occur, given the circumstances of an experiment in a given “probability space.” Simple examples include the fact that the probability of a tossed coin landing heads up is 0.5, and the probability of a six-sided dice roll showing ’3′ is 1/6. Calculating Probability: Points to Consider •The numeric value of a probability is any Real number from zero to one, inclusive. An event with “zero” probability cannot occur; if the probability is ’1′ then the event must occur. As well, the probability that the experiment will yield some event in the probability space is exactly ’1′. •Mathematicians describe a event as “a set of outcomes”. For example, event ‘A’ might refer to a dice roll resulting in an [...]

Tweet

The January 2012 Joint Mathematics Meetings featured an awards presentation in recognition of many outstanding mathematicians, educators and authors. The prizes are awarded by the AMS (American Mathematical Society), the Mathematical Association of America, the Association for Women in Mathematics, and the Society for Industrial and Applied Mathematics. With 33 individual recipients of 19 awards, certificates or prizes, there is simply too much material to fully report. This selection of highlights, with a focus on communications awards, gives a taste of the awards. Persi Diaconis Wins the 2012 Levi L. Conant Prize for Best Expository Paper Persi Diaconis received the Levi L. Conant Prize for an article written for the Bulletin of the American Mathematics Society. Titled “The Markov chain Monte Carlo revolution”, this paper explains the use of Markov chains in modern probability and statistics. Dr. Diaconis was praised for making this article “readable” and “accessible,” with illustrations and numerous examples of the applications for this branch of mathematics. Diaconis has a Ph.D. in mathematical statistics and has previously been the recipient of a MacArthur Fellowship. 2012 Math Awards for Exposition Decoded Science interviewed Allyn Jackson of the AMS to explain more about the awards for exposition. Decoded Science: Persi Diaconis [...]

Tweet

The first article in this series, Introducing Probability Theory without Statistics, noted that probability distribution might be “discrete” or “continuous.” This article builds the foundation for discrete probability functions, by introducing the four axioms and deriving two useful theorems from them. Discrete Probability Functions: The Soul of Discretion The phrase, “probability distribution”, refers to the way the probability of an event is distributed over the outcomes. For example, the chance of a fair coin landing “heads” is 50%, as is the chance of “tails” is an example of a probability distribution. A different example would be the longevity of a specific model of light bulb under test conditions. A “discrete” probability distribution has specific distinct outcomes. Here are two examples: there are 2 possible outcomes for a coin toss; there are 52 different playing cards (excluding jokers) in a standard deck. However, there may be a countably infinite number of distinct outcomes for a discrete probability distribution. A simple example is the number of requests in a queue. Theoretically this could be an infinite number. (It feels infinite to the victim of an Internet “denial of service” attack, where someone maliciously generates requests for a Web page faster than the [...]

Tweet

One marvelous example of the conflict between mathematics and human behaviour is shown in the “Allais Paradox.” Compared to probability theory, in the Allais Paradox, people choose correctly or incorrectly based on irrelevant details. Probability, Payout, Expected Value and Lotteries The mathematical view of “probability” is the likelihood that some specific outcome will occur from an event. Some events might result in a benefit to a participant or observer. A game or lottery has some outcomes classed as “wins” or “losses”. A “win” might indeed result in some benefit, whether it is a cash prize to a participant or the settlement of a gambling bet. The “payout” is the benefit to the winner. This payout may depend on the exact outcome. For example, $1 for every point in the margin of victory in the score between two sports teams. The “expected value” or “expected utility” of an event is the sum of the payout for each outcome multiplied by the probability of that outcome. An example should make this clear. Let’s toss a coin twice, so the outcomes are two heads (HH), two tails (TT) or one head and one tail (HT or TH). Each outcome is equally probable; ‘p’ = [...]

Tweet

This article introduces basic mathematical concepts in probability. Future articles will discuss different aspects, including several paradoxical situations involving probabilities. For those who can’t wait, Solve the Monty Hall Problem using Logic and Mathematics. Probability, Statistics or Likelihood? In mathematics, “probability” is the study of how likely it is for some specific outcome to occur as the result of an event. This is expressed either as a percentage from zero to 100%, or as a number between 0.0 and 1.0 inclusive. “Statistics” is the analysis of the events ruled by probabilities. Often the statistician is concerned with the distribution of probabilities. In the far future, we may delve into statistics. A related word is “likelihood“, which refers to the odds of the outcomes of past events. By contrast, “probability” refers to future events. Some examples might help: If you plan to flip a coin to make a decision, the probability of the “heads” outcome is 0.5, or 50%. Afterward, when you explain why you watched a “Headhunter” movie rather than playing “Pin the Tail on the Donkey”, you would say the “likelihood” of either activity was 50%. From the statistical point of view, a coin toss is well modeled as [...]

