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23. As a project manager, you analyze the amount of time team members will need to spend on a project and the likely ongoing project expenses. What step of the cost benefit analysis are you completing?
26. You expect that a project will bring in $20,000 USD in revenue per year. You estimate it will cost $8,000 up front. You also estimate costs of $150 per month for the first 12 months, which equals $1,800 per year. Using the formula (G-C) ÷ C = ROI, how would you calculate the project’s return on investment (ROI) after the first 12 months?


What makes any day better? Brownies! Add them to a Friday AND a math lesson and you will have some happy students ready to learn. Mrs. Jeannette Bellamy did just that last Friday for her math classes. Ask our KSES 5th grade students if they would like 1/10 of a pan of brownies or 1/100 and I bet they will quickly tell you. What a fun learning experience for our 5th graders and a delicious way to see fractions.


Written primarily for advanced undergraduate and graduate students who are pursuing research careers in economics, Intermediate Statistics and Econometrics offers a broad perspective, bringing together a great deal of diverse material. Its comparative approach, emphasis on regression and prediction, and numerous exercises and references provide a solid foundation for subsequent courses in econometrics and will prove a valuable resource to many nonspecialists who want to update their quantitative skills.
The introduction closes with an example of a real-world data set -- the Challenger space shuttle disaster -- that motivates much of the texts theoretical discussion. The ten chapters that follow cover basic concepts, special distributions, distributions of functions of random variables, sampling theory, estimation, hypothesis testing, prediction, and the linear regression model. Appendixes contain a review of matrix algebra, computation, and statistical tables. Обо всём этом и не только в книге Intermediate Statistics and Econometrics: A Comparative Approach (Dale J. Poirier) скачать Intermediate Statistics and Econometrics: A Comparative Approach, Dale J. Poirier


Solid geometric forms including sphere, ovoid, ellipsoid, cylinder, cube, triangular prism, rectangular prism, cone, square pyramid, triangular pyramid.
A set of wooden or plastic tablets that have the same base or side as the rectilinear solids and the same “shadow” as the curvilinear solids.
A cloth or scarf.
- Take three contrasting geometric solids to a rug.
- With the child watching, explore the shape of the solid in your hands and see the different ways it sits, stands, or rolls on the rug.
- Let the child hold the solid and explore it. Tell the child the name of the solid as he experiences it.
- Repeat this process for all three solids that were selected in the basket.
- Perform a three-period lesson with the solids so that the child learns the names of the solids.
- Repeating the same steps above over the next few days, add two or three more solids per day.
Exercise 1, Presentation
The child doesn’t work independently yet,
Exercise 2, Stereognostic Game
Place many solids that the child knows in the basket. Bring the basket and the scarf to the rug. Cover the basket with the scarf. Say the name of a solid and ask the child to find it without looking, just using his hands. Later, you can put just one solid on a rug underneath the scarf and ask the child to identify what the solid is. You can also place multiple solids under the scarf and play the same game.
Exercise 3, Cards
Bring the geometric solids cards to the rug. Invite the child to explore the material and ask him to see how the various solids match different cards.
Names of the solids: sphere, ovoid, ellipsoid, cylinder, cube, triangular prism, rectangular prism, cone, square pyramid, triangular pyramid.
Control of Error
Refinement of the stereognostic sense.
Raising the child’s awareness of geometric solid shapes in both architecture and the world at large.
3 1/2 +
Some solids have an interesting ability of rotation such as the cone and cylinder, they can roll or spin in ways that a sphere wouldn’t.
Shadows of round – but not sphere shaped – solids such as the ellipsoid and ovoid do in fact match the circle card.


chridd » xkcdsw » Real Edgelord
August 23, 2018
Original: Edgelord (2036)
View in: original forum thread, direct image link/enlarge
Two people are talking to each other. The right one has a white hat and has dots at the ends of his arms and legs, where his legs join his body, at his neck, and various places on his hat (wherever two lines meet, and at the end of any line).
Hatless: So, I hear you're a real edgelord.
So, I hear you're a real edgelord.
White Hat Guy: No!
Caption: How to annoy a graph theory Ph. D.
How to annoy a graph theory Ph. D.
Title text: He's more of a vertexlord
← Previous | Next →| Comments


Thanks to SaM for this:
I don’t know if anybody cares, but I like Binary numbers so I thought I’d
tell everybody what the credit card receipt from CYN said. So here goes,
there’s a couple of ways to interpret them.
The original Binary number: 1110 0100 1001 1101 1001
The converted Decimal number: 936409
Binary number broken into 4-bit words then converted: 12 4 9 13 9
Those numbers converted to equal the alphabet they represent: LDIMI
Don’t know if that’s worth anything, or if anybody cares. Just something
fun I decided to do.


How to Run a Regression Using EViews - YouTube.
Working with Equations. Views of an Equation. Procedures of an Equation. Residuals from an Equation. Storing and Retrieving an Equation. Using Estimated Coefficients. The follow description outlines common views and procedures that are available for an estimated equation. Specialized estimators may support only a subset of these views and procs, while perhaps offering others. Views of an.
Estimation Part B: Estimation Output and Multiple Regression.
Again, this write-up is in response to requests received from readers on (1) what some specific figures in a regression output are and (2) how to interpret the results. Let me state here that regardless of the analytical software whether Stata, EViews, SPSS, R, Python, Excel etc. what you obtain in a regression output is common to all analytical packages (howbeit with slight changes).
Derivation of long run equation. Model One. EVIEWS - YouTube.
How to Run a Regression on Eviews Regression Analysis is quickly becoming more important in all economist’s playbooks. Econometric’s popularity has soared since statistical analysis and regression analysis has become more precise, errors have been rectified and with the push for computer software and applications to ease the once grueling task.
How to write program for series of regressions - EViews.com.
Step-By-Step Guide on Interpreting your Eviews Regression Output When you perform a basic multiple regression analysis on Eviews, your output will generally be of the form of the frame below: Output above is divided into three parts: 1) General Information 2) Relative Statistics 3) Global Statistics The first part contains general information about the nature of the data, method(s) of analysis.
Regression Model. Model Four. EVIEWS - YouTube.
Ordinary least-squares (OLS) regression is a generalized linear modelling technique that may be used to model a single response variable which has been recorded on at least an interval scale. The technique may be applied to single or multiple explanatory variables and also categorical explanatory variables that have been appropriately coded. Key Features At a very basic level, the relationship.
EViews Tutorial: Episode 2 - Transforming Data - YouTube.
A linear regression equation models the general line of the data to show the relationship between the x and y variables. Many points of the actual data will not be on the line. Outliers are points that are very far away from the general data and are typically ignored when calculating the linear regression equation. It is possible to find the linear regression equation by drawing a best-fit.
CrunchEconometrix: Interpreting Regression Output from EViews.
Before you begin with regression analysis, you need to identify the population regression function (PRF). The PRF defines reality (or your perception of it) as it relates to your topic of interest. To identify it, you need to determine your dependent and independent variables (and how they’ll be measured) as well as the mathematical function describing how the variables are related.
How to Run a Regression on Eviews - Economics Wiki.
Understanding the Results of an Analysis. Descriptive Statistics for Variables. NLREG prints a variety of statistics at the end of each analysis. For each variable, NLREG lists the minimum value, the maximum value, the mean value, and the standard deviation. You should confirm that these values are within the ranges you expect. Parameter Estimates. For each parameter, NLREG displays the.
Step-By-Step Guide on Interpreting your Eviews Regression.
Now, it is time to learn how to write a regression equation using spss. We have SPSS regression tutorials that provide insights on the step-by-step procedure of performing linear regression using the SPSS Data Editor Verison 12.0. Through this version, identify the writing regression equation. You can also check manova spss output interpretation or how to write interpretation to get more info.
REGRESSION IN EVIEWS - University of Manchester.
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.
EViews: AutoRegressive Distributed Lag (ARDL) Estimation.
EViews provides three autoregressive spectral methods: OLS, OLS detrending, and GLS detrending, corresponding to difference choices for the data. The following table summarizes the auxiliary equation estimated by the various AR spectral density estimators: where are the coefficient estimates from the regression defined in (30.9).
A Short Introduction to Eviews - University of Pennsylvania.
Example Write-up Correlation and multiple regression analyses were conducted to examine the relationship between first year graduate GPA and various potential predictors. Table 1 summarizes the descriptive statistics and analysis results. As can be seen each of the GRE scores is positively and significantly correlated with the criterion, indicating that those with higher scores on these.


