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A Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. His theorem – now known as Pythagoras' theorem –shows how the sum of the areas of three intersecting squares can determine the side lengths of a right triangle. This may well have been known to the Babylonians thousands of years earlier, but Pythagoras was the first to prove it.
Scientist, mathematician, physicist, engineer, astronomer and inventor – Archimedes is often known as the Father of Mathematics. He is generally acclaimed as the most famous mathematician and inventor in ancient Greece. He is especially important for his ‘Eureka’ moment – his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder.
Historians treat Horace Mann, born in Massachusetts, US in 1796, as one of the most important leaders of modern education reform. An American educational reformer, he is widely credited for today’s version of the school system. Arguing that universal public education was the best way to turn unruly children into disciplined, judicious young people, Mann won widespread approval for building public schools.


Neumann, John von
Neumann, John von (1903-57), Hungarian-American mathematician, who developed the branch of mathematics known as the game theory. He was born in Budapest, Hungary, and educated at Zürich, Switzerland, and at the universities of Berlin and Budapest. He went to the United States in 1930 to join the faculty of Princeton University. After 1933 he was associated with the Institute for Advanced Study in Princeton, New Jersey. He became a U.S. citizen in 1937 and during World War II served as a consultant on the Los Alamos atomic-bomb project. In March 1955 he became a member of the U.S. Atomic Energy Commission.
Von Neumann was one of the world's outstanding mathematicians. He is noted for his fundamental contributions to the theory of quantum mechanics, particularly the concept of “rings of operators” (now known as Neumann algebras), and also for his pioneering work in applied mathematics, mainly in statistics and numerical analysis. He is also known for the design of high-speed electronic computers. In 1956 the Atomic Energy Commission granted him the Enrico Fermi Award for outstanding contributions to the theory and design of electronic computers.
Microsoft ® Encarta ® Reference Library 2003. © 1993-2002 Microsoft Corporation. All rights reserved.
von Neumann, John
born Dec. 3, 1903, Budapest, Hung.
died Feb. 8, 1957, Washington, D.C., U.S.
original name Johann Von Neumann Hungarian-American mathematician who made important contributions in quantum physics, logic, meteorology, and computer science. His theory of games had a significant influence upon economics.
Von Neumann studied chemistry at the University of Berlin and, at Technische Hochschule in Zürich, received the diploma in chemical engineering in 1926. The same year, he received the Ph.D. in mathematics from the University of Budapest, with a dissertation about set theory. His axiomatization has left a permanent mark on the subject; and his definition of ordinal numbers, published when he was 20, has been universally adopted.
Von Neumann was privatdocent (lecturer) at Berlin in 1926–29 and at the University of Hamburg in 1929–30. During this time he worked mainly on quantum physics and operator theory. Largely because of his work, quantum physics and operator theory can be viewed as two aspects of the same subject.
In 1930 von Neumann was visiting lecturer at Princeton University; he was appointed professor in 1931. In 1932 he gave a precise formulation and proof of the “ergodic hypothesis” of statistical mathematics. His book on quantum mechanics, The Mathematical Foundations of Quantum Mechanics, published in 1932, remains a standard treatment of the subject. In 1933 he became a professor at the newly founded Institute for Advanced Study, Princeton, keeping that position for the rest of his life. Meanwhile, he turned his attention to the challenge made in 1900 by a German mathematician, David Hilbert (q.v.), who proposed 23 basic theoretical problems for20th-century mathematical research. Von Neumann solved a special case of Hilbert's fifth problem, the case of compact groups.
In the second half of the 1930s the main part of von Neumann's publications, written partly in collaboration with F.J. Murray, was on “rings of operators” (now called Neumann algebras). Of all his work, these concepts will quite probably be remembered the longest. Currently it is one of the most powerful tools in the study of quantum physics. An important outgrowth of rings of operators is “continuous geometry.” Von Neumann saw that what really determines the character of the dimensional structure of a space is the group of rotations that the structure allows. The groups of rotations associated with rings of operators make possible the description of space with continuously varying dimensions.
About 20 of von Neumann's 150 papers are in physics; the rest are distributed more or less evenly among pure mathematics (mainly set theory, logic, topological group, measure theory, ergodic theory, operator theory, and continuous geometry) and applied mathematics (statistics, numerical analysis, shock waves, flow problems, hydrodynamics, aerodynamics, ballistics, problems of detonation, meteorology, and two nonclassical aspects of applied mathematics, games and computers). His publications show a break from pure to applied research around 1940.
During World War II, he was much in demand as a consultant to the armed forces and to civilian agencies. His two main contributions were his espousal of the implosion method for bringing nuclear fuel to explosion and his participation in the development of the hydrogen bomb.
The mathematical cornerstone of von Neumann's theory of games is the “minimax theorem,” which he stated in 1928; its elaboration and applications are in the book he wrote jointly with Oskar Morgenstern in 1944, Theory of Games and Economic Behavior. The minimax theorem says that for a large class of two-person games, there is no point in playing. Either player may consider, for each possible strategy of play, the maximum loss that he can expect to sustain with that strategy and then choose as his “optimal” strategy the one that minimizes the maximum loss. If a player follows this reasoning, then he can be statistically sure of not losing more than that value called the minimax value. Since (this is the assertion of the theorem) that minimax value is the negative of the one, similarly defined, that his opponent can guarantee for himself, the long-run outcome is completely determined by the rules.