Tweet

What is a power set? The definition of a power set is, “The power set of a given set S is the set of all subsets of S.” The power set of S is shown as P  (S), or 2S. Let’s save a thousand words and just refer to the image. A Refresher about Sets A “set” is simply a collection of elements. These elements may be physical objects, such as the people in your family; or abstractions, such as numbers. The elements in a set may themselves be sets. The special case is the empty set, Ø = {}, which has no elements at all. Sets, Subsets, and the Power Set A subset of a set contains zero, one, or more elements from the original set. It could even be identical to the original set. For example, let’s say that A = {a, b, c}. There is no need to order subsets, but let’s do so anyway in this situation. The subsets of A are A[0] = {Ø} (the set containing the empty set); A[1] = {a}; A[2] = {b}; A[3] = {c}; A[4] = {a, b}; A[5] = {a, c}; A[6] = {b, c}; and A[7] = A [...]

Tweet

The recent article, “Complex Tale of Eisenstein Prime Numbers“, was devoted to the prime numbers found by Ferdinand Gotthold Max Eisenstein. The names of several other mathematicians have become associated with their own sets of primes numbers. This article will introduce some of these very personalized primes. Fermat Numbers and Fermat Primes Pierre de Fermat is perhaps best known for “Fermat’s Last Theorem”. Here, however, the focus is on Fermat numbers and Fermat primes. One calculates a Fermat number as a function of the natural number ‘n’, where ‘n’ is greater than or equal to zero. The formula for a Fermat number is F[n] = 2^(2^n) + 1. Fermat numbers become very large very quickly. To the limits of current computations, only the first five Fermat numbers are primes. In 1650, Fermat had conjectured that each and every Fermat number would be prime. This conjecture is false. Much later, our old friend Eisenstein posed a problem: to prove that there are infinitely many Fermat prime numbers. This problem has neither been proven nor disproven. Gauss found an interesting geometrical use for Fermat prime numbers. The problem is to subdivide a circle, using only a straightedge and a compass. (That’s the [...]

Tweet

Last week’s “Several Different Paths to Prime Numbers” opened with this intriguing image. Unfortunately, there was no room to answer the question “What are Eisenstein primes?” An Explanation of an Eisenstein Prime Number We already know that a “prime number” is a number that can only be evenly divided by itself and the number one. An “Eisenstein Integer” is a complex number constructed with one of the cubic roots of one. An Eisenstein integer have the form “a + b*ω”, where ‘a’ and ‘b’ are integers, and “ω (the lower-case Greek omega) = (1/2)*( i * ( 3^(1/2) ) – 1 )”. In this equation, ‘i’ is the imaginary square root of negative one. Review of Imaginary Numbers and Complex Numbers As noted in “The Definitive Quick Reference Guide to All Types of Numbers“, the imaginary number ‘i’ is defined as “i = (-1)^(1/2)”. Thus, “i^2 = i*i = -1″. The set of Imaginary numbers is the set {y*i} where ‘y’ is a Real number. Complex numbers take the form “x + y*i} where both ‘x’ and ‘y’ are Real numbers. One might prefer to exclude the value “y = zero”, because this leaves a simple Real number. However, Eisenstein [...]

Tweet

Last week’s “Brief Introduction to Prime Numbers” dangled a few teasers – since keen minds are eager to know more, let’s tie up some of the loose ends. Pure Review: What is a Prime Number? A “Prime” number is a natural number greater than one, that is only evenly divisible by itself and one. This infinite but countable set begins as {2, 3, 5, 7, 11…}. Readers might remember that “composite” numbers are the natural numbers that can be composed into factors other than themselves and one. For example, 24 = 2*2*2*3. People have tested prime numbers “by hand”; this work was assigned to early analog computers also, as seen in this image: Are There an Infinite Number of Prime Numbers? Yes, there are infinitely many primes, and the proof goes back to the Greek philosopher Euclid. Suppose we have a set P={2, 3,…p} of prime numbers, and ‘p’ is the highest known prime. Simply calculate p[1] = ( ( 2*3*5*…*p ) + 1 ). Certainly ‘p[1]‘ is not divisible by any of the primes in set P, since there would be a remainder of 1. Assuming that ‘p[1]‘ is divisible by a prime larger than ‘p’. That assumption simply [...]

Tweet

What is the Sieve of Eratosthenes? Last week’s article, A Brief Introduction to Prime Numbers, mentioned the “Sieve of Eratosthenes” – a procedure devised by the clever Greek philosopher Eratosthenes. As a sieve catches fish, but allows water to escape, the Sieve of Eratosthenes retains prime numbers but allows composite numbers to pass through. Essentially, the Sieve is a simple way of listing all prime numbers up to some predetermined number. The image shows an Eratosthenes’ Sieve with an upper limit of 26. Sieve of Eratosthenes to 26 – Image by Mike DeHaan (Click on Image for larger view) “List#1″, the leftmost column, shows all the potential prime numbers from 2 to 26. “List#2″ changes the multiples of ’2′ to red text and puts a light pink background behind them. “List#3″ starts with the next prime, ’3′. Every odd multiple of ’3′ changes to red and receives a darker hue of pink background. As an intelligent human, I did not need to re-eliminate the even multiples of 3, such as ’6′. I saw it had already been eliminated. “List#4″ eliminates those multiples of ’5′ that survived the previous primes. Notice that “List#5″ looks exactly like “List#4″. I was going to [...]