There are days when homeschooling goes just according to schedule. We do Bible, then Math, then Language, then a little Reading and Writing, and finally Science and/or Social Studies. Then there are days like today. :) I actually wonder if days like today turn out even better. I think a lot of times Allie gets more real learning out of these days than the scheduled ones.
Allie decided months ago that she wanted to own a yogurt shop when she grows up. This morning at breakfast, she asked if she could use the graph paper to map out the yogurt shop, showing the tables, chairs, kitchen, bathrooms, etc. I glanced at the clock and realized we had plenty of time and not too much planned today, so I said of course. Two hours later, Allie had the yogurt shop drawn up "to scale", a full menu complete with drawings, and a business plan complete with job descriptions. I would say she learned a lot during that project. :) My little entrepreneur is always trying to set up a shop and sell something outside. She's done everything from lemonade for 25 cents to jumping rope for a dime to whoever will stop and wants entertaining. :) She will be a wonderful business owner one of these days.
After that she spent an hour or so looking over the ukelele books Grannie gave her and teaching herself how to play the ukelele beautifully. She also pretended to have a picnic with Maggie and taught Penelope how to dance.
We did throw Math and Bible in there for good measure, but I don't think today was a lost day at all. :)


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An Introduction to Online Casino Gambling
There are numerous casino games available in land-based casinos. Included in these are bingo, craps, blackjack, poker, roulette, video poker, etc. Some of these games are simple versions of existing games, while some are innovative spin-offs of well-known games. While playing these games in land-based casinos you may wonder which game can be your favorite. To find out, you need to look at the rules, play a variation of the overall game, and decide which game you liked probably the most.
There are three main categories of casino games: table games, gaming machines, and random number cards. The slot machines are the hottest gambling games obtainable in land-based casinos. Slots are made to offer instant payoff in addition to short betting periods. They are often within every casino, from the high rollers in the big city to the people on the strip.
In standard deviation, the casino managers simulate a random number generator. They utilize this generator to look for the expected number of rounds played (the standard deviation is used to look for the selection of expected losses and wins). The worthiness of a typical deviation is calculated using the normal curve (a mathematical equation that describes the probability distribution of outcomes over a range of values, with parameters x and y representing the range of possible outcomes for a random number generator).
When casino games are played on machines that are random and cannot be pre-figured, the casino will base its wins and losses on randomly drawn hands. That is referred to as the ‘house edge’. Standard deviation allows managers to compute the casino’s ‘trading cost’, which is the additional profit a machine might generate if it were successfully run even without any gaming intervention. Standard deviation is often used in spread betting strategies. It is also useful for analyzing the performance of trading systems.
The casino management uses a special sort of mathematical model called the’Monte Carlo technique’ to identify the house advantage. This is attained by fitting a random variable, namely the skill of the players, into a basic probability distribution function. By varying the random variable, the home advantage can be estimated. That is a useful tool in analyzing casino games and the way they are prone to outside influences like skill of the players, house edge, etc.
One of the widely played casino games is roulette, which includes near a million players worldwide. It really is closely related to Blackjack, but roulette has a amount of differences that set it apart. It is played on several tables, that allows the players to adjust the chances according to the number of people at each table. Unlike Blackjack, roulette does not use the deck, and so, there is no known technique for beating the system. The house advantage for roulette is very large; approximately seventy percent.
Another hottest casino games are baccarat, keno, 카지노 쿠폰 poker, slots, and craps. Baccarat is just about the most difficult game to understand, because it combines strategy with chance, despite having luck. It is closely linked to Blackjack, but unlike Blackjack, it uses even smaller tables. Slots are played on one or more revolving machines which allow the players to change between chips until they hit on the winning combination. Craps is played on a single table with an individual die. The players must beat all the other players within a set time period limit to win.
There are many online casinos to choose from when one really wants to play casino games online. Some offer progressive jackpots which will be worth more than the player’s skills improve, while some offer bonuses to encourage players to keep playing. In order to maximize your enjoyment of the gambling experience, remember that you are in control from start to finish. If you make decisions that bring about you becoming frustrated, you can always return back and try again later. But remember which you have already lost if you do not learn from your mistakes.