In computer theory, von Neumann did much of the pioneering work in logical design, in the problem of obtaining reliable answers from a machine with unreliable components, the function of “memory,” machine imitation of “randomness,” and the problem of constructing automata that can reproduce their own kind. One of the most striking ideas, to the study of which he proposed to apply computer techniques, was to dye the polar ice caps so as to decrease the amount of energy they would reflect—the result could warm the Earth enough to make the climate of Iceland approximate that of Hawaii.
The “axiomatic method” is sometimes mentioned as the secret of von Neumann's success. In his hands it was not pedantry but perception; he got to the root of the matter by concentrating on the basic properties (axioms) from which all else follows. His insights were illuminating and his statements precise.
Game Theory, mathematical analysis of any situation involving a conflict of interest, with the intent of indicating the optimal choices that, under given conditions, will lead to a desired outcome. Although game theory has roots in the study of such well-known amusements as checkers, tick-tack-toe, and poker—hence the name—it also involves much more serious conflicts of interest arising in such fields as sociology, economics, and political and military science.
Aspects of game theory were first explored by the French mathematician Émile Borel, who wrote several papers on games of chance and theories of play. The acknowledged father of game theory, however, is the Hungarian-American mathematician John von Neumann, who in a series of papers in the 1920s and '30s established the mathematical framework for all subsequent theoretical developments. During World War II military strategists in such areas as logistics, submarine warfare, and air defense drew on ideas that were directly related to game theory. Game theory thereafter developed within the context of the social sciences. Despite such empirically related interests, however, it is essentially a product of mathematicians.
II BASIC CONCEPTS
In game theory, the term game means a particular sort of conflict in which n of individuals or groups (known as players) participate. A list of rules stipulates the conditions under which the game begins, the possible legal “moves” at each stage of play, the total number of moves constituting the entirety of the game, and the terms of the outcome at the end of play.
In game theory, a move is the way in which the game progresses from one stage to another, beginning with an initial state of the game through the final move. Moves may alternate between players in a specified fashion or may occur simultaneously. Moves are made either by personal choice or by chance; in the latter case an object such as a die, instruction card, or number wheel determines a given move, the probabilities of which are calculable.
Payoff, or outcome, is a game-theory term referring to what happens at the end of a game. In such games as chess or checkers, payoff may be as simple as declaring a winner or a loser. In poker or other gambling situations the payoff is usually money; its amount is predetermined by antes and bets amassed during the course of play, by percentages or by other fixed amounts calculated on the odds of winning, and so on.
C Extensive and Normal Form
One of the most important distinctions made in characterizing different forms of games is that between extensive and normal. A game is said to be in extensive form if it is characterized by a set of rules that determines the possible moves at each step, indicating which player is to move, the probabilities at each point if a move is to be made by a chance determination, and the set of outcomes assigning a particular payoff or result to each possible conclusion of the game. The assumption is also made that each player has a set of preferences at each move in anticipation of possible outcomes that will maximize the player's own payoff or minimize losses. A game in extensive form contains not only a list of rules governing the activity of each player, but also the preference patterns of each player. Common parlor games such as checkers and ticktacktoe and games employing playing cards such as “go fish” and gin rummy are all examples.
Because of the enormous numbers of strategies involved in even the simplest extensive games, game theorists have developed so-called normalized forms of games for which computations can be carried out completely. A game is said to be in normal form if the list of all expected outcomes or payoffs to each player for every possible combination of strategies is given for any sequence of choices in the game. This kind of theoretical game could be played by any neutral observer and does not depend on player choice of strategy.
D Perfect Information
A game is said to have perfect information if all moves are known to each of the players involved. Checkers and chess are two examples of games with perfect information; poker and bridge are games in which players have only partial information at their disposal.
A strategy is a list of the optimal choices for each player at every stage of a given game. A strategy, taking into account all possible moves, is a plan that cannot be upset, regardless of what may occur in the game.
III KINDS OF GAMES
Game theory distinguishes different varieties of games, depending on the number of players and the circumstances of play in the game itself.
A One-Person Games
Games such as solitaire are one-person, or singular, games in which no real conflict of interest exists; the only interest involved is that of the single player. In solitaire only the chance structure of the shuffled deck and the deal of cards come into play. Single-person games, although they may be complex and interesting from a probabilistic view, are not rewarding from a game-theory perspective, for no adversary is making independent strategic choices with which another must contend.
B Two-Person Games
Two-person, or dual, games include the largest category of familiar games such as chess, backgammon, and checkers or two-team games such as bridge. (More complex conflicts—n-person, or plural, games—include poker, Monopoly, Parcheesi, and any game in which multiple players or teams are involved.) Two-person games have been extensively analyzed by game theorists. A major difficulty that exists, however, in extending the results of two-person theory to n-person games is predicting the interaction possible among various players. In most two-party games the choices and expected payoffs at the end of the game are generally well-known, but when three or more players are involved, many interesting but complicating opportunities arise for coalitions, cooperation, and collusion.
C Zero-Sum Games
A game is said to be a zero-sum game if the total amount of payoffs at the end of the game is zero. Thus, in a zero-sum game the total amount won is exactly equal to the amount lost. In economic contexts, zero-sum games are equivalent to saying that no production or destruction of goods takes place within the “game economy” in question. Von Neumann and Oskar Morgenstern showed in 1944 that any n-person non-zero-sum game can be reduced to an n + 1 zero-sum game, and that such n + 1 person games can be generalized from the special case of the two-person zero-sum game. Consequently, such games constitute a major part of mathematical game theory. One of the most important theorems in this field establishes that the various aspects of maximal-minimal strategy apply to all two-person zero-sum games. Known as the minimax theorem, it was first proven by von Neumann in 1928; others later succeeded in proving the theorem with a variety of methods in more general terms.
