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The pedagogical objective of the various degree programs of the Department of Applied Mathematics is to develop student logic using mathematical models and quantitative techniques that can be applied to solve problems encountered in the physical, engineering, biological and behavioral sciences.
As an Applied Mathematics major, you will be trained in classical mathematics, in addition to being exposed to ideas on the cutting edge of current mathematical applications found in a diversity of fields.
Your training will include the use of the latest technological tools, and you will have access to the department's computer labs, equipped with PCs, workstations and the latest mathematical software.
A degree in Applied Mathematics provides an excellent base on which to build a career in virtually any high technology field, and provides a particularly strong foundation for advanced study in science, engineering and finance.
The department also encourages its undergraduate students to pursue a secondary degree (minor degree) from other departments based on their interests.
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Jesús Bonilla holds a degree in mechanical engineering and a MSc in numerical methods in engineering both by the Technical University of Catalonia, UPC. Since September 2014, he is a predoctoral researcher at the Large Scale Scientific Computing team at CIMNE. The objective of his PhD is to develop efficient finite element solvers for fluid models of plasma. He is focused in the development of monotonicity-preserving schemes for hyperbolic problems. He aims to apply these schemes to one and two-fluid magnetohydrodynamic equations. Which model problems of interest in fusion reactors design. All his research is done by developing the LSSC in-house code, FEMPAR, a massively parallel finite element Fortran library.


PINE (AH) 2020 - 2021
Welcome to Pine Class Page.
Teacher: A Hughes
Teaching Assistants: C Moxon, J Winder, C Brookes
We are all wroking hard to be the best we can be.
This half term our Learning Vehicle is Around the World in 80 days.
We will be linking our learning across all subjects to this Vehicle.
In Science we will be looking at animals and their habitats.
In Geography we will be looking at climate around the world and famous monuments from around the world.
In Maths we will be looking at all 4 operations (addition, subtraction, multiplication and division).
In History we will be looking at how Britian has influenced the world.
We will be swimming every Tuesday this half term. Please make sure your child comes to school in their swimming kit. Don't forget to pack a towel.
We will be having the visiting artist working with us on some aboriginal art.
We are also lucky to have some sports coaches coming in to work with us.
All of our work will be posted on class dojo. Don't forget to login to see what we have been up to.
The children have access to many online learning sites to keep them busy at home and in school. Log in and see if you can beat your previous scores on some of the learning games.


Barron's SAT Subject Test Math Level 1
Authors: Ira K. Wolf Ph.D.
Publisher: Barron's Educational Series
Keywords: level, math, test, subject, barron
Category: Sat & Psat, Test Guides - College & University, Education, Reference,
ISBN-10: 0764143557 ISBN-13: 9780764143557
Binding: Paperback (3)
List Price: 14.99 USD
This updated manual opens with an overview of the SAT Math Level 1 test, followed by a section covering the most important tactics for succeeding on the test. Separate chapters review basic arithmetic, fractions and decimals, ratios and proportions, basic algebra, plane geometry, solid and coordinate geometry trigonometry, functions, statistics, and miscellaneous math topics. Three full-length model tests are presented with complete solutions for every problem.
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You can search on LeatherBound to download or purchase an ebook.
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The question of how to grade homework efficiently and effectively is one that many teachers struggle to answer. It is brought up in meetings and textbooks consistently. I currently grade homework based on effort. If the student completes the homework with full effort then they receive full credit, whether or not all problems are correct. Questions may be brought into class and answered by me or the students peers. The main reason I grade homework based on effort is because I want homework to be practice. I do not expect every student to get every question correct the first time around. This method, in my opinion, works, but I was struggling with the challenge of devoting a significant amount of class time to homework questions. Some students needed to see the questions completed and others sat there bored. Also, the answers to the homework questions are in the back of the back and I request that students check their answers. Some students did and others did not, so some knew which questions to ask and others didn't. Another downfall is a student lacks the encouragement from me to revisit incorrect answers and correct them. So, I debated a few options: collect homework and give individual feedback, collect homework intermittently and give individual feedback, continue grading homework based on effort, but don't go over questions in class. I had a few options, but nothing struck the right chord with me. So I researched what other math teachers were doing and found a great resource on National Council of Teachers of Mathematics (NCTM).
The plan for homework is this:
I cannot contain my excitement for this information! I just want to tell every single person I cross paths with about these study habits, primarily retrieval practice.
Us teachers often have a review day and then tell our students to go home and study. Well...what does that mean? I think, and have found that many students do not know. I tutored Calculus 1 in college for 3 years and our focus at the Tutor Center was not the content, but rather the study skills. If we, as learners, know how to learn, then we can master any subject.
I came across these study habits while listening to my favorite podcast on my drive to UMF for my Graduate class. The podcast can be found at https://www.cultof
pedagogy.com/learning-strategies/. Jennifer Gonzalez interviewed Megan Smith and Yana Weinstein on their research about Learning Strategies. While listening, I compared my study habits with their strategies and realized that I naturally did all of them, but many students do not. Many students either don't know where to start or think they are studying when they really aren't. I often think of rereading notes or a textbook. The student thinks they are reviewing and studying, but this does not build new pathways in the brain or bring content into long term memory.
I will provide a brief introduction to each strategy and explain how I use it in the classroom, how your student can use it, and how you can use it with your student. Megan and Yana provide posters discussing each strategy that can be found at https://static1.squarespace.com/static/56acc1138a65e2a286012c54/t/57d03e669de4bbd3567d57a6/1473265254536/All-Color-Posters.pdf. I will be posting these in my room and discussing each strategy with the students.
1. Retrieval Practice
This strategy is the underlying practice in all of the other strategies and the one that I am obsessed with. The main idea is that the student puts away all of their materials and attempts to retrieve content from their memory. The student then checks their materials to see if they were right. This is often seen in flash cards and practice tests.
2. Spaced Practice
With this strategy the student spreads out their studying over a longer period of time compared to a few days or the night before. Yes, cramming can get a passing grade, but that information is only going into short term memory and will not be accessible in the next lessons or for future exams. We all know that math builds on itself!
This one is self explanatory in that the student would elaborate on the concept, asking why and connecting it to other concepts, comparing and contrasting these concepts.
Swap back and forth between concepts rather than doing the some problem over and over. When they discussed this on the podcast I thought of every single math teacher I had and of myself. Homework in math is often the same problem repeated a number of times. I will be changing this in my class!
5. Concrete Examples
For an abstract concept have a concrete example and multiple examples.
6. Dual Coding
Tie visuals to concepts and find different ways to represent information.
This week we really got going on some exciting content!
BC Calculus is cruising through their review of AB topics and have demonstrated mastery on a pretest of Chapter 3 topics including basic derivatives and particle motion. We have covered Chapter 4 and introduced a new BC Topic! We played around with parametric equations and how to take a derivative of such functions.
AB Calculus is underway with Limits. This is their first taste of an abstract concept. Limits have a procedure and set of steps to follow, but to answer questions correctly one must have a true understanding of the concept. It may have caused confusion for some at first, but I believe we are getting the hang of it.
Honors Algebra 2 has PSAT warm up questions to start every class, but the class got to experience a full PSAT test on Wednesday, where we took a break from our Algebra 1 review. We talked about some strategies when taking the PSAT and SAT. They took their first test on Friday over Algebra 1 review.
Honors Advanced Algebra with Trig also starts every class period with PSAT questions as Freshman will be taking the PSAT instead of the NWEA this year. It's always good to get some practice and experience in working with these types of questions. The class took a Quick Quiz this week. These Quick Quizzes are a way for the student to see if they are following along and keeping up with the content in a low stress situation. The Quick Quizzes can be retaken as many times as the student wishes before the Chapter Test. It also provides invaluable retrieval practice. Also, we are moving into new content as well, specifically complex numbers.
BC Calculus will be working on challenge activities to strengthen their understanding of Implicit Differentiation and Chain Rule as well as reviewing for an upcoming test over Chapter 3 and 4.
Both Blue and White day AB Calculus's will be reviewing and testing over Chapter 2.
Honors Algebra 2 will be moving on to new content next week. We will be discussing some of my favorite topics including piece-wise functions and transformations of parent graphs. These concepts are crucial for future courses, primarily Calculus!
Honors Advanced Algebra with Trig will be continuing to learn new material including completing the square and parent graphs!
Some students come into class with a math phobia. I understand that some of you may also dislike math and have an aversion to it. I ask that you please try to be positive when talking about mathematics. I work hard in building a class that students enjoy and find value in the content. I try to build curiosity into my class, so students are interested and buy in to what I am teaching. Ask your students what they learned in math! Encourage them to talk about and get excited about their ability to understand the concepts.