You are currently browsing the tag archive for the ‘teaching maths’ tag.
IB Maths and GCSE Maths Resources from British International School Phuket. Theory of Knowledge (ToK). Maths explorations and investigations. Real life maths. Maths careers. Maths videos. Maths puzzles and Maths lesson resources.
British International School Phuket
Welcome to the British International School Phuket’s maths website. My name is Andrew Chambers and I am currently working at BISP. I am running my site as the school’s maths resources website for both our students and students around the world.
We are a British international school located on the tropical island of Phuket in Southern Thailand. We offer a number of scholarships each year, catering for a number of national and international standard sports stars as well as for academic excellence. You can find out more about our school here.
There are a huge amount of resources to explore – especially for students doing their IAs and for students looking for revision videos. You may also like to try our school code breaking site – where you can compete with over 10,000 students from around the world who have made it onto our school leaderboard.
Pyramid schemes are a very old con – but whilst illegal, still exist in various forms. Understanding the maths behind them therefore is a good way to avoid losing your savings!
The most basic version of the fraud starts with an individual making the following proposition, “pay me $1000 to join the club, all you then need to do is recruit 6 more people to the club (paying $1000 each) and you will have made a $5000 profit.”
There are lots of variations – and now that most people are aware of pyramid selling, now normally revolve around multi-level-marketing (MLM). These are often still pyramid schemes, but encourage participants to believe it is a genuine business by actually having a sales product which members have to sell. However the main focus of the business is still the same – taking money off people who then make their money back after having signed up a set number of new recruits.
The following graphic from Consumer Fraud Reporting is a clear mathematical demonstration why these frauds only end up enriching those at the top of the pyramid:
You can see that if the requirement was to recruit 8 new members, that by the 9th level you would need to have 1 billion people already signed up. Even with the need to recruit just 4 new members you still have rapid exponential growth which very quickly means you will run out of new potential members. For pyramid schemes it is only those in the first 3-4 levels (the white cells) that stand any real chance of making money – and these levels are usually filled by those in on the scam.
Ponzi schemes (like that run by Bernie Madoff) use a similar method. A conman takes money from investors promising (say) 10% annual returns. Lots of investors sign up. The conman then is able to use the lump sum investments to pay the 10% annual returns. This scam can last for years, with people thinking that they are getting a good rate of return, only to find out eventually that actually their lump sum investment has gone.
This is a good topic to look at with graphs (plotting exponential growth), interest rates, or exponential sequences – and shows why understanding maths is an important financial skill.
If you like this topic you might also like:
Benford’s Law – Using Maths to Catch Fraudsters – the surprising mathematical law that helps catch criminals.
Amanda Knox and Bad Maths in Courts – when misunderstanding mathematics can have huge consequences .
NASA, Aliens and Binary Codes from the Star
The Drake Equation was intended by astronomer Frank Drake to spark a dialogue about the odds of intelligent life on other planets. He was one of the founding members of SETI – the Search for Extra Terrestrial Intelligence – which has spent the past 50 years scanning the stars looking for signals that could be messages from other civilisations.
In the following video, Carl Sagan explains about the Drake Equation:
N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);
R* = the average number of star formation per year in our galaxy
fp = the fraction of those stars that have planets
ne = the average number of planets that can potentially support life per star that has planets
fl = the fraction of planets that could support life that actually develop life at some point
fi = the fraction of planets with life that actually go on to develop intelligent life (civilizations)
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space
The desire to encode and decode messages is a very important branch of mathematics – with direct application to all digital communications – from mobile phones to TVs and the internet.
All data content can be encoded using binary strings. A very simple code could be to have 1 signify “black” and 0 to signify “white” – and then this could then be used to send a picture. Data strings can be sent which are the product of 2 primes – so that the recipient can know the dimensions of the rectangle in which to fill in the colours.
If this sounds complicated, an example from the excellent Maths Illuminated handout on codes:
If this mystery message was received from space, how could we interpret it? Well, we would start by noticing that it is 77 digits long – which is the product of 2 prime numbers, 7 and 11. Prime numbers are universal and so we would expect any advanced civilisation to know about their properties. This gives us either a 7×11 or 11×7 rectangular grid to fill in. By trying both possibilities we see that an 11×7 grid gives the message below.
More examples can be downloaded from the Maths Illuminated section on Primes (go to the facilitator pdf).
A puzzle to try:
“If the following message was received from outer space, what would we conjecture that the aliens sending it looked like?”
0011000 0011000 1111111 1011001 0011001 0111100 0100100 0100100 0100100 1100110
Hint: also 77 digits long.
This is an excellent example of the universality of mathematics in communicating across all languages and indeed species. Prime strings and binary represent an excellent means of communicating data that all advanced civilisations would easily understand.
Answer in white text below (highlight to read)
Arrange the code into a rectangular array – ie a 11 rows by 7 columns rectangle. The first 7 numbers represent the 7 boxes in the first row etc. A 0 represents white and 1 represents black. Filling in the boxes and we end up with an alien with 2 arms and 2 legs – though with one arm longer than the other!
If you enjoyed this post you may also like:
Cracking Codes Lesson – a double period lesson on using and breaking codes
Cracking ISBN and Credit Card Codes– the mathematics behind ISBN codes and credit card codes
A maths song sung by current flavour of the month One Direction – follow the lyrics to arrive at the total.
Some of the best maths songs are by Learning Upgrade, such as, videos on circle formulae, fractions, exponents, the quadratic formula and the one below, “Mean, Median and Mode”:
Some other good maths songs:
Westerville South High School in Ohio have made some great rap-based maths songs such as the trigonmentry song, Getting Triggy With It
The Calculus Rhapsody is a fantastic take on Queen’s famous song – and good for IB SL and HL students.
And James Blunt’s Triangle is good for a KS3 shape and space introduction.
Does it Pay to be Nice? Game Theory and Evolution
Golden Balls, hosted by Jasper Carrot, is based on a version of the Prisoner’s Dilemma. For added interest, try and predict what the 2 contestants are going to do. Any psychological cues to pick up on?
Game theory is an interesting branch of mathematics with links across a large number of disciplines – from politics to economics to biology and psychology. The most well known example is that of the Prisoner’s Dilemma. (Illustrated below). Two prisoners are taken into custody and held in separate rooms. During interrogation they are told that if they testify to everything (ie betray their partner) then they will go free and their partner will get 10 years. However, if they both testify they will both get 5 years, and if they both remain silent then they will both get 6 months in jail.
So, what is the optimum strategy for prisoner A? In this version he should testify – because whichever strategy his partner chooses this gives prisoner A the best possible outcome. Looking at it in reverse, if prisoner B testifies, then prisoner A would have been best testifying (gets 5 years rather than 10). If prisoner B remains silent, then prisoner A would have been best testifying (goes free rather than 6 months).
This brings in an interesting moral dilemma – ie. even if the prisoner and his partner are innocent they are is placed in a situation where it is in his best interest to testify against their partner – thus increasing the likelihood of an innocent man being sent to jail. This situation represents a form of plea bargaining – which is more common in America than Europe.
Part of the dilemma arises because if both men know that the optimum strategy is to testify, then they both end up with lengthy 5 year jail sentences. If only they can trust each other to be altruistic rather than selfish – and both remain silent, then they get away with only 6 months each. So does mathematics provide an amoral framework? i.e. in this case mathematically optimum strategies are not “nice,” but selfish.
Game theory became quite popular during the Cold War, as the matrix above represented the state of the nuclear stand-off. The threat of Mutually Assured Destruction (MAD) meant that neither the Americans or the Russians had any incentive to strike, because that would inevitably lead to a retaliatory strike – with catastrophic consequences. The above matrix uses negative infinity to represent the worst possible outcome, whilst both sides not striking leads to a positive pay off. Such a game has a very strong Nash Equilibrium – ie. there is no incentive to deviate from the non strike policy. Could the optimal maths strategy here be said to be responsible for saving the world?
Game theory can be extended to evolutionary biology – and is covered in Richard Dawkin’s The Selfish Gene in some detail. Basically whilst it is an optimum strategy to be selfish in a single round of the prisoner’s dilemma, any iterated games (ie repeated a number of times) actually tend towards a co-operative strategy. If someone is nasty to you on round one (ie by testifying) then you can punish them the next time. So with the threat of punishment, a mutually co-operative strategy is superior.
You can actually play the iterated Prisoner Dilemma game as an applet on the website Game Theory. Alternatively pairs within a class can play against each other.
An interesting extension is this applet, also on Game Theory, which models the evolution of 2 populations – residents and invaders. You can set different responses – and then see what happens to the respective populations. This is a good reflection of interactions in real life – where species can choose to live co-cooperatively, or to fight for the same resources.
The first stop for anyone interested in more information about Game Theory should be the Maths Illuminated website – which has an entire teacher unit on the subject – complete with different sections,a video and pdf documents. There’s also a great article on Plus Maths – Does it Pay to be Nice? all about this topic. There are a lot of different games which can be modeled using game theory – and many are listed here . These include the Stag Hunt, Hawk/ Dove and the Peace War game. Some of these have direct applicability to population dynamics, and to the geo-politics of war versus peace.
If you enjoyed this post you might also like:
Premier League Finances – Debt and Wages
This is a great article from the Guardian DataBlog analysing the finances for last season’s Premier League clubs. As the Guardian says, “More than two thirds of the Premier League’s record £2.4bn income in 2011-12 was paid out in wages, according to the most recently published accounts of all 20 clubs. The Guardian’s annual special report of Premier League clubs’ finances shows they spent £1.6bn on wages last season, most of it going to players.”
The first graph (above) shows the net debt levels for different clubs.
The second graph shows the total turnover:
The third graph shows wages as a proportion of turnover:
and the last one is particularly interesting – as it ranks clubs on their wage bills and their league position. This would be an interesting piece of data to test for the strength of correlation:
I’ve used an online scatter plot to calculate both the regression line and the correlation coefficient:
Which clearly shows a strong positive correlation. This would be an interesting exercise for both IGCSE or IB students (especially Maths Studies).
For even more data, a club by club full breakdown is also provided by the Guardian here. I have also made the data above into a word document to be used as a some A4 posters – and you can download that here: Premier League Debt
If you enjoyed this post you might also like:
This is inspired by a fantastic website – we use math – which has a massive amount of information about different careers using mathematics in a really well laid out format.
According to a comprehensive careers survey by Careers Cast – which looked at over 200 different jobs and ranked them for stress, pay, job stability and work environment – 6 of the top 10 jobs require or strongly prefer maths graduates or those with a good mathematical background.
Meanwhile, a 2009 Survey by the National Association of Colleges and Employers – which looked at American students’ job prospects after leaving university found that – “The top 15 highest-earning college degrees all have one thing in common — math skills.”
“Math is at the crux of who gets paid,” said Ed Koc, director of research at NACE. “If you have those skills, you are an extremely valuable asset.”
From the we use math site there is information (with interviews, maths required, job skills etc) on about 50 maths related jobs – as well as some pretty impressive statistics about the usefulness of mathematics:
These are the A-E jobs included on the site:
There is also an accompanying video with interviews from a large range of mathematicians talking about the jobs they have gone into:
Along with the equally brilliant Maths Careers website there should be enough ammunition to never be stumped by the perennial, “When are we ever going to use this in real life?”
I’ve used the data above to make a poster for display – it’s downloaded from here: Want to be an astronaut?
Michio Kaku – American Professor of Theoretical Physics and fantastic populariser of mathematics and physics takes us through a 40 minute journey on the importance of physics in explaining the universe:
Great stuff – well worth watching!
Graham’s Number – literally big enough to collapse your head into a black hole
Graham’s Number is a number so big that it would literally collapse your head into a black hole were you fully able to comprehend it. And that’s not hyperbole – the informational content of Graham’s Number is so astronomically large that it exceeds the maximum amount of entropy that could be stored in a brain sized piece of space – i.e. a black hole would form prior to fully processing all the data content. This is a great introduction to notation for really big numbers. Numberphile have produced a fantastic video on the topic:
Graham’s Number makes use of Kuth’s up arrow notation (explanation from wikipedia:)
In the series of hyper-operations we have
and so on.
Which clearly can lead to some absolutely huge numbers very quickly. Graham’s Number – which was arrived at mathematically as an upper bound for a problem relating to vertices on hypercubes is (explanation from Wikipedia)
where the number of arrows in each layer, starting at the top layer, is specified by the value of the next layer below it; that is,
and where a superscript on an up-arrow indicates how many arrows are there. In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G = g64 with g63 up-arrows between 3s.
So a number so big it can’t be fully processed by the human brain. This raises some interesting questions about maths and knowledge – Graham’s Number is an example of a number that exists but is beyond full human comprehension – it therefore is an example of a upper bound of human knowledge. Therefore will there always be things in the Universe which are beyond full human understanding? Or can mathematics provide a shortcut to knowledge that would otherwise be inaccessible?
If you enjoyed this post you might also like:
How Are Prime Numbers Distributed? Twin Primes Conjecture – a discussion about the amazing world of prime numbers.
Wau: The Most Amazing Number in the World? – a post which looks at the amazing properties of Wau
Plus Maths has a large number of great podcasts which look at maths ToK topics:
1) An interview with Max Tegmark (pictured above) about why he thinks that the universe is itself a mathematical structure.
2) An interview with physicists David Berman about how many dimensions exist.
3) A talk with cosmologist John Barrow about infinity.
4) A discussion with Roger Penrose about the puzzle of time.
And many more. Well worth a listen!
5) There’s also a good lecture by Professor Ray Monk on the University of Southampton page (see “useful downloads”) looking at the link between philosophy and mathematics – which takes a fascinating journey through the history of maths and the great ideas of great men.
6) Maths for Primates is a fantastic source of podcasts – 14 and counting, on fractals, Zeno, Hilbert’s hotel and more.