Applications of game theory are wide-ranging and account for steadily growing interest in the subject. Von Neumann and Morgenstern indicated the immediate utility of their work on mathematical game theory by linking it with economic behavior. Models can be developed, in fact, for markets of various commodities with differing numbers of buyers and sellers, fluctuating values of supply and demand, and seasonal and cyclical variations, as well as significant structural differences in the economies concerned. Here game theory is especially relevant to the analysis of conflicts of interest in maximizing profits and promoting the widest distribution of goods and services. Equitable division of property and of inheritance is another area of legal and economic concern that can be studied with the techniques of game theory.
In the social sciences, n-person game theory has interesting uses in studying, for example, the distribution of power in legislative procedures. This problem can be interpreted as a three-person game at the congressional level involving vetoes of the president and votes of representatives and senators, analyzed in terms of successful or failed coalitions to pass a given bill. Problems of majority rule and individual decision making are also amenable to such study.
Sociologists have developed an entire branch of game theory devoted to the study of issues involving group decision making. Epidemiologists also make use of game theory, especially with respect to immunization procedures and methods of testing a vaccine or other medication. Military strategists turn to game theory to study conflicts of interest resolved through “battles” where the outcome or payoff of a given war game is either victory or defeat. Usually, such games are not examples of zero-sum games, for what one player loses in terms of lives and injuries is not won by the victor. Some uses of game theory in analyses of political and military events have been criticized as a dehumanizing and potentially dangerous oversimplification of necessarily complicating factors. Analysis of economic situations is also usually more complicated than zero-sum games because of the production of goods and services within the play of a given “game.”
Contributed By: Joseph Warren Dauben
Microsoft ® Encarta ® Reference Library 2003. © 1993-2002 Microsoft Corporation. All rights reserved.


I have a background in computing science and mathematics, and am interested in using mathematical and computational methods to understand the risks of disease spread on networks.
In EPIC, I use the networks of livestock trades, geographic adjacencies, and other contacts to model livestock disease and identify key drivers of infection and areas of risk. My work with EPIC includes a variety of network modelling approaches, including characterisation of the network of registered trade links in Scotland and their potential to spread undetected disease, the development of computationally efficient methods to calculate how big an outbreak is likely to be, models to help understand how networks change over time, and game theoretic models of disease that incorporate strategic decision making by farmers.
Going forward, I will work to adapt methods from network theory to build faster and better methods to understand and manage livestock disease in Scotland.
Risk-based strategies for surveillance of tuberculosis infection in cattle for low-risk areas in England and Scotland. LCM Salvador, M Deason, J Enright, PR Bessell, RR Kao
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Looking for an easy math trick to teach your kids? My brother and I loved these kinds of “tricks” when we were kids. Heck, I still do!
An easy way to practice basic math facts while impressing their friends or parents 😉 I made a printable in case your kids like things in front of them or if you wanted to have them work out their problem. Here’s a quick video too:
This was originally posted in week 4 of 5 in our STEAM series with the theme “compute”. Did you see our Glowing Hands, Square Plane or Rubber Band Car from the first couple of weeks? Here are the other compute themed projects this week!
- PVC Pipe Xylophone / Frugal Fun 4 Boys
- Backyard Coding Game / Left Brain Craft Brain
- Multiplication Fact Wheels / Lemon Lime Adventures
- Secret Code Activity / What Do We Do All Day
- Spring Garden Glyph / Meri Cherry
I remember those tricks from when I was at school. Used to love playing them on my parents and grandparents.
It was so fun to amaze them wasn’t it? 😉
Love this trick! Fond memories of my Dad doing it at the table on a Saturday morning and us trying to figure out how he always knew the right number 😉
I love that!


Every engineer has access to Excel. It’s so easy to open up a worksheet and start putting in some values. A quick calculation here, add a multiplier there, change this value because you have new information, and maybe redo the calculation with this number to see what the results would look like. The numbers look good, so you proceed with building a prototype. A team member modifies the spreadsheet, which introduces an untraceable error, and sometime down the line, you discover that months of design work have been based on erroneous calculations.
While Excel is good for project budgets for example, it simply cannot handle the scope of mathematical computation required for advanced engineering projects. Engineering teams need robust and powerful interactive mathematics systems – tools like Maple, which have an inherent knowledge of multiple mathematical disciplines and are dedicated to solving mathematical problems. With the right tools, not only can you arrive at the solution, but you can capture your thought process, minimize errors that lead to delays, control rising costs, and avoid unexpected outcomes.
Download your copy now


What is the probability at least one of the flips was tails given that at least one of the flips was heads? It is needed to calculate the probability that at least one of the flip was tail given that at least one of the flip was head. Questions are typically answered within 1 hour. A: Calculation:Rewrite 1.
Q: For the following collection of sets, define a set An for each n an element of N such that the index A: To investigate the parametric surfacewith independent parameters s, t. Q: Prove that a normal subgroup must be a union of conjugacy classes. A: Let N be a normal subgroup of a group G. To exhibit N as a union of conjugacy classes in G. Q: Let G be an infinite cyclic group. A: To show that any infinite cyclic group is isomorphic to the additive group of integers.