Craps is one of the most popular dice games in the world, with a rich history that can be traced back to the Roman Empire. The dynamic pace of play and the wide range of betting options are what have caused craps to become an absolute staple in both online casinos and landbased gambling establishments.
Since craps relies heavily on chance and there is no way to affect the outcome of the dice toss, many inexperienced players wrongly assume craps is an easy game to play. This is not necessarily true as the sheer number of betting options will suffice to overwhelm any craps novice. While there is no way for players to predict the outcome of the dice with certainty, they can still turn up a good profit as long as they take the time to learn the bet types and the possible dice combinations. Understanding the house edge, the true odds, and the payouts certainly can work to the advantage of craps players.
Understanding the House Edge
Gambling establishments do not generate profits because the personnel manages to outplay the customers. The casinos’ profits result from the built-in advantage they have over their players. This built-in advantage is referred to as the “house edge” and is the key thing inexperienced players need to understand prior to joining the craps table.
All casino games are tilted in favor of the house as it utilizes the law of averages to gain its edge over the players. The house edge is typically expressed as a percentage and represents the average profit the casino collects from each wager the players make. Please note the house generates a profit on every single bet, regardless of whether it is a winning or a losing one.
As was mentioned earlier, there is a great number of bets you can place in craps. But what is more important, the house edge tends to fluctuate between the different bet types. On some craps bets, the house edge drops to zero while on others, the house’s advantage skyrockets to a two-digit figure. That is why, smart players choose bets with lower house edge which helps them to exploit the game and end their betting session on profit.
Let us demonstrate how the house edge works with an example. Bets on the Pass Line have a low edge of 1.41%. This means that for each $100 players wager on the Pass Line, the casino will collect an average of $1.41. Whether players win or lose is irrelevant – either way, they will lose $1.41 on average per every $100 they wager on the Pass Line. There are players who prefer to place Proposition bets because the latter have more substantial payouts. However, this is not always a good idea since the house edge for Proposition bets ranges between 5.56% and 16.67%.
Pass/Don’t Pass Line bets and Come/Don’t Come bets are considered a smarter option, especially for inexperienced players. The tilt in favor of the casino is smaller with such bets and the probability of the shooter rolling a winning dice combination for these bets is much greater. Because of this, such bets have lower payout ratio and pay even money.
Players, who wager on the Pass Line are allowed to lay or take odds on their bets. Once the shooter rolls his point, players can collect money on their Pass Line bets but will also be paid at true odds. This causes the house edge to drop to 0% and is the only instance in which the casino does not have an advantage over craps players. How winning bets are paid also influences the house edge. When paying out winning multiple-roll bets, the casino typically rounds the sum down to the nearest number.
Dice Combinations in Craps
In order to calculate the house edge on craps bets, players are required to acquaint themselves with the dice combinations and the probability of each number being rolled. As we know, craps is played with two dice and each die is a cube with six equal-sized sides. There are 11 possible outcomes of a two-die toss, namely 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.
The number of possible combinations is calculated by multiplying the number of the sides of the two dice. Thus, we get 6 x 6 = 36, from which it follows there are 36 combinations that all will total one of the eleven outcomes we have listed above. You can check the dice combinations and the totals they add up to below.
|Dice Combinations in Craps|
|Outcome||Possible Combinations with Two Dice||Ways to Rolls the Outcome|
|2||1 – 1||1|
|4||1-3, 3-1, 2-2||3|
|5||1-4, 4-1, 3-2, 2-3||4|
|6||1-5, 5-1, 2-4, 4-2, 3-3||5|
|7||1-6, 6-1, 2-5, 5-2, 3-4, 4-3||6|
|8||2-6, 6-2, 3-5, 5-3, 4-4||5|
|9||3-6, 6-3, 4-5, 5-4||4|
It certainly is not difficult to notice there are more combinations for some of the outcomes. There are more ways for the shooter to roll out 7 than any other number as becomes evident by the diamond shape of the middle column. Since there are 6 combinations of the dice that total 7, the chances of this particular number being rolled are the greatest. This is yet another thing players need to consider carefully when deciding which types of bets to place at the craps table.
How to Calculate the House Edge in Craps
As craps is a negative expectation game, it is mathematically impossible for players to gain advantage over the house. The only exception is when players are taking or laying odds on their bets. However, understanding the probability of specific numbers being rolled and the house edge on different bet types is essential as it would enable players to make better-informed decisions when placing their bets.
The two dice used in the game of craps allow for a total of 11 outcomes with totals from 2 to 12. There are 36 different ways in which the shooter can roll these 11 outcomes. From this, it follows that the total of 3 can be rolled in two different ways meaning that the probability of this number being thrown is 2 to 36. Similarly, there are six different combinations that add up to seven, so the chances of throwing this number are 6 in 36. The probability of rolling any of the other numbers can be calculated in the same fashion.