Tweet

Would you like a simple introduction to primes? This article introduces a series about prime numbers by answering the following basic questions: What are prime numbers? Are primes important? How many prime numbers are there? What is a Prime Number? Prime numbers are those Natural numbers that are greater than one and can only be evenly divided by the numbers one and itself. As a set, P={all ‘i’ in N such that “i>1″ and, for ‘j’ in N, where “i/j” is in N only if ‘j’ equals either one or ‘i’}. The set of prime numbers begins as {2, 3, 5, 7, 11, 13, 17…}. What are Non-Prime Numbers Called? Non-prime numbers greater than 2 are called “composite” numbers. These are positive integers that are “composed of” the product of primes. The set of composite numbers begins as {4, 6, 8, 9, 10…}. Why are Zero and One Commonly Excluded From Lists of Prime Numbers? Zero: Since zero divided by anything is still zero, with no remainder, one could say that zero is not a prime number. Zero may be excluded from lists of prime numbers because no-one ever found an acceptable reason to include it in a discussion of [...]

Tweet

Background to this Reference Guide to Numbers Several previous articles introduced a variety of numbers, including Natural Numbers, Integers, Rational Numbers, Real Numbers, Imaginary Numbers, Irrational Numbers, Infinite Numbers, and others. This quick reference guide is intended to provide additional information and a summary of the main types of numbers. The Sets of Numbers The below list is organized so that each set of numbers in the list includes all the sets listed above it. For example, Integers, second on the list, includes all Natural Numbers.  The exceptions are noted briefly within the list. Further notes are listed below. N = Natural Numbers = {0, 1, 2, …}. These are also known as the “counting numbers”. Sometimes zero is excluded from this set, but included in the “Whole” numbers. Z = Integers = {…, -2, -1, 0, 1, 2, …}. This set is the union of Natural Numbers and their counterparts, negative numbers. Q = Rational Numbers = {i/j} for all ‘i’ and ‘j’ in the Integers, excluding any fraction with j=0. There is a lot to be said in the section below. Irrational Numbers are all “Real” numbers that cannot be expressed as a ratio of integers. This is [...]

Tweet

Neodymium magnets are powerful enough to demonstrate the subtle effects of diamagnetism, the weak repulsion that magnets can exert on some substances. What is Diamagnetic Repulsion? Magnetism is an attractive force that sticks a fridge magnet onto a refrigerator. However, “like” magnetic poles repel; the “North” ends of two magnets push each other apart. These are the effects of “ferromagnetism” (from “ferrous”, meaning “concerning iron”). Standard magnetism is both an attractive and repulsive force, depending on the circumstances. Many substances show no obvious attraction to magnetism. We don’t expect a magnet to hold a wood or plastic Yo-Yo. However, some materials are weakly repelled by magnetism. This effect is called “diamagnetism“, or “diamagnetic repulsion“. The Power of Neodymium Magnets Some materials are more powerful than standard iron or steel magnets. Neodymium magnets are among the most powerful permanent magnets. They may be called “rare earth” magnets. If you don’t have a neodymium magnet, feel free to try this experiment with whatever magnets you might have handy. A second choice is to wind your own electromagnet, powered by a quick-discharge battery. The winding might get quite warm, since there is little resistance in the circuit; it could also damage a rechargeable [...]

Tweet

Last week’s article introduced N, the set of Natural numbers, the numbers with which we count. This week resumes with a few functions for Natural numbers, which will extend them beyond their wildest dreams. Functions and Identity Numbers in Natural Numbers We had introduced the successor function, using the Greek lower-case “sigma.” In simple terms, σ(i) = i+1 for any Natural number ‘i’ in N. Since we defined N as the set of all these successors, the result of simply adding ’1′ to a Natural number stays inside the set N. Addition and Multiplication as Natural Successor Functions We can repeat the successor function, much like counting on one’s fingers and toes. The normal addition operation, ‘+’, is just a way of expressing this repeated successor function. Just as “1 = σ(0)” (“one is the successor of zero”), and “2 = σ(σ(0))” (“two is the successor of the successor of zero”), we can write “i = j+k” to show that ‘i’ is the result after “j plus k” successor functions. Likewise, we can define multiplication as a sequence of identical additions. In both cases, the result remains in the set N. We say “i = j*k” if successively adding ‘j’ [...]