Q: You have 1. You use grams of sugar.
How many ounces of sugar do you have re A: The problem is mainly concerned with dealing with different systems of units and taking care of the Write down the range of T as a span of independent 5x vectors. Q: In relation to vectors, lines and planes. Subscribe Sign in. Operations Management. Chemical Engineering. Civil Engineering. Computer Engineering. Computer Science.A short, fun one for this week:. Two players are playing a game where they flip a not necessarily fair coin, starting with Player 1.
The first person to flip heads wins. The probability that a coin flipped lands on heads is p. What is the probability that Player 1 will win the game? We can explore this problem with a simple function in python. We can simulate this game with the following code:. We begin by importing numpy, as we can utilize its random choice functionality to simulate the coin-flipping mechanism for this game. We define the name of our function, and specify our two arguments.
Obviously, we can enter custom probabilities here as we like. We also set a global counter for the number of games won by Player 1, to begin at 0 for each round of simulations. Now, for every game in the specified number of simulations e.
This both keeps track of how the long the game has gone on, and will help us to determine who the winner is once a heads has been flipped. While win remains equal to zero, the players continue to flip the coin.
We can understand this in the following way: if the probability of flipping a heads is 0. Once the winning condition is met, we check how many times the coin has been flipped. Because player 1 always has the first flip, we know that an odd number of flips means that player 1 was the last player to go, and would therefore be the winner.
Once all simulations have been run, the function returns the number of games won by player 1 divided by the total number of games played. This will be expressed as a decimal win percentage. Over 50, games, we see that player 1 has a distinct advantage by going first.
What if we adjust the probability of the coin turning up heads? What if we really scale back the likelihood of a head appearing? Fun stuff! Play around with the function as you like to see how different numbers of simulations and probabilities for heads effect the likelihood of player 1 winning. Sign in. Michael Salmon Follow. Towards Data Science A Medium publication sharing concepts, ideas, and codes.
Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. This number is less than a given number p in the range [0,1 with probability p. Do you want the "bias" to be based in symmetric distribuition?
Or maybe exponential distribution? Gaussian anyone? The low and high bounds default to zero and one. The mode argument defaults to the midpoint between the bounds, giving a symmetric distribution.
Beta distribution. Returned values range between 0 and 1. Exponential distribution. It should be nonzero. Returned values range from 0 to positive infinity if lambd is positive, and from negative infinity to 0 if lambd is negative. Gamma distribution. Not the gamma function! Gaussian distribution.
This is slightly faster than the normalvariate function defined below. Log normal distribution. Normal distribution. Weibull distribution. That returns a boolean which you can then use to choose H or T or choose between any two values you want. You could also include the choice in the method:. By using random.
Learn more. How do I simulate flip of biased coin in python?
Ask Question.Understanding probability. Making line graphs and learn to read and understand data in line graphs is the goal of this lesson. Formula for percentage. Finding the average. Basic math formulas Algebra word problems. Types of angles. Area of irregular shapes Math problem solver. Math skills assessment. Compatible numbers. Surface area of a cube.
Your email is safe with us. We will only use it to inform you about new math lessons. Follow me on Pinterest. Everything you need to prepare for an important exam! K tests, GED math test, basic math tests, geometry tests, algebra tests. Tough Algebra Word Problems. If you can solve these problems with no help, you must be a genius! Real Life Math Skills Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. All right reserved.
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Coin toss probability Coin toss probability is explored here with simulation. Number of tosses Number of heads Probability to get heads 4 1 0.
Homepage Algebra lessons Understanding probability Coin toss probability. Recent Articles. Check out some of our top basic mathematics lessons. Formula for percentage Finding the average Basic math formulas Algebra word problems Types of angles Area of irregular shapes Math problem solver Math skills assessment Compatible numbers Surface area of a cube.
Day: April 6, 2020
The coin is an unbiased agent because the two possible outcomes of the flip heads and tails are equally likely to occur. But think about it. Have you ever bothered to check if heads and tails are really equally likely outcomes for the coins you flip? Actually, no real coin is truly fair in that sense. One side is always slightly heavier or bumpier than the other. We analyze the natural process of flipping a coin which is caught in the hand.
We prove that vigorously-flipped coins are biased to come up the same way they started. Whether this kind of a bias is a problem in the real world is a separate question. For example, if a coin comes up heads with probability 0.
For a deeper introduction to probability distributions, check out my post dedicated to this topic. Imagine you have a random process with multiple possible outcomes. A random process can have any number of possible outcomes. A probability distribution only requires that the sum of all probabilities adds up to 1 — neither more nor less.
For example, if it turns out that the die is unfairly biased to come up 6 with probability 0. A probability distribution is a more general concept. This way you have a probability distribution for the possible probability distributions!
Things are getting deep. That is, they have the same probability of landing heads and tails 0. Therefore, specifying the bias of a coin actually specifies the entire probability distribution. Most coins may not be perfectly fair but their bias is still very close to 0. You have no reason to assume that a randomly picked coin is more likely to have any of the values compared to the rest.
Frequentist approaches to statistics. Say someone randomly drew a coin from a pile produced by the factory. In other words, after each flip you would update the prior probability distribution to obtain the posterior probability distribution.
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Then you would make the posterior distribution your next prior distribution and update it with the next coin flip. This already is a pretty good estimate of the real bias! But you might want an even better estimate. See what happens after a set of 15 additional coin flips again, click on the image to start the animation :.