Education: Zeno Math
Zeno powers up a love of math in low-income communities.
June 10, 2019
By Cynthia Flash. Read this story and more in Volume 4 (pdf) of Seattle Foundation's Heart & Science magazine.
When four-year-old Austin Lau plays with the cards,
blocks and trucks he got from the teacher who visited
his house, he doesn’t know he’s doing math. He
simply sorts the pieces. Or counts the wheels. These
are fun games he enjoys nearly every day with his
mother Heidy on the floor of their south Seattle home.
“He loves it. He can count to 100,” says Heidy, who
came to Seattle from China two years ago. “For the
blocks he builds different shapes, castles. He has a
match game. Every week we get an activity.”
Those activities and games are distributed to Austin
and other preschoolers by Zeno, a small nonprofit
focused on improving kids’ math confidence and
abilities with engaging and interactive games.
The goal is to excite young children about math
before they enter grade school – to make sure they
are ready to learn more and have a positive attitude
about math from the beginning.
Zeno focuses on low-income communities. In its
MathWays for Early Learning program, Zeno partners
with family service providers such as preschools,
home visiting programs and other community
organizations to provide math game kits to families.
Zeno partners with communities
with the greatest needs to engage
kids in positive math experiences
early in life. They work through
preschools and family service
organizations to support fun and
engaging math activities.
For more information, visit
It hosts Family Math Nights in schools it partners
with and hosts semi-annual MathFests, a celebration
of math play open to the public.
Just as literacy programs aim to get books into
homes to support parents reading to their children,
Zeno works to bring math play and activities into
homes, says executive director Julie Marl.
Zeno’s MathWays for Early Learning program is
seeing strong demand. Last year, this program
served 500 families of preschool-age children. It
expects to serve 1,500 families this year – mainly in
King, Pierce and Snohomish Counties.
“It’s amazing that she remembers those shapes that
I couldn’t even say when I was that age,” says Helen
Altamurano-Tang, whose three-year-old daughter
Phoebe plays Zeno games with her older sisters.
“Can you say `parallelogram'?” Helen asks Phoebe.
“Parallelogram,” Phoebe answers proudly before
returning to her Zeno-provided transportation toys.
“Zeno helps her to familiarize those shapes, those
colors, those numbers,” Helen says. “She’s really
excited to be more active in learning.”
Research shows that early math skills are a better
predictor of later academic success than literacy
or social emotional skills. “It’s about building
a kid's math foundation, so they can have the
opportunity for greater success throughout their
academic career," said Marl. "By building a strong
foundation in math, kids have so many more
doors open to them down the road.”
Jesse Gilliam, communications director for
Washington STEM, agrees. “We work with Zeno
because we feel young people throughout
Washington state really deserve and need the
tools to prepare for their future,” he said. “Because
our economy and world are increasingly digitally
focused and focused on using STEM (Science,
Technology, Engineering and Mathematics) skills,
we believe that STEM education is needed to get
these folks there. It’s not only for the economy, but
for life and interacting with the world.”
Heart & Science Magazine
Read Heat & Science magazine Vol. 4 for more on how philanthropists, community organizations and Seattle Foundation are working to create a healthy community through supporting education and health & wellness.
Interested in making your giving more impactful? Connect with one of our expert philanthropic advisors.
Heart and Science Magazine,
Children and youth,
Low income households,


When I go shopping, I am often asked if I’d like to join the particular stores points program.
You know what you get with points at most of these places — you get points. I never join anything for points.
Personally I do not need points.
However, the other day my friend Grace, gifted me ONE MILLION points, out of the goodness of her heart. It was her birthday and I think she was feeling generous.
ONE MILLION POINTS – to do with what ever I want.
As a thank you, I gifted Grace 2,000 points right back at her – then and there.
But, then, it struck me, if my math is right (and I have zero confidence in my math skills) I have 980,000 points at my beck and call.
I’ve decided I’d like to cash in ALL my points and put them in one basket. I’m kind of hoping 980,000 points maybe enough to get Ron James to my next birthday party!
Also, if anyone has any points they aren’t using and you think it will help to get Ron James to Sechelt, I’d happily take them, just in case I need more.