Once players understand how to calculate probability, they can proceed to calculate the house edge. It would be best to provide an example to make things as clear as possible. The Craps 2 bet wins whenever a total of 2 comes up on the next roll of the dice. In other words, there are 35 ways to lose and a single way to win with this type of bet. If it wins, the payout will be 30 to 1. Since there are only two outcomes for this type of bets, win or lose, its expected value can be calculated in the following way: (1/36)x30 – (35/36) = -5/36 = -13.89%.
Other wagers, like the Any Craps bet, allow for more winning combinations. The Any Craps bet pays out 7 to 1 whenever numbers 2, 3 or 12 are rolled. There is one combination for number 2, two combinations that add up to number 3, and one combination that adds up to 12. Thus, there are 4 ways to win out of 36 possible combinations, so the probability of collecting a payout with this bet can be expressed like this: 4/36 = 1/9. This means that out of every nine Any Craps bets, one will win and the other eight will lose. The expected return for this bet can be expressed as follows: (4/36)x7 – (32/36) = -4/36 = -1/9 = -11.11%.
As you can see, the house edge in craps tends to fluctuate greatly depending on the type of bet you place. With some bet types where the outcome is determined by multiple rolls of the dice, calculating the built-in house advantage is a mean feat. Players, who experience difficulties performing such complex calculations, can try to learn the house edge of craps bets by heart.
Understanding the True Odds
You may have noticed there is a discrepancy between the probability of winning with specific bets and the way the winnings are paid out. This discrepancy is exactly what gives the house its edge. The house succeeds in maintaining an advantage over players by paying less for winning bets than the true odds would dictate. For instance, bets on the Pass Line pay even money and have the lowest house edge of 1.41% only, which means players will lose $1.40 on average per every $100 they wager at the craps table. Basically, the only exception to this rule is when free odds are taken or laid on the bet – in this case, the house holds no advantage over players because winnings are paid at true odds.
You may wonder why players persist in betting on craps when the game is obviously tilted against them. The truth of the matter is smart players exploit a phenomenon, called distribution variance, which is precisely what causes the hot and cold streaks when the dice are rolled repeatedly, because perfect, even distribution is something that occurs rarely, if ever, in nature.
Also, it is important to remember that it takes prolonged periods of time for the odds to balance out so that the house can make a profit. Thousands of hundreds of dice rolls are required for the odds to follow their natural path and maintain equilibrium. However, many players remain at the craps table for short periods of time only. It is precisely during these fleeting moments of time that variance creeps in and allows craps players to turn up a profit. At least, if they bet smartly, manage their bankroll properly, and understand the probabilities of winning with different types of bets.
The Payouts of Craps Bets
What causes confusion among inexperienced players is the overwhelming number of bets they can place at the craps table. Needless to say, as the probability of rolling out specific numbers varies, the payouts for different craps bets also differ. Since the chances of winning with bets on the Pass/Don’t Pass Line and Come/Don’t Come bets are the greatest, these wagers pay even money. The chances of winning with Proposition bets are smaller, so their payouts are more significant. You will be able to find the payouts and the house edge for all bets in craps in the table below.
|Craps Bets Payout and House Edge|
|Type of Bet||Payout||True Odds||House Edge|
|Pass Line/Come Bet||1 to1||251 to 244||1.41%|
|Don’t Pass/Don’t Come Bet||1 to 1||976 to 949||1.36%|
|Free Odds Bet on the Pass Line||2 to 1 (4 or 10), 3 to 2 (5 or 9), 6 to 5 (6 or 8)||Same as Payout||0.00%|
|Free Odds on Don’t Pass Bets||1 to 2 (4 or 10), 2 to 3 (5 or 9), 5 to 6 (6 or 8)||Same as Payout||0.00%|
|Free Odds on Come Bets||2 to 1 (4 or 10), 3 to 2 (5 or 9), 6 to 5 (6 or 8)||Same as Payout||0.00%|
|Free Odds on Don’t Come Bets||1 to 2 (4 or 10), 2 to 3 (5 or 9), 5 to 6 (6 or 8)||Same as Payout||0.00%|
|Place Bets on 4 and 10||9 to 5||2 to 1||6.67%|
|Place Bets on 5 and 9||7 to 5||3 to 2||4.00%|
|Place Bets on 6 and 8||7 to 6||6 to 5||1.52%|
|Place Bets to Lose 4 and 10||5 to 11||3.03%|
|Place Bets to Lose 5 and 9||5 to 8||2.50%|
|Place Bets to Lose 6 and 8||4 to 5||1.82%|
|Field Bets on 3, 4, 9, 10 and 11||1 to 1||5.56%|
|Field Bets on 2 and 12||2 to 1||5.56%|
|Hardway Bets on 6 or 8||9 to 1||10 to 1||9.09%|
|Hardway Bets on 4 or 10||7 to 1||8 to 1||11.11%|
|Big 6 or 8||1 to 1||9.09%|
|Lay Bets on 4 and 10 (5% Commission)||1 to 2||2.44%|
|Lay Bets on 5 and 9 (5% Commission)||2 to 3||3.23%|
|Lay Bets on 6 and 8 (5% Commission)||5 to 6||4.00%|
|Buy Bets on 4 and 10 (5% Commission)||2 to 1||4.76%|
|Buy Bets on 5 and 9 (5% Commission)||3 to 1||4.76%|
|Buy Bets 6 and 8 (5% Commission)||6 to 5||4.76%|
|Big Red/Seven Bets||4 to 1||5 to 1||16.67%|
|Any Craps Bets||7 to 1||8 to 1||11.11%|
|Proposition Bets on 2 and 12||30 to 1||35 to 1||13.89%|
|Proposition Bets on 3 and 11||15 to 1||17 to 1||11.11%|