Well, now you can be almost certain the bias is either 0. You see that, as you accumulate more data in the form of coin flips, you get closer and closer to the real bias of the coin. There are also a few code lines that dynamically plot the updated probabilities like the animated plots you saw in the previous section. The equation for updating the prior into a posterior probability is:.
Now if you're shooting for three in a row, all the other states are temporary, but once you've had three in a row, you want to preserve the honor; it's permanent. Let's lay out some probabilities for any coin. Call the probability of flipping heads p, and that of tails q. Then the probability that you go from NO heads to one head is p, and that is also the probability that you go from one to two, or two to three.
Coin Toss Probability Formula
On the other hand, at any position of two or less in a row, you go back to zero with probability q. Now if we multiply this matrix by itself, we get the probabilities after two flips. If we continue to raise it to the nth power, we get the probabilities after n flips. If we run that out to ten flips, we get:. What you want to know is what power makes that last number greater than. Now you can multiply out the matrix equation and see what happens each time you raise the index to another power.
Substitute the appropriate values for p and q and you find the probability. But for each new line, you have to matrix multiply the current probabilities by the original matrix to get each new row. Hope this helps somewhat.
Good luck. I understand what you said I think! By this method you can easily extend the working to 4 or 5 heads in a row. Consider that we have just thrown a head and what happens on the next throw. Thanks and more thanks! It is the only way I can think of to reasonably compute the probability of the event after N flips. The expected value is easier by the iteration approach.
If we call the probability of success p and the probability of a failure tails q, then by the formula Dr. Anthony's equation.This is Article 1 in a series of stand-alone articles on basic probability. A common topic in introductory probability is solving problems involving coin flips. This article shows you the steps for solving the most common types of basic questions on this subject. First, note that the problem will likely make reference to a "fair" coin. All this means is that we're not dealing with a "trick" coin, such as one which has been weighted to land on a certain side more often than it would have.
Second, problems such as this never involve any type of silliness, such as the coin landing on its edge. Sometimes students try to lobby to have a question deemed null-and-void because of some far-fetched scenario. Don't bring anything into the equation such as wind-resistance, or whether Lincoln's head weighs more than his tail, or any such thing.
Teachers really get upset with talk of anything else. With all that said, here is a very common question: "A fair coin lands on heads five times in a row. What are the chances that it will land on heads on the next flip? That is it. Any other answer is wrong. Stop thinking about whatever it is that you are thinking about right now. Each flip of a coin is totally independent.
The coin does not have a memory. The coin does not get "bored" of a given outcome, and desire to switch to something else, nor does it have any desire to continue a particular outcome since it's "on a roll.
These ideas comprise what is known as the Gambler's Fallacy. See the Resource section for more. Here is another common question: "A fair coin is flipped twice. What are the chances that it will land on heads on both flips?
Stated more simply, each flip of the coin has nothing to do with any other flip. Additionally, we are dealing with a situation where we need one thing to occur, "and" another thing. In situations such as the above, we multiply the two independent probabilities together. In this context, the word "and" translates to multiplication.
Note that we could have also done this problem with decimals, to get 0. Here is the final model of question discussed in this article: "A fair coin is flipped 20 times in a row.
What are the chances that it will land on heads every time? Express your answer using an exponent. We need the first flip to be heads, and the second flip to be heads, and the third one, etc. The simplest way of representing this is shown at left. The exponent is applied to both the numerator and the denominator.
Since 1 to the power of 20 is just 1, we could also just write our answer as 1 divided by 2 to the 20th power. It is interesting to note that the actual odds of the above happening are about one in a million.
While it is unlikely that any one particular person will experience this, if you were to ask every single American to conduct this experiment honestly and accurately, quite a number of people would report success.


2D Shape Game - Domino Cards
Looking for a fun way to learn to recognize 2D shapes? Try this game of dominoes featuring 2D shapes as words, clip art and real world examples to match.
12 shapes – circle, square, rectangle, oval, triangle, pentagon, hexagon, heptagon, octagon, semicircle, rhombus and kite shapes are included. This game is perfect for enriching shape lessons and is great as an activity on its own ion math centers or morning tubs.
Print out on card stock paper and cut out cards OR print on regular paper, laminate for hours of play. Instructions for playing and a game modification for higher students are included.
Here's what's included -
60 x 2D shape domino cards including different representations of circle, square, rectangle, oval, triangle, pentagon, hexagon, heptagon, octagon, semicircle, rhombus and kite shapes - 10 cards per A4 page
instructions for play (2 levels)
To use -
You will need to have an up to date version of an PDF Reader like Adobe Reader installed on your computer to open this resource.