You can do more with bits than just turning them on or off. This episode will show you how to shift bits left or right for either really quick multiplication or division or to maneuver them into place.
If you start out with the value 0x5a and want to first isolate the 5 digit, the previous episode explains how to use masking to accomplish this. After masking, you have 0x50 which is not quite the value 5 that you might need. If you want to convert 0x50 into 0x05, then you’re going to have to shift it to the right.
Shifting either left or right is based on binary digits not hexadecimal digits. Because each hexadecimal digit is composed of four binary digits, then to shift a hexadecimal value by one of its own digits, you’ll need to shift the underlying bits by four positions.
Let me show you this in binary with the value 0x50:
0101 0000 shifting this to the right four times gives:
0010 1000 this is after a single shift
0001 0100 this is after two shifts
0000 1010 this is after three right shifts
0000 0101 and finally after four right shifts.
Notice how as the digits shifted to the right, we filled in the vacated positions on the left with zeros. The same thing happens if we shift to the left except that now the vacated positions will be on the right.
Maybe you want to swap hexadecimal digits and turn 0x5a into 0xa5. The example just now shows how to move 0x50 into 0x05. We didn’t actually need to mask out the other digit though. Because when shifting, you know that vacated spots get filled in with zeros but it’s also possible for digits to fall off the other side. Let me show you what happens if we shift 0x5a to the left this time by four shifts.
0101 1010 this is 0x5a before any shifts
1011 0100 this is after the first shift to the left
0110 1000 this is another shift to the left and the first one has dropped
1101 0000 this is the third left shift
1010 0000 after four left shifts, the “a” is now in the left digit and the “5” is gone.
Listen to the full episode or read the full transcript below.
The previous episode showed you how you could isolate, set, and clear bits either individually or in a group. If you have a byte with the value 5a in hexadecimal, you know how to isolate each half now and with appropriate masks, you can easily turn this into either 50 or 0a. But what if you wanted to work with just that 5 part? In it’s current position, even after masking, it’s not 5 it’s 50. I’ll show you how you can shift this so it is just 5.
We going to use some operations called shift left and shift right to accomplish this. These are bitwise operations and allow you to shift bits a certain number of positions. Remember that in hexadecimal each digit actually represents 4 binary digits. So to shift the value 50 so it becomes 05, it’ll need to be shifted to the right by 4 positions.
There’s actually a lot more to these operations than might first seem likely. the first thing to note is that each language can define the details of these shift operations as they see fit. The basic operations will be the same but the edge cases could be different between languages.
For example in C++, if you try to shift either left or right by more positions than your type has available, then you may not get the results you expect. This is getting into undefined territory so make sure that when you shift bits, that your shift amounts make sense. Another good way to cause problems is to try shifting by a negative number of locations. Your compiler might generate code that behaves differently than another compiler in these situations.
Another aspect to be aware of when shifting to the right in C++ has to do with negative values. I’m not talking about trying to shift by a negative amount. That already doesn’t make any sense. I’m talking about what happens when you have a negative number and try to shift it to the right. Most implementations that I’ve seen will divide the initial negative number by a power of two. Don’t worry, I’ll explain this more in just a moment. So if you start out with the value -8 and shift this to the right by 1 position, you’ll end up with -4. Just be aware that working with negative numbers like this is technically up to your C++ implementation to define.
And if you look at how shifting is done in C#, it’s actually much more defined.
Before we get into this though, let me explain more about that division and I’ll also show you how you can use shifting to perform multiplication. Let’s do the multiplication first since it’s the easiest. Actually, now’s a good time to take a short break for this message from our sponsor.
( Message from Sponsor )
I’m going to explain shifting in terms of decimal first since we’re all familiar with that. Imagine for a moment that computers understood decimal numbers directly and that a byte consisted not of 8 binary values of zero and one but of 8 decimal digits from 0 through nine. We could count all the way from 0 through 99,999,999. If I’m losing you here, just pretend that 8 digits is the highest anybody can count and that there is no number greater than 8 digits. so the number 100 million or anything greater would not exist.
Alright, lets start with the value 1. We need something other than all zeros or shifting in any direction just has no effect. This applies to computers too. If we take the value 1 and shift it to the left by a single location, what value should we put into the position that the 1 used to occupy? We’ll fill in vacated positions with 0. So shifting the value 1 to the left and then filling in the empty location with a zero gives us the value 10. If we shift it again, this time, we’re shifting the 1 over a location and we’re shifting the 0 over a single location too. then we fill in the empty spot that the 0 used to occupy with another 0. This gives us the value 100 now. Each time we shift left, the value is multiplied by 10. We all learned this in grade school, if you want to multiply by ten, just add a zero on the right. That’s a roundabout way of describing a left shift operation. Or maybe you consider my explanation the more roundabout way and consider it simpler to just think of adding a zero. Whatever works for you.
The end result is that each time you shift a value to the left, you end up putting a new zero on the right. If you shift the value 12 to the left one position, then you just place a single zero on the right and you have 120.
Now what about shifting to the right? Most grade school teachers never went this far in their explanation. Shifting to the right without getting into fractional values means that the right digit is going to fall of the edge. It actually doesn’t matter which way a number is shifted, something’s going to fall off the side being shifted towards. It wasn’t obvious when we shifted 12 to the left to get 120. But what if we have the value 12,345,678 and try shifting that to the left? Since all 8 of our digits are being used, the 1 digit will get cut off. This doesn’t happen in real life because we don’t have a fixed set of digits and our numbers can just keep getting as big as they need.
Shifting the value 12 to the right by one position will cause the 2 digit to fall of and we’re left with just 1. Again, this doesn’t happen in real life because we can start using decimal places and can start growing numbers as small as we want in the other direction. In the real world, shifting 12 to the right by one position will result in 1.2.
Remember that computers don’t have decimal places and don’t have unlimited digits. When programming and shifting, you have to expect digits will fall off the edge. I think I got a bit sidetracked there. Let’s start with 120 and shift that to the right by one position. the 0 digit will fall of the edge and we’ll have 12. this divided the value by 10.
If you can understand what I just described about shifting decimal numbers to the left dropping the leftmost digits while adding zeros on the right vs. shifting decimal numbers to the right and dropping the rightmost digits while adding zeros on the left, then you already understand shifting in binary too.
In many ways, shifting binary is a lot easier. You only have zeros and ones to worry about. Let’s imagine a very small computer that can only work with just 4 bits. If we start with the value 0011 and shift it to the left one, we get 0110. shifting it to the left again gives us 1100. One more time to the left gives us 1000. This caused a 1 digit to fall of the edge and now we only have one bit that has a value of 1. If we did this left shift again, then the realist would be 0000 because now all the 1 digits fell of the edge. Once we’re at all zeros, shifting left or right makes no difference anymore. Let’s go back to the starting value of 0011 and this time shift it to the right. A digit falls off the edge this time too and we have 0001. Shifting that result to the left will not bring back the 1 digit that fell off. It’s gone forever. So shifting 0001 to the left just gives us 0010.
The last thing I want to mention today is how shifting in binary also causes multiplication or division. Just like in decimal if you want to multiply by ten, you add a zero. This works in binary with a slight change. Remember binary is base two. So if you want to multiply by 2 in binary, you just add a zero. And to divide by 2, you just remove the digit on the right. As long as the digit that gets removed is a zero, then you have a nice even division by 2. If it wasn’t a zero, then you have almost divided by 2.
The same thing happens in decimal. If we start with the value 10 and you want to divide by 10, just remove the zero and you have 1. That’s a perfect division by ten. But if you started with 12 and just removed the 2 digit, then your answer will be in the general area but not exact.
In binary, the value 100 is four. Removing the rightmost zero or shifting it to the right depending on how you want to look at it give you the result of 10 which is 2. That’s a perfect division by 2. But if you start out with the value 101 which is 5 and do the same thing, you still end up with 10 which is 2. Now 5 divided by 2 should be 2 and a half but we can’t easily express that without doing something entirely different.


Hale Academy is hosting a series of math workshops geared toward the parents of students in PreK4 through 5th grade. This is the first in the series. It is about how students are learning the “why” before the algorithm to gain a true understanding of math and not just a memorization technique.
Students were given the opportunity to shop at Publix using their math skills. Hale students were exposed to the many difficult decisions parents have to make each week while grocery shopping for their family. Students realized very quickly that math never goes away. It is a useful tool throughout life.


