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 […]

## Examples of Turing Machines: Loops, Halts, and Rewriting

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, […]

## The Turing Machine: A Brief Introduction

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 […]

## Potential New Algorithm to Calculate the Cube Root of a Number

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 […]

## Repercussions from the Richard Paradox: Math Rules

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 […]

## Conditional Probability is Not Commutative: Formulas and Examples

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 […]

## An Introduction to Calculating Conditional Probability in Mathematics

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 […]

## A Taste of the 2012 Joint Mathematics Awards and Prizes

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 […]

## Axioms and Two Useful Theorems of Discrete Probability Functions

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 […]

## The Probability of the Allais Paradox in Lottery Preferences

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 […]