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Finish the rings quicker and quicker
- & Enjoy Maths -
Times Tables & Math for Kids is a free software application from the Kids subcategory, part of the Games & Entertainment category.
The app is currently available in English and it was last updated on 2016-03-21. The program can be installed on Android.
Times Tables & Math for Kids (version 6.9) has a file size of 8.07 MB and is available for download from our website.
Just click the green Download button above to start. Until now the program was downloaded 0 times.
We already checked that the download link to be safe, however for your own protection we recommend that you scan the downloaded software with your antivirus.
Here you can find the changelog of Times Tables & Math for Kids since it was posted on our website on 2016-10-08.
The latest version is 6.9 and it was updated on soft112.com on 2018-03-29.
See below the changes in each version:
posted on 2016-03-21
If you fail, a bubble appears with the correct answer.,So you learn from your mistakes ;),Do yourself a math genius .,Test " MATHS RINGS" now !!


If you are visiting from China, it might be slow to load the pictures, but eventually you will get to see my lovely face. 😜
I am currently a quantitative researcher at JQ Investments, and an industrial advisor at School of Finance, Southwestern University of Finance and Economics. I received my Ph.D. in economics from Stanford University, advised by Alvin Roth. Before coming to Stanford, I got my Bachelor's degrees in mathematics and statistics from UC Berkeley. My research interests lie primarily in game theory, market design, algebraic combinatorics and stochastic systems. In particular, I am interested in understanding the dynamics of reaching a market equilibrium. In my free time I enjoy playing video games and that's where I "put game theory to the test". I have always been fascinated with strategic reasoning, and to find the winning strategies is the reason why I decided to get a Ph.D. in game theory in the first place. Some of my past gaming "achievements" can be found in the Trivia page.
Some academic trivia:
Originally I came to Stanford pursuing an Operations Research Ph.D., and later transferred to Economics. I remained in both programs for 4 years and finished the degree requirements for both. In the end I was free to choose which Ph.D. to get. (As it turns out, getting two Ph.D.s from Stanford would require 270 non-overlapping units, which is 9 years worth of coursework.) I was fair to both subjects and published two papers in each field (Mathematics of Operations Research + Operations Research Letters vs 2 x Games and Economic Behavior).
My Ph.D. dissertation is Market Structure and Dynamics. And my dissertation defense committee consists of Alvin E. Roth (primary adviser), Itai Ashlagi, Fuhito Kojima, Robert B. Wilson (chair) and Zeyu Zheng. So two Nobel laureates wasted 90 minutes listening to my nonsense (－‸ლ).
My Erdős number is 4. Erdős-Hoffman-Rothblum-Roth-Wu and also Erdős-Rubel-Blair-Roth-Wu. (Thanks Al!)
David Hilbert is my advisor^6. So I am luckily an academic descendent of both Euler and Gauss (picture credit to the Mathematics Genealogy Project). My profile in the project can be found here.


Free Sample Algebra 2 Worksheets. pdf; This PDF can be accessed. or for answers to any of your homework questions. Download your FREE.
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QHS geometry students work the angles to build tiny house
For five years, classes of Quitman sophomores have been putting their geometry skills into practice by building a tiny house.
John Herring and Chris Hood’s class, “Geometry in Construction,” learns about geometry then builds a 12 foot by 36 foot tiny house.
Instead of regular geometry, students have the option of participating in a hands-on experience where they alternate between learning geometry one class period then work on the house next time.
“We take what we learn in geometry class and apply to the world of construction,” Herring said.
The class began the project in October, and it should be complete by the end of May.
In the past, the house was built by the students and then auctioned. However, this time a woman bought the house at the beginning of the year, and the students are building the home for her.
Students are building the home on campus and it will be moved to the location the buyer chooses.
A previous principal found the class from a school in Colorado five years ago, and they’ve continued ever since, Herring said.
Herring, who also teaches woodshop classes, said the house is fully functional and ready to move in once the students finish construction. The home has one bedroom and one bathroom.
He added that each of the students brings different levels of construction skills from beginners to those with prior experience.
“They apply geometry skills outside and learn to work together,” Herring said. “The school is very supportive to help us out.”


How much will state college cost in 2035?
When you look 18 years ahead to 2035, colleges could have a tuition of $54,070 per year, and private colleges could be looking at a tuition of $121,078 per year. Let’s do a little quick math here. The projected cost of college in 2035 of $121,078 multiplied by four…that’s $484,312 for a four-year degree.
What will the cost of college be in 18 years?
At this rate, when babies burn today are turning 18, a year of higher education at a private school — including tuition, fees, and room and board — will cost more than $120,000, Vanguard said. Public colleges could average out to $54,000 a year.
How much will it cost to go to college in 2030?
College could cost up to $100,000 per year by 2030.
What is the average cost of college in 2020?
For the 2020-2021 academic year, the average price of tuition and fees came to: $37,650 at private colleges. $10,560 at public colleges (in-state residents) $27,020 at public colleges (out-of-state residents)
How much does the cost of college increase each year?
On average, tuition tends to increase about 8% per year. An 8% college inflation rate means that the cost of college doubles every nine years.
How much will a 4 year college cost in 2024?
By the end of those proposed steps, estimated tuition and systemwide mandatory fees for California residents in the entering class of 2024 would be $15,414, not including housing, food and books, according to plan. The figures could change somewhat because they will be partly linked to inflation.
What will college cost 2039?
How much will it cost to send your child to college in 18 years and how much do you need to save? A 4 year degree is estimated to be priced at $442,697.85 for students enrolling in 2039 if tuition increases average 7% per year until then.
How do you calculate future cost of education?
The formula used is: Future Value = Present Value (Expected returns+Rate of interest) *N where ‘N’ is the duration of the time period of the investment required.


Retail math is a fundamental aspect of the retail industry that involves various mathematical calculations and analysis related to inventory, sales, pricing, profit margins, and other financial metrics. It is an essential tool for retailers to make informed decisions and optimize their operations to maximize profits and efficiency.
In the context of clothing, footwear, and fashion accessory stores, retail math is particularly crucial due to the industry's dynamic and competitive nature. Here's why retail math is important in this specific sector:
Understanding Retail Math:
Retail math encompasses a range of calculations, including but not limited to:
Gross Margin: Calculating the difference between the cost of goods and the selling price, crucial for determining profitability.
Sales per Square Foot: Evaluating the efficiency of retail space utilization.
Sell-Through Rate: Measuring the percentage of inventory sold in a given time period.
Importance to Clothing, Footwear, and Fashion Accessory Stores:
In the fashion industry, trends change rapidly, and consumer preferences can shift swiftly. Retail math allows these stores to:
Optimize Inventory Levels: Calculating sell-through rates helps in managing inventory effectively by restocking popular items and reducing slow-moving merchandise.
Pricing Strategy: Determining the right price points based on cost, competitor pricing, and desired profit margins to stay competitive yet profitable.
Seasonal Planning: Utilizing historical data and trends to plan for seasonal variations and stock accordingly.
Importance for Clothing Store Buyers:
Buyers in clothing stores play a vital role in the success of the business. Retail math equations empower them to:
Make Informed Purchasing Decisions: By analyzing historical sales data and using retail math, buyers can accurately forecast demand and purchase inventory that aligns with consumer preferences.
Negotiate with Suppliers: Understanding cost structures and profit margins helps buyers negotiate better deals with suppliers, ensuring competitive pricing and improved profit margins.
Relevance for Aspiring Clothing Buyers:
For those aspiring to become clothing buyers, a solid grasp of retail math is invaluable. Here's why:
Effective Decision-Making: Retail math equips buyers to make data-driven decisions, optimizing inventory purchases and maximizing profitability for the store.
Competitive Edge: In a competitive industry like fashion, having the ability to analyze data and understand retail math sets aspiring clothing buyers apart, making them more attractive to employers.
Career Growth: Proficiency in retail math opens doors to higher-level positions within the retail industry, enhancing career prospects and opportunities for advancement.
Retail math is an indispensable tool in the clothing, footwear, and fashion accessory retail sector, aiding in efficient inventory management, pricing strategies, and overall business profitability. Aspiring clothing buyers should prioritize understanding and mastering retail math to succeed and thrive in this dynamic industry.
Learn about retail math equations.


Just for a fun challenge, I decided to learn to count to 10 in 10 languages, besides English. In most languages, if you know 1-10, you can say just about any number. Eleven, for example, being 'ten-one', literally.
I've also learned a few greetings: 'hello', 'goodbye', 'please', 'thank you' and 'sorry' in each language.
Now I have accomplished this in:
French, Spanish, Greek, German, Italian, Russian
Chinese, Korean, Arabic, Japanese
(Originally, I had planned to learn 20 languages, but I have decided that 10 is enough, for now.)