History of math. The most ancient mathematical activity was counting. The
counting was necessary to keep up a livestock of cattle and to do business.
Some primitive tribes counted up amount of subjects, comparing them various
parts of a body, mainly fingers of hands and foots. Some pictures on the
stone represents number 35 as a series of 35 sticks - fingers built in a
line. The first essential success in arithmetic was the invention of four
basic actions: additions, subtraction, multiplication and division. The
first achievements of geometry are connected to such simple concepts, as a
straight line and a circle. The further development of mathematics began
approximately in 3000 up to AD due to Babylonians and Egyptians.
BABYLONIA AND EGYPT
Babylonia. The source of our knowledge about the Babylon civilization are
well saved clay tablets covered with texts which are dated from 2000 AD and
up to 300 AD . The mathematics on tablets basically has been connected to
housekeeping. Arithmetic and simple algebra were used at an exchange of
money and calculations for the goods, calculation of simple and complex
percent, taxes and the share of a crop which are handed over for the
benefit of the state, a temple or the land owner. Numerous arithmetic and
geometrical problems arose in connection with construction of channels,
granaries and other public jobs. Very important problem of mathematics was
calculation of a calendar. A calendar was used to know the terms of
agricultural jobs and religious holidays. Division of a circle on 360 and
degree and minutes on 60 parts originates in the Babylon astronomy.
Babylonians have made tables of inverse numbers (which were used at
performance of division), tables of squares and square roots, and also
tables of cubes and cubic roots. They knew good approximation of a number
[pic]. The texts devoted to the solving algebraic and geometrical
problems, testify that they used the square-law formula for the solving
quadratics and could solve some special types of the problems, including up
to ten equations with ten unknown persons, and also separate versions of
the cubic equations and the equations of the fourth degree. On the clay
tablets problems and the basic steps of procedures of their decision are
embodied only. About 700 AD babylonians began to apply mathematics to
research of, motions of the Moon and planets. It has allowed them to
predict positions of planets that were important both for astrology, and
In geometry babylonians knew about such parities, for example, as
proportionality of the corresponding parties of similar triangles,
Pythagoras’ theorem and that a corner entered in half-circle- was known for
a straight line. They had also rules of calculation of the areas of simple
flat figures, including correct polygons, and volumes of simple bodies.
Number [pic] babylonians equaled to 3.
Egypt. Our knowledge about ancient greek mathematics is based mainly on two
papyruses dated approximately 1700 AD. Mathematical data stated in these
papyruses go back to earlier period - around 3500 AD. Egyptians used
mathematics to calculate weight of bodies, the areas of crops and volumes
of granaries, the amount of taxes and the quantity of stones required to
build those or other constructions. In papyruses it is possible to find
also the problems connected to solving of amount of a grain, to set number
necessary to produce a beer, and also more the challenges connected to
distinction in grades of a grain; for these cases translation factors were
But the main scope of mathematics was astronomy, the calculations connected
to a calendar are more exact. The calendar was used find out dates of
religious holidays and a prediction of annual floods of Nile. However the
level of development of astronomy in Ancient Egypt was much weaker than
development in Babylon.
Ancient greek writing was based on hieroglyphs. They used their alphabet. I
think it’s not efficient; It’s difficult to count using letters. Just think
how they could multiply such numbers as 146534 to 19870503 using alphabet.
May be they needn’t to count such numbers. Nevertheless they’ve built an
incredible things – pyramids. They had to count the quantity of the stones
that were used and these quantities sometimes reached to thousands of
stones. I imagine their papyruses like a paper with numbers ABC, that
equals, for example, to 3257.
The geometry at Egyptians was reduced to calculations of the areas of
rectangular, triangles, trapezes, a circle, and also formulas of
calculation of volumes of some bodies. It is necessary to say, that
mathematics which Egyptians used at construction of pyramids, was simple
and primitive. I suppose that simple and primitive geometry can not create
buildings that can stand for thousands of years but the author thinks
Problems and the solving resulted in papyruses, are formulated without any
explanations. Egyptians dealt only with the elementary types of quadratics
and arithmetic and geometrical progressions that is why also those common
rules which they could deduce, were also the most elementary kind. Neither
Babylon, nor Egyptian mathematics had no the common methods; the arch of
mathematical knowledge represented a congestion of empirical formulas and
THE GREEK MATHEMATICS
Classical Greece. From the point of view of 20 century ancestors of
mathematics were Greeks of the classical period (6-4 centuries AD). The
mathematics existing during earlier period, was a set of the empirical
conclusions. On the contrary, in a deductive reasoning the new statement is
deduced from the accepted parcels by the way excluding an opportunity of
Insisting of Greeks on the deductive proof was extraordinary step. Any
other civilization has not reached idea of reception of the conclusions
extremely on the basis of the deductive reasoning which is starting with
obviously formulated axioms. The reason is a greek society of the classical
period. Mathematics and philosophers (quite often it there were same
persons) belonged to the supreme layers of a society where any practical
activities were considered as unworthy employment. Mathematics preferred
abstract reasoning on numbers and spatial attitudes to the solving of
practical problems. The mathematics consisted of a arithmetic - theoretical
aspect and logistic - computing aspect. The lowest layers were engaged in
Deductive character of the Greek mathematics was completely generated by
Plato’s and Eratosthenes’ time. Other great Greek, with whose name connect
development of mathematics, was Pythagoras. He could meet the Babylon and
Egyptian mathematics during the long wanderings. Pythagoras has based
movement which blossoming falls at the period around 550-300 AD.
Pythagoreans have created pure mathematics in the form of the theory of
numbers and geometry. They represented integers as configurations from
points or a little stones, classifying these numbers according to the form
of arising figures (« figured numbers »). The word "accounting" (counting,
calculation) originates from the Greek word meaning "a little stone".
Numbers 3, 6, 10, etc. Pythagoreans named triangular as the corresponding
number of the stones can be arranged as a triangle, numbers 4, 9, 16, etc.
- square as the corresponding number of the stones can be arranged as a
From simple geometrical configurations there were some properties of
integers. For example, Pythagoreans have found out, that the sum of two
consecutive triangular numbers is always equal to some square number. They
have opened, that if (in modern designations) n[pic] - square number,
n[pic] + 2n +1 = (n + 1)[pic]. The number equal to the sum of all own
dividers, except for most this number, Pythagoreans named accomplished. As
examples of the perfect numbers such integers, as 6, 28 and 496 can serve.
Two numbers Pythagoreans named friendly, if each of numbers equally to the
sum of dividers of another; for example, 220 and 284 - friendly numbers
(here again the number is excluded from own dividers).
For Pythagoreans any number represented something the greater, than
quantitative value. For example, number 2 according to their view meant
distinction and consequently was identified with opinion. The 4 represented
validity, as this first equal to product of two identical multipliers.
Pythagoreans also have opened, that the sum of some pairs of square numbers
is again square number. For example, the sum 9 and 16 is equal 25, and the
sum 25 and 144 is equal 169. Such three of numbers as 3, 4 and 5 or 5, 12
and 13, are called “Pythagorean” numbers. They have geometrical
interpretation: if two numbers from three to equate to lengths of
cathetuses of a rectangular triangle the third will be equal to length of
its hypotenuse. Such interpretation, apparently, has led Pythagoreans to
comprehension more common fact known nowadays under the name of a
pythagoras’ theorem, according to which the square of length of a
hypotenuse is equal the sum of squares of lengths of cathetuses.
Considering a rectangular triangle with cathetuses equaled to 1,
Pythagoreans have found out, that the length of its hypotenuse is equal to
[pic], and it made them confusion because they tried to present number
[pic]as the division of two integers that was extremely important for their
philosophy. Values, not representable as the division of integers,
Pythagoreans have named incommensurable; the modern term - « irrational
numbers ». About 300 AD Euclid has proved, that the number [pic]is
incommensurable. Pythagoreans dealt with irrational numbers, representing
all sizes in the geometrical images. If 1 and [pic]to count lengths of some
pieces distinction between rational and irrational numbers smoothes out.
Product of numbers [pic]also [pic]is the area of a rectangular with the
sides in length [pic]and [pic].Today sometimes we speak about number 25 as
about a square of 5, and about number 27 - as about a cube of 3.
Ancient Greeks solved the equations with unknown values by means of
geometrical constructions. Special constructions for performance of
addition, subtraction, multiplication and division of pieces, extraction of
square roots from lengths of pieces have been developed; nowadays this
method is called as geometrical algebra.
Reduction of problems to a geometrical kind had a number of the important
consequences. In particular, numbers began to be considered separately from