Tweet

By singular request, if not by popular demand, today’s article presents the first in a series of guides to different types of numbers. It promises a roller coaster ride, with a slow uphill start and a mad rush to the finish. This article begins with “Sets and Successors: A Natural Foundation“. All Set for Successors It is tempting but premature to discuss numbers by saying, “Let’s start by counting 1, 2, 3…”. We need to cobble two shoes before we can walk on this path. Set the Stage: What is a Set? The first “shoe” is the “set”. A set is a collection of elements. The “empty set” has no elements; it is shown as “{}”. Let’s associate the number of elements in the empty set with the number zero. A set’s elements could be numbers, groups of numbers, other sets, or even odder entities. Examples of the first two cases are: A set consisting of numbers may look like this: {1, 2, 3}. A set consisting of groups of numbers may look like this: {(1, 1), (2, 4), (3, 9)}. Normally we name sets with just a capital letter; for example, let A = {1, 2, 3}, or B [...]

Tweet

On Sept. 22, 2011, scientists at the European Organization for Nuclear Research reported that they had measured neutrinos traveling faster than light. This discovery has taken about two years, and is based on the OPERA experiment at CERN. The Speed of Light and of Neutrinos Let There be Light Since light is the “cosmic speed limit,” the speed of light should be measured “in a vacuum.” Light slows down as it passes through various materials. This change in speed causes refraction, but that whole subject is ignored when discussing the relative speed of neutrinos. The speed of light in a vacuum is 299,792,458 m/s (meters per second). The speed of light is symbolized as the English letter c. Light may be considered as a set of fast-moving particles with no “rest mass” (no weight). The particles are called “photons”. Light may also be thought of as a wave; we see different wavelengths as different colors. Speeding Neutrinos A neutrino is a “little neutral” particle with almost no “rest mass”, and no electrical charge. It is very rare for a neutrino to interact with any other matter. However, the sheer number of neutrinos created by some quantum reactions allows some interactions [...]

Tweet

Last week’s article, “Cantor Defeated Galileo in the Battle of Infinite Numbers“, noted: “…the square root of two is not rational. No fraction can exactly equal 2^(1/2).” 2^(1/2) is, as you know, the square root of two. This claim deserves some proof; let’s ask Pythagoras and Euclid for their expert opinions. Quick Reviews of Different Types of Numbers This article almost always deals with positive numbers, since the “square root of two” problem first arose when finding the length of lines in geometry. It would also stretch the article to repeatedly include: “…when both integers are positive or when both integers are negative.” Natural Numbers and Integers The Natural numbers are often defined as the positive integers. In the previous article, we also included zero as a natural number. The natural numbers are sometimes called the “counting numbers.” Integers, of course, include negative numbers. Sometimes called “whole” numbers because they are not fractions, the integers simply extend the set of natural numbers by including the set formed by multiplying each natural number by ‘-1‘. Rational Numbers The Rational numbers are the fractions made by dividing one integer by another. We exclude “division by zero”. Usually the division operation is defined as the [...]

Tweet

While Cantor and Galileo did not squabble over their infinite sets, Cantor’s set of Real numbers is indeed larger than Galileo’s set of Natural numbers. How can one infinity exceed another? Building a Foundation for Cantor’s Infinities Understanding how Georg Cantor’s “Real” infinity is larger than the infinities in Galileo’s paradox requires dealing with set theory. We start with some definitions. A Cardinal is More than Just a Bird While a “cardinal” is a favorite red bird, the word “cardinality” in set theory has nothing to do with birds. When the elements of two sets can be placed in a one-to-one mapping, the two sets have the same “cardinality“. The cardinality of a finite set, such as the letters of the alphabet, is a specific finite natural number. For an infinite set, the cardinality is a trickier concept. For any set A, the cardinality of A is shown as |A|. You Can Count on the Natural Numbers Define N to be the infinite set of Natural numbers {0, 1, 2,…}. (Thomas Jech’s article starts at zero; some of my previous articles excluded zero from the set of Natural numbers). Any infinite set S is “countable” if |S| = |N|. That [...]

Tweet

Besides astronomy, Galileo provided a paradox about the infinite series of square numbers. His paradox gave great insight into a thorny problem in the mathematics of set theory. Defining Galileo’s Infinite Series of Square Numbers Paradox Galileo’s paradox is that there are “more” natural numbers than square numbers; but yet there must be the same number of each. How could Galileo do more with the same? Prepare for Galileo with a Set of Coins Let’s start with a set of the coins commonly circulated in Canada. The set C={1 cent, 5 cents, 10 cents, 25 cents, 100 cents (which is the $1 Loonie), 200 cents (or the $2 Toonie)}. This set has 6 elements. Let’s make a set with only the values of these coins. The set V={1, 5, 10, 25, 100, 200}. We can define two subsets, based on whether the number is even or odd. So VO={1, 5, 25} and VE={10, 100, 200}. Of course there are many other ways to define subsets of V. Let’s define a much more interesting subset of V as the numbers which happen to be the square of a number found in V. So VS = {1, 25, 100} because 1^2=1, 5^2=25 [...]

