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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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. [...]

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 [...]

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, [...]

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 [...]

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 [...]

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 [...]

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, [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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” [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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 [...]

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. [...]

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 [...]

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. [...]

“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 [...]

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 [...]