geometry because to work with incommensurable divisions it was possible
only with the help of geometrical methods. The geometry became a basis
almost all strict mathematics at least to 1600 AD. And even in 18[pic]
century when the algebra and the mathematical analysis have already been
advanced enough, the strict mathematics was treated as geometry, and the
word "geometer" was equivalent to a word "mathematician".
One of the most outstanding Pythagoreans was Plato. Plato has been
convinced, that the physical world is conceivable only by means of
mathematics. It is considered, that exactly to him belongs a merit of the
invention of an analytical method of the proof. (the Analytical method
begins with the statement which it is required to prove, and then from it
consequences, which are consistently deduced until any known fact will be
achieved; the proof turns out with the help of return procedure.) It is
considered to be, that Plato’s followers have invented the method of the
proof which have received the name "rule of contraries". The appreciable
place in a history of mathematics is occupied by Aristotle; he was the
Plato’s learner. Aristotle has put in pawn bases of a science of logic and
has stated a number of ideas concerning definitions, axioms, infinity and
opportunities of geometrical constructions.
About 300 AD results of many Greek mathematicians have been shown in the
one work by Euclid, who had written a mathematical masterpiece “the
Beginning”. From few selected axioms Euclid has deduced about 500 theorems
which have captured all most important results of the classical period.
Euclid’s Composition was begun from definition of such terms, as a straight
line, with a corner and a circle. Then he has formulated ten axiomatic
trues, such, as « the integer more than any of parts ». And from these ten
axioms Euclid managed to deduce all theorems.
Apollonius lived during the Alexandria period, but his basic work is
sustained in spirit of classical traditions. The analysis of conic sections
suggested by him - circles, an ellipse, a parabola and a hyperbole - was
the culmination of development of the Greek geometry. Apollonius also
became the founder of quantitative mathematical astronomy.
The Alexandria period. During this period which began about 300 AD, the
character of a Greek mathematics has changed. The Alexandria mathematics
has arisen as a result of merge of classical Greek mathematics to
mathematics of Babylonia and Egypt. Generally the mathematics of the
Alexandria period were more inclined to the solving technical problems,
than to philosophy. Great Alexandria mathematics - Eratosthenes, Archimedes
and Ptolemaist - have shown force of the Greek genius in theoretical
abstraction, but also willingly applied the talent for the solving of
practical problems and only quantitative problems.
Eratosthenes has found a simple method of exact calculation of length of a
circle of the Earth, he possesses a calendar in which each fourth year has
for one day more, than others. The astronomer the Aristarch has written the
composition “About the sizes and distances of the Sun and the Moon”,
containing one of the first attempts of definition of these sizes and
distances; the character of the Aristarch’s job was geometrical.
The greatest mathematician of an antiquity was Archimedes. He possesses
formulations of many theorems of the areas and volumes of complex figures
and the bodies. Archimedes always aspired to receive exact decisions and
found the top and bottom estimations for irrational numbers. For example,
working with a correct 96-square, he has irreproachably proved, that exact
value of number [pic] is between 3[pic] and 3[pic]. Архимед has proved also
some theorems, containing new results of geometrical algebra.
Archimedes also was the greatest mathematical physicist of an antiquity.
For the proof of theorems of mechanics he used geometrical reasons. His
composition “About floating bodies” has put in pawn bases of a
Decline of Greece. After a gain of Egypt Romans in 31 AD great Greek
Alexandria civilization has come to decline. Cicerones with pride approved,
that as against Greeks Romans not dreamers that is why put the mathematical
knowledge into practice, taking from them real advantage. However in
development of the mathematics the contribution of roman was insignificant.
INDIA AND ARABS
Successors of Greeks in a history of mathematics were Indians. Indian
mathematics were not engaged in proofs, but they have entered original
concepts and a number of effective methods. They have entered zero as
cardinal number and as a symbol of absence of units in the corresponding
category. Moravia (850 AD) has established rules of operations with zero,
believing, however, that division of number into zero leaves number
constant. The right answer for a case of division of number on zero has
been given by Bharskar (born In 1114 AD -?), he possesses rules of actions
above irrational numbers. Indians have entered concept of negative numbers
(for a designation of duties). We find their earliest use at Brahmagupta’s
(around 630). Ariabhata (born in 476 AD-?) has gone further in use of
continuous fractions at the decision of the uncertain equations.
Our modern notation based on an item principle of record of numbers and
zero as cardinal number and use of a designation of the empty category, is
called Indo-Arabian. On a wall of the temple constructed in India around
250 AD, some figures, reminding on the outlines our modern figures are
About 800 Indian mathematics has achieved Baghdad. The term "algebra"
occurs from the beginning of the name of book Al-Jebr vah-l-mukabala
-Completion and opposition (Аль-джебр ва-л-мукабала), written in 830
astronomer and the mathematician Al-Horezmi. In the composition he did
justice to merits of the Indian mathematics. The algebra of Al-Horezmi has
been based on works of Brahmagupta, but in that work Babylon and Greek math
influences are clearly distinct. Other outstanding Arabian mathematician
Ibn Al-Haisam (around 965-1039) has developed a way of reception of
algebraic solvings of the square and cubic equations. Arabian mathematics,
among them and Omar Khayyam, were able to solve some cubic equations with
the help of geometrical methods, using conic sections. The Arabian
astronomers have entered into trigonometry concept of a tangent and
cotangent. Nasyreddin Tusy (1201-1274 AD) in the “Treatise about a full
quadrangle” has regularly stated flat and spherical to geometry and the
first has considered trigonometry separately from astronomy.
And still the most important contribution of arabs to mathematics of steel
their translations and comments to great creations of Greeks. Europe has
met these jobs after a gain arabs of Northern Africa and Spain, and later
works of Greeks have been translated to Latin.
MIDDLE AGES AND REVIVAL
Medieval Europe. The Roman civilization has not left an appreciable trace
in mathematics as was too involved in the solving of practical problems. A
civilization developed in Europe of the early Middle Ages (around 400-1100
AD), was not productive for the opposite reason: the intellectual life has
concentrated almost exclusively on theology and future life. The level of
mathematical knowledge did not rise above arithmetics and simple sections
from Euclid’s “Beginnings”. In Middle Ages the astrology was considered as
the most important section of mathematics; astrologists named
About 1100 in the West-European mathematics began almost three-century
period of development saved by arabs and the Byzantian Greeks of a heritage
of the Ancient world and the East. Europe has received the extensive
mathematical literature because of arabs owned almost all works of ancient
Greeks. Translation of these works into Latin promoted rise of mathematical
researches. All great scientists of that time recognized, that scooped
inspiration in works of Greeks.
The first European mathematician deserving a mention became Leonardo
Byzantian (Fibonacci). In the composition “the Book Abaca” (1202) he has
acquainted Europeans with the Indо-Arabian figures and methods of
calculations and also with the Arabian algebra. Within the next several
centuries mathematical activity in Europe came down.
Revival. Among the best geometers of Renaissance there were the artists
developed idea of prospect which demanded geometry with converging parallel
straight lines. The artist Leon Batista Alberty (1404-1472) has entered
concepts of a projection and section. Rectilinear rays of light from an eye
of the observer to various points of a represented stage form a projection;
the section turns out at passage of a plane through a projection. That the
drawn picture looked realistic, it should be such section. Concepts of a
projection and section generated only mathematical questions. For example,
what general geometrical properties the section and an initial stage, what
properties of two various sections of the same projection, formed possess
two various planes crossing a projection under various corners? From such
questions also there was a projective geometry. Its founder - Z. Dezarg
(1593-1662 AD) with the help of the proofs based on a projection and
section, unified the approach to various types of conic sections which
great Greek geometer Apollonius considered separately.
I think that mathematics developed by attempts and mistakes. There is no
perfect science today. Also math has own mistakes, but it aspires to be
more accurate. A development of math goes thru a development of the
society. Starting from counting on fingers, finishing on solving difficult
problems, mathematics prolong it way of development. I suppose that it’s no
people who can say what will be in 100-200 or 500 years. But everybody
knows that math will get new level, higher one. It will be new high-tech
level and new methods of solving today’s problems. May in the future some
man will find mistakes in our thinking, but I think it’s good, it’s good
that math will not stop.
Ван-дер-Варден Б.Л. «Пробуждающаяся наука». Математика древнего Египта,
Вавилона и Греции. МОСКВА, 1959
Юшкевич A.П. История математики в средние века. МОСКВА, 1961
Даан-Дальмедико А., Пейффер Ж. Пути и лабиринтыю Очерки по истории
математики МОСКВА, 1986
Клейн Ф. Лекции о развитии математики в XIX столетии. МОСКВА, 1989