Tweet

Anyone who washes a cup after coffee dried in it has seen a “coffee ring”. Is the ring caused by the cup’s convex bottom? Several experiments get to the bottom of this phenomenon. Introducing the Coffee Ring Home Lab Experiment This is a simple lab experiment, both safe and interesting for anyone. More accurately, it is an open-ended set of tests that suitable for individuals or whole families. The basic test is to let a small puddle of coffee dry on a flat surface, such as a plate. This will leave a darker ring around the outside, with a much lighter centre. The set of experiments is to determine if all liquids behave that way, or is there something magical about coffee? The Author’s Flirtation with Coffee Rings Each image shows regular coffee in the upper left, tea in the upper right, soya sauce in the lower right, and instant decaffeinated coffee in the lower left. The same source was used for each plate. The “soap” plate had just a touch of well-diluted dish soap added to each sample. Lab Equipment for the Coffee Ring Home Experiment The above image shows the all the equipment except for the pots of [...]

Tweet

Why is the first elevator you see more likely to be going in the wrong direction? Two physicists reported this phenomenon in 1958. It is sometimes called the “elevator paradox,” but “elevator puzzle” better expresses the surprise or annoyance that everyone feels when faced with an uncooperative elevator that always seems to be going the wrong direction. No, this is not a paranoid delusion. Just follow the mathematics into probability theory, or skip to the logic at the end. Defining the Elevator Puzzle It seems likely that a person on a lower floor of a multistory building wants to go up, since there are more floors above than below. Similarly, a person on a higher floor has more potential destinations below than above. If the likelihood of the desired direction of travel depends on the relative starting position, then it seems reasonable that the elevators are more likely to be going in that direction. However, if you request both an “up” and a “down” elevator, you would observe that the first elevator to open its doors is going in the direction where there are fewer floors! The Simplified Elevator Puzzle The simplest version of the elevator puzzle is an abstraction [...]

Tweet

Monty Hall, the host of “Let’s Make a Deal”, asks a contestant to choose the one door to a valuable prize, rejecting the other two doors leading to junk. The host then opens one of the rejected doors, revealing a junk prize. The “Monty Hall problem” now is: should the contestant stay with the original choice, or switch to the other unopened door? Setting the Stage for the Monty Hall Problem Let’s label the doors ‘A’, ‘B’ and ‘C’, even though the game may have called them ’1′, ’2′ and ’3′. We will see why in the next sub-section. For a science-oriented experiment, let’s ensure that Monty Hall has no prior knowledge about what is hidden behind which door. That way, he cannot reveal anything to an observant contestant. While we’re stipulating conditions, let’s also say that there is neither cheating nor ESP: the contestant has no prior knowledge, supernatural insight or extra assistance to help with the selection process. The final condition is that Monty Hall and his team do not cheat by moving the prizes around after the contestant has made a choice… especially not after the second choice! To Simplify Discussing the Monty Hall Problem For the [...]

Tweet

The Leidenfrost Effect is also known as “film boiling.” This one effect makes water dance on a frying pan, flow uphill, and protects a sausage from burning in hot lead! Public Safety Message for Testing the Leidenfrost Effect The Leidenfrost Effect deals with steam and heat. These experiments involve hot objects. Carelessness can lead to personal injury from burns or scalding, or even fires. Do not perform these experiments while impaired by alcohol, drugs or sleep deprivation. Minors must have parental permission and supervision. If the heat source has an open flame, be sure that your clothes won’t brush past the flames. Long flowing hair, long loose sleeves, or scarves are all hazards to be eliminated before starting these experiments. Some types of frying pans will scorch or burn if you heat them with nothing in the pan, so use an old skillet that is no longer the household favorite for this experiment. A heavier frying pan will heat or cool more slowly than a lighter one of the same metal. Simple Leidenfrost Experiments in a Frying Pan The minimum lab equipment for this home science experiment is: A skillet or frying pan A stove top or other heat source [...]

Tweet

L’ Hôpital’s Rule to Cure Unhealthy Functions Calculus is a powerful tool for mathematicians and scientists, but it does have an Achilles’ heel. The function in question must be smooth, continuous and defined at the point or interval being examined. A function with a “zero divided by zero” result is undefined. L’ Hôpital’s Rule cannot break the laws of calculus, but it can bend them… in some examples. For the linguists among us, the French name “L’ Hôpital” was “L’ Hospital” but the ‘os’ has been replaced by the ‘o’ with a circumflex accent. Some now write it “L’ Hopital”, especially in Internet URLs. L’ Hôpital’s Rule Only Treats Certain Ailments Previous articles introduced differential calculus and Riemann sums for integral calculus. An earlier article found its limits; more accurately, this article found the value of a function at the limit as the independent variable approached a specific X-value. One particular problem for any real-number function arises for “division by zero”. For example, let F(x) = ((x + a)*(x – a))/(x – a) = (x^2 – a^2)/(x – a); then F(x) is undefined at x=a. But where does that value go on a graph? Two L’ Hôpital’s Rules for the [...]