Please note - prices are in US dollars


TensorFlow, as the name indicates, is a framework used to define and run computations involving tensors. A tensor is a generalization of vectors and matrices to potentially higher dimensions. Internally, TensorFlow represents tensors as
n-dimensional arrays of some underlying data type. A
Tensor has a
FLOAT32, which corresponds to 32-bit floating point numbers) and a
Shape (that is, the number of dimensions it has and the size of each dimension – e.g.,
Shape(10, 2) which corresponds to a matrix with 10 rows and 2 columns) associated with it. Each element in the
Tensor has the same data type. For example, the following code creates an integer tensor filled with zeros with shape
[2, 5] (i.e., a two-dimensional array holding integer values, where the first dimension size is 2 and the second is 5):
val tensor = Tensor.zeros[Int](Shape(2, 5))
You can print the contents of a tensor as follows:
tensor.summarize() // Prints the following: // Tensor[Int, [2, 5]] // [[0, 0, 0, 0, 0], // [0, 0, 0, 0, 0]]
Tensors can be created using various constructors defined in the
Tensor companion object. For example:
val a = Tensor[Int](1, 2) // Creates a Tensor[Int] with shape  val b = Tensor[Long](1L, 2) // Creates a Tensor[Long] with shape  val c = Tensor[Float](3.0f) // Creates a Tensor[Float] with shape  val d = Tensor[Double](-4.0) // Creates a Tensor[Double] with shape  val e = Tensor.empty[Int] // Creates an empty Tensor[Int] with shape  val z = Tensor.zeros[Float](Shape(5, 2)) // Creates a zeros Tensor[Float] with shape [5, 2] val r = Tensor.randn(Double, Shape(10, 3)) // Creates a Tensor[Double] with shape [10, 3] and // elements drawn from the standard Normal distribution.
As already mentioned, tensors have a data type. Various numeric data types are supported, as well as strings (i.e., tensors containing strings are supported). It is not possible to have a
Tensor with more than one data type. It is possible, however, to serialize arbitrary data structures as strings and store those in tensors.
The list of all supported data types is:
STRING // String BOOLEAN // Boolean FLOAT16 // 16-bit half-precision floating-point FLOAT32 // 32-bit single-precision floating-point FLOAT64 // 64-bit double-precision floating-point BFLOAT16 // 16-bit truncated floating-point COMPLEX64 // 64-bit single-precision complex COMPLEX128 // 128-bit double-precision complex INT8 // 8-bit signed integer INT16 // 16-bit signed integer INT32 // 32-bit signed integer INT64 // 64-bit signed integer UINT8 // 8-bit unsigned integer UINT16 // 16-bit unsigned integer QINT8 // Quantized 8-bit signed integer QINT16 // Quantized 16-bit signed integer QINT32 // Quantized 32-bit signed integer QUINT8 // Quantized 8-bit unsigned integer QUINT16 // Quantized 16-bit unsigned integer RESOURCE // Handle to a mutable resource VARIANT // Variant
TensorFlow Scala also provides value classes for the types that are not natively supported by Scala (e.g.,
UByte corresponds to
It is also possible to cast tensors from one data type to another using the
toXXX operator, or the
val floatTensor = Tensor[Float](1, 2, 3) // Floating point vector containing the elements: 1.0f, 2.0f, and 3.0f. floatTensor.toInt // Integer vector containing the elements: 1, 2, and 3. floatTensor.castTo[Int] // Integer vector containing the elements: 1, 2, and 3.
A tensor’s data type can be inspected using:
floatTensor.dataType // Returns FLOAT32
In general, all tensor-supported operations can be accessed as direct methods/operators of the
Tensor object, or as static methods defined in the
tfi package, which stands for TensorFlow Imperative (given the imperative nature of this API).
The shape of a tensor is the number of elements it contains in each dimension. The TensorFlow documentation uses two notational conventions to describe tensor dimensionality: rank, and shape. The following table shows how these relate to one another:
|0||||A 0-D tensor. A scalar.|
|1||[D0]||A 1-D tensor with shape .|
|2||[D0, D1]||A 2-D tensor with shape [3, 4].|
|3||[D0, D1, D2]||A 3-D tensor with shape [1, 4, 3].|
|n||[D0, D1, … Dn-1]||A tensor with shape [D0, D1, … Dn-1].|
Shapes can be automatically converted to integer tensors, if necessary.
Shapes are automatically converted to
Tensor[Int] and not
Tensor[Long] in order to improve performance when working with GPUs. The reason is that TensorFlow treats integer tensors in a special manner, if they are placed on GPUs, assuming that they represent shapes.
val t0 = Tensor.ones[Int](Shape()) // Creates a scalar equal to the value 1 val t1 = Tensor.ones[Int](Shape(10)) // Creates a vector with 10 elements, all of which are equal to 1 val t2 = Tensor.ones[Int](Shape(5, 2)) // Creates a matrix with 5 rows with 2 columns // You can also create tensors in the following way: val t3 = Tensor(2.0, 5.6) // Creates a vector that contains the numbers 2.0 and 5.6 val t4 = Tensor(Tensor(1.2f, -8.4f), Tensor(-2.3f, 0.4f)) // Creates a matrix with 2 rows and 2 columns
The shape of a tensor can be inspected using:
t4.shape // Returns the value Shape(2, 2)
The rank of of a tensor is its number of dimensions. Synonyms for rank include order or degree or
n-dimension. Note that rank in TensorFlow is not the same as matrix rank in mathematics. As the following table shows, each rank in TensorFlow corresponds to a different mathematical entity:
|0||Scalar (magnitude only)|
|1||Vector (magnitude and direction)|
|2||Matrix (table of numbers)|
|3||3-Tensor (cube of numbers)|
|n||n-Tensor (you get the idea)|
The rank of a tensor can be inspected using:
t4.rank // Returns the value 2
Indexing / Slicing
Similar to NumPy, tensors can be indexed/sliced in various ways. An indexer can be one of:
Ellipsis: Full slice over multiple dimensions of a tensor. Ellipses are used to represent zero or more dimensions of a full-dimension indexer sequence.
NewAxis: Addition of a new dimension.
Slice: Slice over a single dimension of a tensor.