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Solution Steps. 5 a + 2 a – 3 b – a + 5 b. 5 a + 2 a − 3 b − a + 5 b. Combine 5a and 2a to get 7a. Combine 5 a and 2 a to get 7 a. 7a-3b-a+5b. 7 a − 3 b − a + 5 b. Combine 7a and -a to get 6a. Combine 7 a and − a to get 6 a.
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how to determine concrete yield
how to determine concrete yield
Mail: [email protected]
How to determine concrete yieldTo calculate yield the following formula should be used: Y = s___ , in which n Y = yield of concrete produced per 94-pound bag of cement, in cubic feet; S = volume of concrete produced per batch in cubic f e e t ; N = number of bags of cement in the batch. Re l a t i v e yield, which is the ratio of actual volume of
Mar 01, 1986 · YIELD CALCULATION BASED ON ABSOLUTE VOLUME Concrete mixes are designed by the absolute volume technique. In a batch of concrete, the sum of the absolute volumes cement, aggregate and water plus the volume of air is equal to the volume of concrete produced per batch.Yield Line Theory For Slab Design - Assumptions, Methods how to determine concrete yieldThe yield line theory of analysis is a factored or ultimate load method of analysis. The yield line theory is conducted based on the bending moment of the structural element at its collapse state.The yield analysis was proposed by Ingerslev in 1923. The importance of the theory is more reflected in the analysis of slabs.See more on theconstructor.orgWhat is yield? - American Concrete InstituteWhat is yield? A. Yield is the volume of fresh concrete produced in a batch, expressed in cubic meters. It is calculated by dividing the total mass of the material by the measured density of the freshly mixed concrete. References: SP-1 (02)
Nov 09, 2009 · A: Yield measures volume (in cubic feet) of concrete. A better number to know is relative yield. It's important because you expect a load of concrete to fill a certain volume in your forms or slabs. If the ready-mix producer says he sent 8 cubic yards (216 cubic feet) and you receive 210 cubic feet, you can use relative yield to find out why.Standard Test Method for Density (Unit Weight), Yield, and how to determine concrete yield1.1 This test method covers determination of the density (see Note 1) of freshly mixed concrete and gives formulas for calculating the yield, cement content, and air content of the concrete.Yield is defined as the volume of concrete produced from a mixture of known quantities of the component materials.Related searches for how to determine concrete yieldhow to determine yield ratehow to determine yield percentagehow to determine yield strengthconcrete yield formulasconcrete yield calculatoryield how to calculaterelative yield concreteconcrete yield calculation excel
steel reinforcement at the yield stress and concrete at the concrete design strength for reinforced concrete beam design M u = maximum moment from factored loads for LRFD beam design n = modulus of elasticity transformation coefficient for steel to concrete n.a. = shorthand for neutral axis (N.A.) pH = chemical alkalinityHow to determine concrete yieldTo calculate yield the following formula should be used: Y = s___ , in which n Y = yield of concrete produced per 94-pound bag of cement, in cubic feet; S = volume of concrete produced per batch in cubic f e e t ; N = number of bags of cement in the batch. Re l a t i v e yield, which is the ratio of actual volume ofHow to calculate Volume of concrete for Slabs, Columns how to determine concrete yieldApr 25, 2018 · Calculation of Volume of concrete for Slabs: A slab is in the shape of a rectangle, In order to calculate the volume of concrete required for the slab, find out the surface area of the slab and then multiply it with the depth/thickness of slab as shown in fig. From the above fig Length = 6m, Breadth = 5m and thickness/depth of the slab = 0.15m
To calculate yield, you will divide the number of tons of material by the depth to be paved. This will tell you the square yards that amount of material will pave. Take that numberthe square yardsand multiply it by 9 to get the square footage and divide that by the paving width to learn the linear feet the material should pave.How to Determine Concrete Yield| Concrete Construction how to determine concrete yieldOct 01, 1968 · When the concrete producer receives the approved mix design, he should check it for yield. This can be accomplished by applying the yield formulas. If the mix design is incompatible with correct yield, this fact should be called to the attention of the appropriate party and a revised design obtained which will produce correct yield.How to Calculate Density of ConcreteDensity is thus the necessary information to look for in the product data sheet to calculate the consumption in kg for an expected wet thickness. But be careful with the term density in product data sheets! Density or mixed density? The difference is crucial.
Feb 26, 2018 · An 80-pound bag yields approximately 0.60 cubic feet of concrete. 60-pound bags yield 0.45 cubic feet and 40-pound bags just 0.30 cubic feet. The actual yield is approximate because the amount of water added to the mix may vary. 60 to 80 pounds is a lot of weight to lift, carry and handle.HOW TO ESTIMATE YIELD OF CONCRETE FOR VOLUME Aug 12, 2015 · 1. By Rule of ThumbThe rule of thumb may be used to estimate the approximate yield of concrete from a given concrete how to determine concrete yield 2. By Absolute Volume of IngredientsFOP AASHTO T 121 (20) - WAQTCD = density of the concrete mix . V. m = volume of measure (Annex A) Yield m 3 3 = . Where: Y m 3 3= yield (m of the batch of concrete) W = total mass of the batch of concrete
Mar 31, 2019 · It is difficult to directly test the tensile strength of concrete, so indirect methods are used. The most common indirect methods are flexural strength and the split tensile strength. The split tensile strength of concrete is determined using a split tensile test on concrete cylinders.ESTIMATING CONCRETE MASONRY MATERIALS - NCMAestimating concrete masonry units Probably the most straightforward material to estimate for most masonry construction projects is the units themselves. The most direct means of determining the number of concrete masonry units needed for any project is to simply determine the total square footage of each wall and divide by the surface area how to determine concrete yieldDiscrepancies in Yield What, Why, How? Nevada Ready Concrete yield is defined as the volume of freshly mixed concrete from a known quantity of ingredients. Ready mixed concrete is sold on the basis of volume of fresh, unhardened concrete in cubic yards (yd3) or cubic meters (m3) as discharged from a truck mixer.The basis for calculating the volume is described in the ASTM C 94, Specification for Ready Mixed Concrete. The volume of fresh mixed and unhardened concrete in a given batch is determined by dividing the total weight of the materials b how to determine concrete yieldSee more on nevadareadymix how to determine concrete yield
The concrete block calculator estimates how many bags of concrete is necessary to cover the given area. Use Sakretes concrete calculator to measure how much concrete you need. The concrete block calculator estimates how many bags of concrete is necessary to cover the given area. how to determine concrete yield Calculate Where to Buy. Product. Sand Mix Countertop Mix how to determine concrete yieldConcrete Mix Design CalculationsCalculating sand & stone to yield 27ft3 of concrete Assume this concrete needs to haveAssume this concrete needs to have Sand / Aggregate ratio of 0 42Sand / Aggregate ratio of 0.42 Volume of Sand 042 Total Volume of Aggregate = 0.42 Volume of Sand Volume of Sand = 7.68 ft3 18.28 ft3Concrete CalculatorThis free concrete calculator estimates the amount of concrete necessary for a project, and can account for different shapes and amounts of concrete. Explore other calculators related to housing or building, as well as hundreds of other calculators addressing finance, math, fitness, health, and more.
Calculate volumes for concrete slabs, walls, footers, columns, steps, curbs and gutters. Enter dimensions in US units (inches or feet) or metric units (centimeters or meters) of your concrete structure to get the cubic yards value of the amount of concrete you will need to make this structure.Concrete Calculator - How Much Do I Need? | QUIKRETE how to determine concrete yieldConcrete Mixes Enter the size of the slab you want to build in square feet. The calculator will indicate the number of 60 lb bags you will need to build a 4 inch or 6 inch slab. (all yields are approximate and do not allow for uneven substrate, waste, etc.) Enter Number of Square Feet Desired:Concrete Calculator - How Much Concrete Do I Need? - The how to determine concrete yieldApr 23, 2018 · For concrete, the formula for volume is as follows: length x width x thickness. To determine how many bags of concrete you will need, divide the total cubic yards needed by the yield. Use the following yields per each bag size: 40 pound bag yields .011 cubic yards. 60 pound bag yields .017 cubic yards.
Jun 30, 2014 · The yield of concrete has led to many disagreements between producers and contractors. The starting point is recalling that the total concrete batch volume shown on the ticket is computed and not measured. The computed volume is calculated from the batch weight divided by the density of each ingredient, plus the best estimate of actual total how to determine concrete yieldChapter 6. Compression Reinforcement - Flexural Members6.5. Example of analysis of a reinforced concrete section having compression reinforce-ment. f c=5,000 psi Determine the nominal moment, Mn, and the ultimate moment capacity, Mu, of the reinforced concrete section shown below. f y = 60,000 psi Solution As= 7.62 in2 22.2 As= 3.8 in2 2.5 12Calculate the stresses in the steel and in the concrete at how to determine concrete yieldOur procedure for analyzing a reinforced concrete section in flexure starts by assuming a strain profile: 1) Assume a strain profile (a linear profile is defined by two numbers) 2) Calculate the concrete and steel stress distributions (elastic moduli are given) 3) Calculate stress resultants (also called internal forces) 4) Check for equilibrium with external forces (caused by loads).
Yield is used to determine if the concrete supplier has delivered the amount of concrete ordered.For example, if the contractor orders 7 yards and the concrete pour comes short of filling the structure, a yield test can be performed to determine if 7 yards were actually delivered. To calculate yield, follow these simple steps. Note that you must calculate density prior to calculating yield.You will take the total mass of all materials batched (aggregate, water, cementthis figure will be give how to determine concrete yieldSee more on sicerts how to determine concrete yieldACI ASTM C138 Flashcards | QuizletASTM C138 is used for what test method for Density(unit weight), Yield, and Air Content(gravimetric) of concrete A lower density may indicate what 1)materials have changed[lower specific gravity], 2)higher water content, 3)higher air content, 4)a change in proportions of (PDF) 23 Yield Line Analysis for Slabs | WAQAR SALEEM how to determine concrete yieldFor slabs, the corresponding mechanism is the yield line. For the overloaded slab, the resisting moment per unit length measured along a yield line is constant as inelastic rotation occurs; the yield line serves as an axis of rotation for the slab segment. Figure 23.1a shows a simply supported, uniformly loaded reinforced concrete slab.