Garden of Eden FACT SHEET for the United States
1) A ¼ acre piece of land produces an average of about 7.3 pounds of food per day, for one person,
for one year, with 47 common crops grown in the United States. (See Crop Yield Verification
2) According to Random House Dictionary, an acre of land is 43,560 square feet, so ¼ of an acre is ¼ of that, or 10,890 square feet. If a quarter-acre piece of land is square, it is about 105 feet long on each side of the square. If we double ¼ acre, we get ½ of an acre. This is 21,780 square feet. This allows ¼ acre for food and ¼ acre for a residence, recreation, etcetera.
3) According to Wikipedia.com, the population of the U.S. in 2018 was 327,096,265 people. If you gave everyone in the nation ½ of an acre of land per person, how much land would that be?
1) A ¼ acre piece of land produces an average of about 7.3 pounds of food per day, for one person, for one year, with 47 common crops grown in the United States. (See Crop Yield Verificationpaper.)
(327,096,265 people) X (½ acre land/person) = 163,548,132 acres.4) There are 640 acres in one square mile. How many square miles is 163,548,132 acres?
163,548,132 acres ¸ 640 acres/square mile = 255,544 square miles5) If 255,544 square miles was one, big, square piece of land, how long would one side of the square be?
answer: Ö 255,544 = 505 miles/ side of square6) The United States is 3,531,905 square miles, in size. What percent of this land is 255,544 square miles?
255,544 square miles ¸ 3,531,905 square miles = 0.072THAT MEANS THAT ALL AMERICANS COULD EAT ON ABOUT 7% OF OUR LAND WITH THE GARDEN OF EDEN LIFESTYLE!!!
converting to percentage: 0.072 X 100 = (approximately) 7.2%
937,000,000 acres ¸ 640 acres/sq. mi. = 1,464,062 square miles8) What percentage of the total land area of the U.S. is 1,464,062 square miles?
1,464,062 square miles ¸ 3,531,905 square miles = .4145 = 41.45%THIS MEANS THAT RIGHT NOW THE UNITED STATES USES ALMOST 6 TIMES AS MUCH LAND AS WE WOULD NEED TO LIVE IN A GARDEN OF EDEN LIFESTYLE!


The great thing about 2-year-olds is they don't live by their calendars. When you pull out their just-completed Valentine's Day gift on February 23rd, they don't whip out their palm pilots, pull up their calendars, and give you a smug look that says, "what took you so long?" They take the gift, smile, and start to play. At least, that was the response from my 2-year-old when I presented her with her new puzzle game at breakfast this morning.
Audrey loves puzzles. She can also count to ten (occasionally - and sometimes frequently - leaving out the number eight). Lately, number books have been a big hit, especially one she received from her Grammy for Christmas. I've noticed that most of the books show the number spelled out (five), the numeric number (5), and a picture of some objects whose number corresponds with the numeric and spelled out number (5 balls). The book from Grammy takes things one step further with numbered buttons on a side panel that say the number when you push them. Audrey has started to master pushing the right numbered buttons, but each number is written in a separate color, so I don't know if she recognizes the numbers or is just matching the up the colors and shape of the numeric numbers. Also, most of the number books we have use different objects to illustrate the numbers on each page.
I started thinking about how I could simplify the idea of matching numeric numbers up with counted objects and came up with the idea of these Valentine puzzle pieces. Now, the colors and objects are the same, so the only thing that makes one puzzle coupling different from the rest is the numeric number and number of hearts. The puzzles are made out of a medium-thickness cardboard box that used to house a yoga mat, but more recently found itself on top of the to-be-recycled heap. I traced some puzzle pieces we already own onto the cardboard then used an exacto knife to cut them out. I painted them white using two coats of acrylic paint left over from another project then used red acrylic paint to freehand the numbers and hearts (again, two coats). I thought of this project at the beginning of February, when it seemed like it would take a week, tops, to carve out niches of time to complete it. I finally finished it at 10 pm last night. What can I say, turns out I'm not a speed demon with the exacto knife or paintbrush. Luckily, two-year-olds don't care.


Ashritha learned to play chess at the age of 7. At 13, she became a national master under the training of Bulgarian Grandmaster Dejan Bojkov at the NorCal House of Chess. In 2015, she took first place in the U.S. Girls’ Junior Championship with a score of 6.5/9. The very next day, she flew to Colombia for the 2015 Pan-American Youth Chess Championship where she earned bronze through a tie-break. She has played in the World Youth Championships four times from 2012-2015, and the 2020 US Women’s Championship will be her fourth appearance in her chess career.
Currently, she is a junior studying at University of California, Berkeley and is majoring in Applied Mathematics and Data Science. She is also the current co-president at the chess club at Berkeley and is a data science enthusiast. Her hobbies include playing the piano, painting, and singing.


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Science - A Mathematical Mystery Tour
Fascinating BBC Horizon programme from 1984 looking at the greatest unsolved problems in mathematics including Fermats Last Therom (since solved), The Goldback Conjecture, The Riemann hypothesis, the P=NP Problem, and more. Featuring interviews with many modern mathematicians, this documentary also looks at the history of maths and some of if its major players from Euclid to Bertrand Russell. In discussing the fact that proof requires more than calculated verification reference is made to the Merten Conjecture. A conjecture which was disproved in 1984, despite holding firm for 99 years and having been verified on computer up to the first 10 billion numbers. Afact that makes us think about the things in our own lives which, with far less evidence, we consider to be certainties. Documentary about pure mathematics, featuring the Japanese mathematicians who had just broken their own world record calculating pi to 16 million decimal points a $25000 prize for the first person to prove there are an infinite number of twin primes and why it took Bertrand Russell 362 pages to prove that one plus one equals two.


Hello and welcome, this is Tim Phillips with Revenade,
and today I want to share with you an idea called the time value of money. I
know many of you have studied financial and investment analysis and in that we've all learnt about the concept of the time value of money that a dollar
today is worth more than a dollar tomorrow and it is the basis for all
compounding and also the discounting of future cash flows and with it we can
make sound investment decisions when we have competing alternatives.
Read the rest of this entry »


50 Mathematical Ideas You Really Need to Know
- 3rd April 2008
- ISBN: 9781847240088
Just the mention of mathematics is enough to strike fear into the hearts of many, yet without it, the human race couldn't be where it is today. By exploring the subject through its 50 key insights - from the simple (the number one) and the subtle (the invention of zero) to the sophisticated (proving Fermat's last theorem) - this book shows how mathematics has changed the way we look at the world around us.
Tony Crilly is Reader in Mathematical Sciences at Middlesex University, having previously taught at the University of Michigan, the City University in Hong Kong, and the Open University. His principal research interest is the history of mathematics, and he has written and edited many works on fractals, chaos and computing. He is the author of the acclaimed biography of the English mathematician Arthur Cayley.
Quercus staff memberBack To Top ^