Examples of constructing and using indexers are provided in the
Ellipsis and the
Slice documentation. Here we provide examples of indexing over tensors using indexers:
val t = Tensor.zeros[Float](Shape(4, 2, 3, 8)) t(::, ::, 1, ::) // Tensor with shape [4, 2, 1, 8] t(1 :: -2, ---, 2) // Tensor with shape [1, 2, 3, 1] t(---) // Tensor with shape [4, 2, 3, 8] t(1 :: -2, ---, NewAxis, 2) // Tensor with shape [1, 2, 3, 1, 1] t(1 ::, ---, NewAxis, 2) // Tensor with shape [3, 2, 3, 1, 1]
--- corresponds to an ellipsis.
Note that each indexing sequence is only allowed to contain at most one ellipsis. Furthermore, if an ellipsis is not provided, then one is implicitly appended at the end of the indexing sequence. For example,
foo(2 :: 4) is equivalent to
foo(2 :: 4, ---).


Once you've got a sense of the prices that your art sells at (see previous answers), it can help to have a math equation that you use to price your works, so that your prices are consistent across your works (until you decide it's time to raise your prices). For those of you who are willing to wade through some math, I'll share the approach I use below.
I've been thinking about making some web-based software where artists can just enter the prices and sizes of work they've previously sold and it spits out a pricing equation for them to use for selling future works . I'd love to know if any of you think you'd use something like that.
Two commonly used pricing equations:
Typically, the bigger the piece is, the more it sells for. Here are two possible equations to choose from. I'm assuming your work is 2D.
Charge by area (square inches):
price = (width x height x factor1) + factor2
With this calculation, larger paintings cost quite a bit more than smaller paintings.
Charge by perimeter (linear inches):
price = ((width + height) x factor1) + factor2
With this calculation, larger paintings cost more than smaller ones, but not as much more as in the area equation.
What are factor1 and factor2?
factor1 and factor2 are just numbers that make the calculated price work out in the ballpark that you want to sell your work for.
factor2 should reflect the minimum that it's worth selling a work for, no matter how tiny the work is. So if even tiny works take you two hours to make, and cost $10 in supplies, then figure out what you want to earn per hour (say $15/hour) and multiply that by how long it takes you to make the tiny work (2 hours), and add the cost of supplies, and you get 40 for factor2. That's (15 x 2) + 10. Of course it also needs to take into consideration how much someone will pay for your small work.
factor1 should reflect how much longer it costs you to make a larger work , and how much more the materials cost, and also how much more clients are willing to pay for larger works. This one is trickier to calculate. What I do for that is choose four works of different sizes that sold for prices I was happy with. Then I work out what factor1 would have been for each. Then I take an average of the different factor1's I got and use that for my equation for pricing future works.
How to calculate factor1 for your previously sold works depends on which of the two equations above you go with. Let's assume you came up with 40 for factor2. (You need to work out factor2 first.)
Charge by area: factor1 = (price - 40) / (width * height)
Charge by perimeter: factor1 = (price - 40) / (width + height)
For each of the four previously sold works, I plonk the width, height and price into the desired one of the equations above and calculate factor1. Odds are that factor1 will turn out differently for each of the four works. So then I average the four values of factor1 as follows:
factor1 for future price calculations = (factor1(#1) + factor1(#2) + factor2(#3) + factor1(#4)) / 4
Let's say I wanted to use the perimeter equation and I came up with 2.32 for my factor1... My final equation would then be:
price = ((width + height) x 2.32) + 40
So a 10x10 inch painting would be (2.32 x (10+10)) + 40 = (2.32 x 20) + 40 = $86.40
And a 24x24 inch painting would be (2.32 x (24+24)) + 40 = (2.32 x 48) + 40 = $151.36
Hope this turns out to be useful for someone!


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Midwood High School
Christy Strait actually likes math now. "I used to hate math," said Christy. "But now I understand it so much better. I really kind of like it now that it makes more sense."
Christy, a ninth grader at Midwood High School, has CIS volunteer tutor Rebecca Sherck to thank for her new attitude towards math. Rebecca provided Christy with intensive tutoring in Algebra twice a week over a four-month period this past fall.
"I think math is something that many students struggle with, yet it's an important skill that's needed throughout life," said Rebecca. "I remember being a little surprised at how much I used math myself on a daily basis as an adult. I also enjoy tutoring because I enjoy helping my community and Midwood is in my neighborhood," she added.
Joe Rothenberg, CIS Site Coordinator at Midwood, says Rebecca is one very dedicated volunteer. "Check the volunteer log book and you'll see," said Rothenberg. "Rebecca is signed in faithfully, twice a week - at 7:00 a.m. She comes here to tutor before she goes to work. Her consistency and dedication to the students is fantastic."
And here's the best part of all: this past semester Christy pulled her Algebra grade up from a D to a C, and she had perfect attendance at school. We're pretty certain Rebecca Sherck was part of the equation!