Tweet

How Much Area is Hiding Under that Curve? Last week’s introduction to differential calculus explained how to find the slope of the tangent at any point along a simple curved function. This week we begin working at finding the area underneath a curve. “Why bother?” is the first question that people usually ask. “It’s the important concept to introduce integral calculus” is not a very satisfying answer. The Importance of the Area Under a Curve Integral calculus is all about knowing the rate at which something happens, and then calculating the total amount of “something” that has happened. A simple example comes when travelling. If you can bicycle at a steady 20Km per hour, how far will you get in 3 hours? The trivially simple answer is “60Km” = 20Km/hour * 3 hours. A similar question is, “How many products can the factory make in a day if it makes 2.7 units per minute”? These are simple because the rate is a constant per unit of time. The problem is more complex if the rate changes. The classic science question is “How far does an object fall in a given amount of time in Earth’s gravity, assuming no wind resistance”? [...]

Tweet

The Main Question in Differential Calculus “Differential calculus” is a big phrase but a very useful part of mathematics. Several previous articles have built a foundation, and now the first floor will be erected. The question that differential calculus asks is: What is the slope of a function at a given point? What Do “Slope” and “Function” Mean? To Define a Function: For this article, a function relates one variable to another; it is often written as “y = f(x)”. Calculus uses real numbers rather than integers or rational numbers. (Rational numbers include every fraction, like “7/15,” but exclude irrational numbers such as PI and the square root of two). Let’s ignore imaginary numbers for now, too. Again, for this article, the best way to think of a function is that it prescribes a line or curve graphed on a Cartesian plane. Three other necessary features of a function for calculus are “smooth” and “continuous” and “well defined”. A smooth function does not have a sudden change. A continuous function does not have missing pieces. A “well-defined” function has only a single ‘y’ value for any ‘x’ value. Here are some examples. A function: “y = x unless ‘x’ is [...]

Tweet

Why Mathematics Needs Limits   In algebra, sometimes we cannot simply plug a value into a formula and calculate the answer. A common example occurs if we need to “divide by zero”. Remember, if we say “a=b/c”, it means that “b=a*c”. But if ‘c’ has the value of zero, and ‘b’ is non-zero, then there is no number ‘a’ to satisfy “b=a*c”. We then say that “a=b/c” is undefined when ‘c’ has the value zero. In some situations we can sneak up on a useful answer, however. One method is to use “limits”. Karl Weierstrass Set the Limits Although Newton, Leibniz and Cauchy may be more famous for their work with infinitesimally-small numbers and, therefore, for the mathematics of calculus, it was Weierstrass who generally gets the credit for “epsilon-delta” limits. Karl Weierstrass (1815-1897) had an interesting life. His father sent him to the University of Bonn to study finance and law. Weierstrass instead engaged in fencing and drinking … presumably in that order, since he survived his sword-play. Later he officially switched to his preferred subject, mathematics, and became a teacher. The University of Königsberg rewarded his paper on Abelian functions with an honorary doctorate in 1854. He soon [...]

Tweet

This article deals with calculating the areas of squares and triangles. It is an easy introduction to a series eventually leading to calculus, the mathematics of Newton and Leibniz. But let’s not put the cart before the horse. On second thought, we do need to start with Descartes. The Cartesian Plane   René Descartes, a French citizen who died in Sweden, lived from 1596 to 1650. He developed the “Cartesian plane”, the familiar X-Y coordinate system mapped onto a flat plane. This is a blend of geometry and algebra, and will become very useful later in this series of articles. The Cartesian plane has a horizontal X-axis and a vertical Y-axis. Any point in the plane has an X-Y coordinate value such as (x=1, y=1). Usually this is shown as (1, 1). To discuss any arbitrary set of points, notation such as “(x1, y1), (x2, y2), (x3, y3)…” is used. The plane is usually drawn with the X-axis and Y-axis meeting in the centre, at coordinate (0,0). However, this article will focus on the first quadrant, where both X > 0 and Y > 0. This next image shows a two-tone square, with each side 6 units long. The corners [...]