There are several factors that one should consider when selecting the best school math program for the young ones. Young children are likely to find math as a source of anxiety. There are several programs such as curriculum and the child schools who use nothing like math and remember from school. There is necessary information about math instruction, whereby maths seem f to connect and enhance the learning skills amongst the children. The majority of the schools are likely to use things such a pattern, blocks, and cubes. They get expected to reinforce the kids learning in the classrooms.
One of the universal math programs is that it will teach the concepts in the small bites. The best ideas will help the children in understanding the old ideas. For instance, learning of the new ideas and review of the practices. There is a need to review the information about the word problems that should make real-world connections. It is essential to make it easy for the hidden who are likely to find understanding math a robust plan.
There is an application that helps the children to master the concepts build on the form. For instance, there will be making of the pictures on the graphs using the formula. You will set the information that means the kids have the skill learned. There are a lot of issues for handling and solving the activities and problems that get implied in the real world. There would need to assure that there are different adapted needs since the small group work s will allow the different skills to get imparted to the young ones. The children will learn the practical concepts that work to come up with a more significant focus send the mental assessment of the issues.
There are lessons designed to meet all the grade-level standards. Some factors would make it simple for the young ones who might be struggling to pay attention to the vocabulary. There is a need to make sure that you set the right method for solving the personal issues one would be suffering from. There is a proper review of the information on how the activities that get undertaken. It is essential to get the right services and solve the issues and eliminate the fears the young ones might be suffering from.
It is essential to get the best professional care and helps the children in thinking and reason around the subject areas. It will make sure that the children get the best care in terms of the programs used in their real life. It will make use of the actual objects and models they see daily. The feature will make sure that the children have a language-based curriculum. It will enhance the level of awareness amongst the learners within the setting.
There is the use of the material from the different programs that will help in combining them to build the units for every student. You will assure that children have the ability and skill to learn at their own pace. It will ensure that you can easily keep track of the set records and help children learn math fast.


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Employee Spotlight Kaviraj Singh Jon
March 12, 2021
Posted by Tantus Solutions Group
Kavi has over 3 years of experience as a Data Analyst and Analytical Consultant working in the Banking and Financial Services industry. He has worked on the development and implementation of AML Solutions and IFRS9. He has a great passion for Data Science because of his interest in Mathematics and Statistics. Meanwhile, in his spare time, Kavi would love to spend time with friends, playing football and cricket.


Many analysts in the fields of audit, forensic and accounting sciences have heard of Benford’s Law. I have not seen many practical cases where this law has been applied to identify fraud or manipulation of data but, in the few cases where I did, it almost always delivered interesting but to some extent confusing results.
I believe the confusion stems from a lack of understanding the mathematical basis of Benford’s Law. As an auditor I grasp the concept and how it can assist in the identification of data manipulation but I lack the mathematical understanding to effectively interpret the results of a Benford analysis. In fact, from a review of articles written by mathematicians, it seems even they believe the exact reason why certain natural numbers follow this law and others not, still require further study and description.
But let us start at the beginning. According to an article published in the American Scientist (T.P Hill – 1998), the astronomer and mathematician Simon Newcomb published a two-page article in the American Journal of Mathematics (1881) reporting a mathematical phenomenon. Newcomb described his observation that books of logarithms in the library were quite dirty at the beginning and progressively cleaner throughout. From this he inferred that fellow scientists using the logarithm tables were looking up numbers starting with 1 more often than numbers starting with 2, numbers with first digit 2 more often than 3, and so on.
After a short heuristic argument, Newcomb concluded that the probability that a number has a particular first significant digit (that is, first non-zero digit) d can be calculated as follows:
Prob (first significant digit= d) = log10 (1 + 1/d), d = 1,2, … ,9
In particular, his conjecture was that the first digit is 1 about 30 percent of the time and 9 only about 4.6 percent of the time. That the digits are not equally likely to appear comes as something of a surprise, but to claim an exact law describing their distribution is indeed striking.
Newcomb’s article went unnoticed, and 57 years later General Electric physicist Frank Benford, apparently unaware of Newcomb’s paper, made exactly the same observation about logarithm books and also concluded the same logarithm law. Evidence indicates that Benford spent several years gathering data, and the table he published in 1938 in the Proceedings of the American Philosophical Society was based on 20,229 observations from such diverse data sets as areas of rivers, American League baseball statistics, atomic weights of elements and numbers appearing in Reader’s Digest articles. Benford’s article received much wider attention and led to the name “Benford’s Law”
I first got interested in Benford’s Law at a conference in Germany where I met accounting professor Mark Ngrini, originally from South Africa but based in the United States at the time. He did his Phd thesis on Benford’s Law and delivered a very convincing presentation of the practical value of the law in accounting data. Since then I have always looked for opportunities to use Benford’s law in data analysis. I use Arbutus software that provides probably the most advanced Benford functionality.


Why all the fuss about multiplication testing?
Some teachers have been up in arms about the new Year 4 multiplication tests and for good reasons. What’s the point of telling teachers what they already know about their own classes?
But there are other considerations.
Here’s the other thing: in Year 4, children are expected to learn “multiplication and division facts for multiplication tables up to 12×12”.
But wait, the content of the new statutory tests is all multiplication – what about poor old division? Why isn’t she invited to the party?
Division has been shunned, told it’s not welcome and left out of the equation.
This means that by not including the all-important related division facts some schools will teach to the test and not bother with them but just focus on ‘tables’.
Knowledge of division as the inverse operation of multiplication is as important as understanding the relationship between addition and subtraction. Without this crucial understanding children have to learn the division facts independently of the multiplication facts and that is a very tall order that many find hard to do.
I’m not convinced that the DfE understand the crucial distinction between the two types of multiplication table, e.g. the ‘table of 4s’ and the ‘4 times table’.
The table of 4s collects together sets that have a cardinality of 4, whereas the 3 times table is all about the total number in four sets. Lots of maths textbooks fail to make the difference between the two equivalent ideas and quite often confuse the two.
The multiplication tests are already starting to look like a bit of a mess.
There is a solution to all this: always read the question.
What do I mean? Let me explain. I’ll ask you a few questions:
4 x 5 =
4 x 6 =
4 x 12 =
This could come up in the new tests.
Now let’s mark the test as taken by Neil,
4 x 5 = 12
4 x 6 = 13
4 x 12 = 19
How did Neil do? Well, if the DfE were marking then he’d score zero.
If I were marking then I’d give him 3 out of 3.
You see, this is a ‘real-life’ incident.
Neil was a Y6 boy I taught some 25 years ago and the only maths ‘genius’ I have ever taught in that time. I’ve taught plenty of talented mathematicians but not of Neil’s calibre – he is one of life’s savants and a huge fan of Lewis Carroll.
When I told Neil he’d got 0/3 he challenged me and I ended up revising my marks in light of his comments.
He told me that the questions I asked didn’t specify which base to answer in so he answered according to the following logic:
4 x 5 = 12 on base 18
4 x 6 = 13 on base 21
4 x 12 = 19 on base 39
Come on, this is just base-ic maths.
Wow! Point taken!
And there’s the solution….if the new tests expect children to answer on a base 10 understanding then they’d better make that explicit at the top of the test: please answer in base 10.
If they don’t then they cannot make assumptions that children don’t know their times tables because they might be thinking (as Neil was) in completely different ways.
It’s all about the base….and the cost. The tests are going to cost around £5.2 million – imagine how else that money could be spent instead.