Tweet

Zeno, Achilles and the Tortoise Around 450 BCE, the Greek philosopher Zeno of Elea spread the vicious rumour that Achilles was unable to catch a tortoise. This story is often called “Zeno’s Paradox”, but it is only one of several attributed to him. The Math Problem for Achilles and the Tortoise Let’s put the story into the form of a modern math problem. Achilles can run ten times faster than the tortoise, which has a head start of 100m. (For those in the USA, “100 yards” is just as good a number). They both begin running at the same time. How far must Achilles run to exactly catch up to the tortoise? The Paradox for Achilles and the Tortoise After 100m, when Achilles reaches the tortoise’s starting point, he sees that the chelonian is now 10m farther ahead. When Achilles reaches the 110m mark, the tortoise has gained another metre. Each time, in fact, that Achilles reaches the place where the tortoise had been, it still is 10% farther ahead. Zeno’s “paradox” is that the swift Achilles cannot catch the plodding tortoise. The Simple Modern Solution for Achilles and the Tortoise How far must Achilles run to catch the tortoise? [...]

Tweet

Benford’s Law, More than a Mere Statistical Quirk Benford’s Law is a bit unusual in mathematics, in that it started with a statistical quirk rather than a logical deduction from axioms through proofs. It also comes as a surprise to most people. It is also rare in science for a law to be named for the second of three “discoverers”. Usually the first publisher claims the prize. Just what is this law? A Quick Statement of Benford’s Law Benford’s Law states that, for a large collection of numbers with a wide range of values, the probability that the leading digit is n, is log[base 10]((n+1)/n). The Graph of Benford’s Law This spreadsheet shows the graph of Benford’s Law. As you can see, it shows that the number “1″ would be the leading digit about 30% of the time; “2″ over 17%; declining to “9″ at under 5%. Who Developed Benford’s Law? Three people share the honours for developing Benford’s Law: Newcomb, Benford and Pinkham. Simon Newcomb Suggested the Formula Simon Newcomb (1835-1909) simply noticed that the well-used book of logarithm values was more “tatty” at the front than at the back. He assumed the reason was that the wear and [...]

Tweet

Lothar Collatz proposed the “Collatz problem” in 1937. It is still an unproven conjecture. Although it has not been disproved, neither is there an accepted proof. Defining the Collatz Function and Series A neat and tidy mathematical description of the problem is: Let “n”, “i”, “j” and “m” be any positive integers, and define a function f(n): f(n)=3*n + 1, if “n” is odd (( and greater than 1 )) f(n)=n/2, if “n” is even Then define a series a[j] of positive integers, where a[1]=m for an arbitrary positive integer “m”, and a[i+1]=f(a[i]) What Does The Collatz Conjecture Really Mean? Here is an English description. First, we define a function on a positive integer, “n”, that has two rules. If “n” is odd, multiply “n” by three and then add one. If “n” is even, divide “n” by two. Let’s call this whole function “f(n)”. Second, we create a series of positive integers. Choose any positive integer as the starting number. Create the next number in the series by applying that two-rule function “f(n)” to the current number in the series. It would be decent of us to point out that the series is really boring if you start with [...]

Tweet

Two of my previous articles had proofs using mathematical induction. After publishing the second article, I realized that some might not know exactly what mathematical induction is, or how it normally is used. The least technical description would be that it is like a row of standing dominoes. First, demonstrate that one standing domino falls after it is pushed. Second, demonstrate that pushing over one standing domino will knock down the second. Essentially, that proves that a row of standing dominoes will fall if the first is pushed. Mathematical induction is useful when dealing with proofs about “natural numbers”: the set {1, 2, 3…}. This is exactly the set of positive integers. The “Principle of Mathematical Induction” itself is an axiom of mathematics. It is a statement assumed to be true, so it can be a building block in a mathematical system. The Definition of Mathematical Induction Peano’s Fifth Axiom is the “Principle of Mathematical Induction”, which has two practical steps. A proof using mathematical induction must satisfy both steps. The principle also starts with a quick definition. Let P(n) be the function or relationship about the number “n” that is to be proven. [1] Prove that “P(1) is true”. [...]

Tweet

Thales’ Theorem states the following: Using the diameter of a circle as the base of a triangle with the apex on that circle, means that this triangle will be a right triangle and the diameter will be the hypotenuse of that triangle. As John Page says, “The diameter of a circle subtends a right angle to any point on the circle”. (The word “subtends” means that it “creates an angle at a distant point”). The diagram shows it more clearly. Position ‘O’ is the center of the circle. Line AB is the diameter of the circle. Point ‘c’ is on the circumference. Thales’ Theorem says that the angle ACB of the triangle ACB is a right angle…regardless of where ‘C’ may be placed on the circumference. To Prove Thales’ Theorem To prove Thales’ Theorem, let’s start by adding another line, CO, to the diagram. Preparing for the Proof of Thales’ Theorem The point ‘O’ is the center of a circle with radius of length ‘r’. AB is a diameter with ‘O’ at the center, so length(AO) = length(OB) = r. Point C is the third point on the circumference. We have triangles OCA and OCB, and length(OC) = r also. [...]