Arithmetic Tutors in Eden Prairie, MN
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Tim has been a tutor and mentor to our middle school son for nearly a year now. Our son’s attitude, work ethic and grades have improved beyond our expectations. Tim has a way of connecting and relating that facilitates a great learning experience. I thank our lucky stars we found him and highly recommend him.
Mr. Freeburg was my sophomore geometry teacher. Before this my toughest classes have always been math- I just couldn’t grasp certain subjects that are essential to moving forwards into Algebra 2 and Pre-Calc. When the SAT rolled around, I did terribly on the math section and I decided to give Mr. Freeburg a call. He helped me to finally grasp those concepts that had given me trouble and I found my confidence in math growing alongside my understanding. My grade in Physics even shot up and I finished the year off with A’s in both subjects. Not to mention my SAT score improved ? . I greatly appreciate everything that Mr. Freeburg has done for me and I highly recommend him to anyone who needs help!
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We all love magic and tricks. So today we are going to create a little bit of magic together by performing some simple actions. This test is a special combination of numbers and colors. It’s not difficult! The only thing that you need to remember is the color that belongs to your number.
We at Bright Side prepared a test for you. Take a seat and have some rest from your daily routine with our magic test that can read your mind.
For example, we picked 15.
There is number 15 in this table, so we noted the yellow color.
There is also the number 15 here. We noted the orange color.
Yes! Again we found 15! The purple color is noted.
And, again the red color has been noted.
Here we didn’t find our number 15.
So we had yellow — 1, orange −2, purple — 4, and red — 8. Let’s add up the numbers.
1 + 2 + 4 + 8 = 15!
That is our number that we picked at the beginning!
How did the magic test work for you? Please, share this article with your friends and tell us about your results below.


Joshua Lee Padgett
University of Arkansas
Department of Mathematical Sciences
Fayetteville, Arkansas 72701
Office: SCEN 0225
I am currently an Assistant Professor in the Department of Mathematical Sciences at the University of Arkansas, an Affiliated Faculty member at the Center for Astrophysics, Space Physics, and Engineering Research, and a Research Partner/Honorary Adjunct Faculty in the Department of Mathematics and Statistics at Texas Tech University. I received my Ph.D. in Mathematics from Baylor University in August 2017 under the advisement of Qin "Tim" Sheng and completed my postdoctoral training in the Department of Mathematics and Statistics at Texas Tech University. Before that I received my B.S. in Mathematics from Gardner-Webb University, where I also was a member of the Track and Field team (I competed in the javelin and hammer throw). My undergraduate studies also included cellular biology, which led to several semesters of research into various aspects of cancer metabolism (in particular, the Warburg effect).
Here is a copy of my most recent Curriculum Vitae [updated January 2022].
Joshua Lee Padgett's Google Scholar profile.
Joshua Lee Padgett's ResearchGate profile.
A link with the information regarding the XVIII Red Raider Minisymposium I am co-organizing, may be found here.
A link with the information regarding the JMM Special Session I co-organized in 2020, may be found here.
With the support of Erica Graham, Candice Price, and Shelby Wilson, an amazing (former) colleague of mine, Raegan Higgins, maintains the website Mathematically Gifted and Black (which can be found by following the link). Please visit this webpage in order to learn more about the issue that black scholars face in academia and mathematics. The page also contains details regarding how one can support and nominate black scholars.
Note: I am currently looking for graduate students. Interested students should email me or fill out the form here. While I am interested in a wide range of topics (and, in general, am willing to learn about others), potential research students should take a look at my research descriptions below or my recent publications to have a better idea of the topics I am likely to consider or propose.
Below you can find some (what I believe to be) important recent updates regarding myself or my research.
I am co-PI on an NSF collaborative research grant with an interdisciplinary research team composed of myself and members from Baylor University and the University of Deleware. This exciting new project explores a novel method for studying turbulent behavior in dusty plasmas while also exploring the use of non-local operators. This project combines theoretical mathematical analysis, computational mathematics, high-performance computing, mathematical modeling, theoretical physics, and experimental physics.Learn More
I am co-PI on an NSF conference grant with an interdisciplinary team composed of myself and members from Texas Tech University's Departments of Mathematics and Statistics and Biological Sciences. This grant allows us to continue the long tradition of hosting a Red Raider Minisymposium and also allows for the provision of travel funds to interested early-career researchers. This year's focus in on the inclusion of heterogeneity in biologically relevant mathematical models.Learn More
My paper studying novel methods for approximating certain high-dimensional partial differential (PDEs) equations has been recently submitted for publication and may be found on arXiv (see link below). This paper provides a novel method for approximating high-dimensional PDEs while avoiding suffering from the so-called curse of dimensionality (CoD). This result provides the foundation needed to prove that deep neural networks possess the expressive power to provide CoD-free approximate solutions to high-dimensional PDEs (an important result which validates the use of deep learning techniques when approximating PDEs).
M. Hutzenthaler, A. Jentzen, B. Kuckuck, and J. L. Padgett, Strong Lp-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
arXiv version: arXiv:2110.08297
My paper studying properties of objects which generalize classical strongly continuous semigroups has been accepted for publication in the recent book From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory. These generalizations include solution operators arising from certain singular integral equations, as well as those from the classical setting. Moreover, the analysis is carried out in abstract Banach spaces which guarantees that the results hold for numerical approximations, as well. This article appears in a book which has been published in honor of Lance Littlejohn's 70th birthday.
T. F. Jones, J. L. Padgett, and Q. Sheng, Intrinsic properties of strongly continuous fractional semigroups in normed vector spaces
arXiv version: arXiv:21012.11092
Official book version: https://doi.org/10.1007/978-3-030-75425-9_14
My paper studying discrete non-local operators and their explicit series representations has been accepted for publication in the Journal of Mathematical Analysis and Applications. This exciting publication is the result of a long standing collaboration across several fields with the ultimate goal of improving our ability to model realistic physical phenomena with a high level of accuracy.
T. F. Jones, E. G. Kostadinova, J. L. Padgett, and Q. Sheng, A series representation of the discrete fractional Laplace operator of arbitrary order
arXiv version: arXiv:2101.03629
Official JMAA version: https://doi.org/10.1016/j.jmaa.2021.125323
My primary research interests lie in the areas of numerical analysis, applied mathematics, and computational mathematics. I am particularly interested in problems arising in biology and physics whichexhibit nonstandard computational challenges—such as problems with singular, nonlocal, or stochastic influences. My work has employed a variety of mathematical techniques, with a particular focus on combining computational techniques with those from operator theory, spectral theory, and Lie group theory. My most significant contributions have been the development of the abstract numerical analysis for such problems, which allows for the obtained results to have a wider range of applications. Such efforts allow for the construction of qualitatively and quantitatively superior computational algorithms. Moreover, it allows for the results to be applied to numerous physical problems of interest, such as those arising in mathematical biology, combustion theory, and plasma physics.
My recent research efforts have been focused on machine learning and deep artificial networks, with a particular emphasis on how these tools may be employed to efficiently approximate high-dimensional partial differential equations. This area is of great importance in the scientific community and garners interest from a wide array of academic disciplines. My current focus is on the theoretical and mathematical considerations of deep learning; i.e., my focus is on proving theorems regarding deep artificial networks. The mathematics for this field is still in its infancy, and as such, there are a great deal of exciting problems to be pursued in this direction.
Topics of interest:
The following is a list of grant proposals that have either been funded.
Below are some pictures of my two dogs: Murphy (an Australian Shepherd) and Memphis (a Goldendoodle). Murphy is now approximately five years old and Memphis is now approximately two and a half years old.


The videos below are lessons to help with various “challenging” proofs in PACE 1110, the 2nd in the ACE Geometry course. I try to clarify important concepts, pronounce terminology, explain why certain steps must be included, and help students “see” the solution to some of the puzzling problems. I trust these videos will be helpful. If there are other trouble spots in this PACE, feel free to use the “contact” link above to let me know.
NOTE: I made all these videos using a program and wireless mic on my notebook computer and used the built-in webcam — sorry the video quality isn’t the greatest, but I think you can still see and hear what you need to get help
Mr. Chris Walters, a math teacher at an ACE School, is making videos for this course and uploading them to his channel. You may find them to be helpful and he gave permission for me to share this link with you.


Pick a random real number between zero and one. The number you choose, with probability one, will contain an encoding of all of the books in the US Library of Congress. This sounds absurd, but real numbers require infinite precision and every time you deal with the infinite, things get absurd. Infinities, including the infinite number of digits to express almost every real number, don’t exist. Curiously then, real numbers are not real.
How do we choose a random number between zero and one? The easiest way to explain is using binary decimals. The binary number 0.1000… with zeros forever denotes the number ½ or, in base 10 notation, 0.5. The binary decimal 0.01000… with zeros forever is the number ¼. Keep repeating the 01 pattern in 0.101010… forever gives 2/3. In base 10, this is the same as sixes forever, or 0.666666…
Decimal expressions for real numbers like pi and the square root of two go on forever without repeating.
So here’s how a real number is chosen randomly on the interval 0 to 1. A repeated coin flip is used. Flip a coin. If the result is heads, it’s a one. A tails is a zero.
Here’s the process for generating a random number between 0 and 1:
A coin is flipped and we see a heads. The first bit in the randomly chosen random number is therefore one, or 0.1. Flip the coin again and get another heads and the new number is 0.11. Then a tails for 0.110. Another heads for 0.1101, etc. To generate a real number, the flipping goes on forever so that we get the infinite precision required for a real number.1 Random numbers generated in programs like MatLab or Excel have a finite number of digits and therefore do not have the required infinite precision to be a real randomly chosen number.
Assume a random number is generated by coin flipping. Here are some questions: If we go long enough, will there ever be a sequence of ten 1’s in a row? Obviously yes. How about a hundred one’s in a row? We might have to wait a while but, eventually, yes. At this point, things start to get weird.
Text can be coded as a string of zeros and ones. The standard ASCII assigns a binary string to every English letter, number and punctuation symbol used in this article. You can therefore encode this article into a binary sequence of one’s and zeros. Do you think that, in generating a binary random number, you would eventually run across a bit sequence identical to the ASCII encoding of this article? It will take a long time, but eventually yes. Also, at some time, the entire binary encoding of the King James Bible will be seen. In fact you will eventually generate the binary code for all of the books in the United States Library of Congress.
This might sound unbelievable but remember there is an infinite resource available when flipping a coin an infinite number of times. That’s a lot of coin flips.
Here is what is even more remarkable. If I randomly generate a real number and you randomly generate a random number by coin flipping, both of our numbers will contain the US Library of Congress somewhere. In fact, almost all reals generated by coin flipping will contain the library.1
There are obvious examples where this is not the case, like 0.1000… or 0.101010… . But the chances of generating such numbers by random coin flipping are improbable to the point of being dismissed. Mathematicians would say the entire US Library of Congress and everything else is encoded in every randomly chosen number between 0 and 1 with probability one. “Everything else” includes all your passwords and the binary version of all computer programs ever written. This is pretty astonishing for a single randomly chosen real number.
The absurdity of this conclusion is evidence that real numbers don’t exist because the infinite doesn’t exist and almost every real number requires an infinite number of digits. Real numbers of infinite precision do not occur in reality. In the real world, if you say a voltmeter reads pi volts, you mean 3.14159 volts. Infinite precision is not possible in reality.
There’s a lot of information contained in a single generated random number between 0 and 1. A lot of time will be spent looking for where the encoding of the Bible starts – but it’s there. So are all versions of the Bible with one or more flaws. Instead of “In the beginning” we might read “In the beginninx” followed by the rest of the Bible.
For one number, such claims seem ridiculous in reality. Most real numbers require infinite precision and the noted mathematician David Hilbert (1862–1943) noted “the infinite is nowhere to be found in reality.”2 It only exists as an idea in our minds.
Therefore, curiously, most real numbers are not real.
1 I am not the original author of the idea of generating a random real number but, unfortunately, I do not have a reference at present for the original.
2 David Hilbert, “On the infinite” in Benacerraf, Paul, and Hilary Putnam, eds. Philosophy of mathematics: Selected readings. Cambridge University Press, 1984.
You may also enjoy: Can we add new numbers to mathematics? We can work with hyperreal numbers using conventional methods. It could start in high school. (Jonathan Bartlett)


Mathematics is a core subject studied by all pupils to GCSE level. Our aims are for all students to become confident and proficient in numeracy and functional Mathematics skills and to develop enthusiasm for furthering their knowledge and understanding of the subject in order to achieve their full potential at GCSE.
Key Stage 3 (Year 7 – to February of Year 9)
Initially Year 7 students are taught in three mixed ability sets with one small support set. During the first three weeks they have a circus of Mathematical tasks with different members of the department in order to observe and asses their ability. Pupils are allocated sets commensurate with their ability. Sets are reviewed regularly and parents are informed of any changes.
The Key Stage 3 programme of study follows the requirements of the four main areas of the National Curriculum
• Using and applying mathematics
• Number and Algebra
• Shape, space and measures
• Handling Data
Key Stage 4 - GCSE Mathematics
Students continue to be taught in sets and follow the Edexcel modular GCSE, specification 2MB01.
Pupils sit three module examinations:
Availability of Unit examination for current Year 11 pupils (September 2012 – June 2013)
Unit 1 November 2012, March 2013, June 2013
Unit 2 November 2012, March 2013, June 2013
Unit 3 June 2013.
Current Year 10 (September 2012 – June 2013)
Pupils study the Edexcel GCSE specification 2MB01. However the three unit examinations will only be available in June 2014. As all GCSE examination must be taken at the end of the course with effect from June 2014.