Tweet

What are “triangular numbers” and “square numbers”? How can you change one into the other? The basics of triangular and square numbers are not as complex as you’d think. How to Create Triangular Numbers Using Pennies The easiest way to think of triangular numbers is to start placing objects into the shape of a triangle. Check the first image: 10 pennies in the shape of a triangle. Each side has 4 pennies, and the center is filled with a penny. Let’s call this a “4-10″ triangle, because there are 4 pennies on a side and 10 pennies in total. “10″ is the fourth triangular number. In the rather strange and arbitrary way that mathematicians start things, one could say that a triangle with 1 penny on each side forms a triangle with 1 penny in total. This would be a “1-1″ triangle. In any case, the first triangular number is “1″. Again, check that first picture and just look at the bottom penny. Put 2 pennies on each side to get a total of 3 pennies. “3″ is the third triangular number. How to Create Triangular Numbers Using Mathematics The rule to create the [n]th triangular number, T, is: T(n) [...]

Tweet

What is the Golden Ratio? Refer to the first image. The long blue line is broken into two unequal lengths “A” and “B”, so the total length is “A+B”. It happens that the ratio of A:B is exactly equal to the ratio (A+B):A. This is the “Golden Ratio”, which also goes by the name “Golden Mean”. Of course, a line can be broken into any pair of lengths…preferably values from the “positive real” set of numbers. It happens that this is special for any number of reasons. The numeric value is approximately 1.618, but there is much more to it than that. The History of the Golden Ratio The natural world provides examples of this approximate ratio. Consider a person’s height compared to the area in which the waist “cuts” the torso where a person bends at the waist. The head-to-navel length is “B”, the navel-to-toe length is “A”. For most people, the A:B ratio is nearly equal to the (A+B):A ratio. In other words, full height is to toe-to-navel length as toe-to-navel length is to navel-to-scalp length. Greek architects and sculptors found that this is an aesthetically pleasing way to create buildings and art. In particular, columns and buildings [...]

Tweet

The phrase, “the Fibonacci sequence” usually refers to a set of numbers that starts as {0, 1, 1, 2, 3, 5, 8, 13, 21…}. At least, “The Life and Numbers of Fibonacci” starts with “zero and one”. Many others would skip the zero and simply start with “one and one”, but that does not matter. This sequence is created by following two rules: The first two numbers are 0 and 1. The next number is the sum of the two most recent numbers.   Actually, one must perform rule #2 over and over again, until one runs out of time, patience, paper or ink in the pen. So, constructing the Fibonacci sequence starts from 1=1+0, then 2=1+1, then 3=2+1, then 5=3+2, 8=5+3, and so on. Why is the Fibonacci Sequence Important? As with many mathematical concepts, the Fibonacci sequence started as the solution to a contest. Then people began to realize it crops up in many places. Leonardo Fibonacci Won a Contest Leonardo Fibonacci invented his sequence to win a competition sponsored by Emperor Frederick II in 1225. The contest question was: Start with a pair of rabbits. Every month, every pair of rabbits who are over a month old [...]

Tweet

“Image of a Turmeric Flower or Curcuma longa” by sophiea Curcumin is found in turmeric, the primary spice in curry. Both modern medicine and Asian tradition make claims for the health benefits of curcumin. What really is known about this spice derivative? Traditional Claims of the Health Benefits of Turmeric Turmeric powder is made from the rhizome of the turmeric plant. As noted in “curcuma longa“, traditional East Indian medical uses for turmeric starts with ingesting it for stomach ache. It has also been used to treat skin infections, muscle aches and arthritis. One traditional Chinese treatment is using a turmeric derivative as a topical analgesic colic. Internally, the Chinese have used turmeric to treat chest pain, colic, hepatitis and ringworm. Recent Medical Claims for Curcumin Modern medicine usually extracts the active curcumin from the flavourful turmeric. It might be applied externally, taken orally or injected in a solution. The claim for being a “blood thinner” seems true; this makes curcumin potentially beneficial for people in danger of heart attack or stroke. In “Turmeric“, Medline Plus warns of the danger of combining curcumin with other anti-coagulants, such as ibuprofin or warfarin. The blood-thinning properties of curcumin make it inadvisable for [...]

Tweet

A tesseract is the equivalent of a cube in four dimensions; in other words, it is a four-dimensional hypercube. As G. Olshevsky comments in Glossary for Hyperspace, a hypercube fits into “n-space” for any value of “n”. Human laziness leads few people to think in five dimensions or beyond. Therefore a hypercube is often considered to refer to merely four dimensions. To Develop a Tesseract A good place to start developing a tesseract is with this image. This paragraph will follow the series of five drawings, going from top to bottom. Start with a single point, which has no “size”. A point is not a tiny circle, because a circle has the length of its circumference and the area it covers. If one could hammer down the point at its starting location, and then drag it along in one direction, it would leave a trail called a “line”. That is, if it were large enough to be nailed down, and still have some surface left over to grab. It really does not have those properties, so it is easier to just draw a line with a pencil. Still, a line is the extension of a point in one dimension. In [...]

Tweet