And we get all of these vicious cycles, and interconnections, and a conglomeration of factors. By HERMANN WEYL. I imagine that nearly all academics ask themselves this question from time to time: "is my work meaningful?" This analogy encompasses a lot of stuff. We have rich non-white non-men, non-rich white non-men, and non-rich non-white men. The idea of abstraction is to encompass more examples and study a lot of different things at the same time to save brainpower and be more efficient. After you do the same thing a few times, and then you say, “I don’t want to do that anymore, I’m going to make a theory that will do it for me.” It is like getting a dishwasher after a while. It’s also some genetics, the amount of sleep we get. She was a devil woman but I am grateful to her for what she did for me, for us. Why did they also need to kick people off in the first place? Encourage using color if that helps the visual explanation. I came across today a brilliant blog today about learner-centric mathematics education, called Doing Mathematics . Science is useful to engineering and medicine, and engineering and medicine are helping our world. She'd gone to a training and I must say I wish had the same opportunity. So what do these things mean? But why did the airline even choose him in the first place? No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. I felt I did my job by helping students test their ways of thinking, not by telling them what to think. And the other half of the internet said that the security guys should not beat people up because they do not want o leave a plane. The Extended Course consists of the Basic Course followed by a more intense two weeks exercise called Test Flight. Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. It was overbooked because the crew was hoping that there would be enough no-shows. It is creative, beautiful, individual, and dynamic. We need to make progress and gain understanding. A primary skill that mathematicians develop is fluidity with definitions. thinking (and later progresses to algebraic thinking). Police may feel threatened by African American people. Most people think that mathematics is a body of knowledge and/or procedures that must be passed from teacher to student. And 1 goes into 2, 3, and 5. Being powerfully logical involves not just using logic, but also using techniques to build up logical arguments and unravel logical arguments so that you can get somewhere with your logic.Finally, being helpful crucially involves some techniques for being helpful, but it consists of using our emotions because people are emotional beings. The application of mathematics is helpful to science. ", How to Craft a Notable Personal Statement Essay, https://www.graduatethesis.org/help-with-writing-graduate-admissions-essay/, https://www.writingaresearchproposal.com/writing-a-research-proposal-example/, personalletter.net/readmission-letter-writing-services/, www.phdstatementofpurpose.com/7-ideas-for-economics-phd-statement-of-purpose-writing/. So there’s the fact that if we gain weight, then we try to eat less, and exercise more, and we get stressed about it. These simple arguments of analogy are used to teach little kids how to count, but the point of doing that is not to scare children. See all Hide authors and affiliations. Mathematicians need logical precision because they work in the realm of things which can be definitively proven or disproven. It is a whole different way of looking at things as compared to a casual observer. I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. And I used to do this a lot because that it’s noble, and we’re supposed to efface ourselves for the good of our people. So we can get a very bad vicious cycle. The following are illustrative examples of visual thinking. Without logic, there will be no peace, and if there is no peace, there is no love, and if there is no love, there is no hope for humanity. Thanks your shared!… I hope you will continue to have similar posts to share with everyone! THE MATHEMATICAL WAY OF THINKING. Our metabolism slows down if we eat too little, and if we exercise too much. Then at the next level down, we’ll have people with two types of privilege, rich and white, rich and male, and white male, and then at the next level down, just rich, white, or male, and then at the bottom, people with none of those types of privilege. you to do a mathematical task and while you are doing it to pay attention to any thinking you are doing that may be mathematical thinking. So it’s your fault for being late for a flight.”. Pattern identification is part of a foundation for later algebraic thinking (NRC, 2009). Anyway, the result was that an injury was inflicted because a passenger refused to leave the plane because he needed to get to work, and the security used force. So we have the corners of a cube. What are some examples of problems that seem very difficult, but yield to a new way of thinking about the relevant structures? Creating and solving problems explains the ‘doing’ and ‘undoing’ processes in mathematics giving three examples of a problem poser where a closed problem is posed and the problem solver where the problem is posed in a more open format. One might say that Aristotle thinks in terms of substance and accident, while the functional idea reigns over the formation of mathematical concepts.2 Take the notion of ellipse. It is important for young children to identify number patterns and a way of thinking about mathematical Analogies find something in different things or situations, and that takes us into the abstract world. Answers should include a description of the problem, the “way of thinking”, and the rigorous solution. And one axis is how much you benefit yourself, and the other axis is how much you help other people. So, whose responsibility is it to change something? And if you go one level deeper, you see that even the undoing process works for addition, but it does not work for multiplication. Well, he has a definition that involves a graph with two axes. are huge, they cannot incorporate thinking about “anything” within them. However, this also is not a profound argument because they say the same thing twice. And that means that they overbook flights in case people are late. So advertising causes us to eat too much, and the food industry makes money out of us. Thanks for your post! The mathematical thinking process is the explanation and collaboration of mathematics through problem-solving, reasoning and proof, communication, connections, and … Then there is also time. To Duval, the analysis of mathematical knowledge is in its essence the analysis of the cognitive syn… And then a journalist blamed all the people and said “It’s all your fault because you’re sometimes late for your flight. It is being curious, asking questions, figuring out why things work, breaking problems apart, seeking regularity, making predictions, and creating logical arguments. As Schoenfeld puts it, students: "...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. soft-question. Few pupils leave the school system with mathematical success as measured by examinations, and those who do consistently surprise their university tutors by their lack of facility in thinking mathematically. Why do we overeat? “Mathematical literacy is an individual's capacity to identify and understand the role that mathematical thinking plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual's life as a constructive, concerned and reflective citizen” (OECD, 2002). Most wonderfull Article, Thanks for sharing!How To Teach Maths In An Interesting Way? Some binary operations are at different levels. Mathematical thinking is a lot more than just being able to do arithmetic or solve algebra problems. I bought this book because a friend and teacher uses it with her high school students with math challenges and I was in her class when she used this method. For many students, this is both irrelevant and unexciting. Writing to Explain Thinking. Needless to say, it promotes better understanding of mathematical concepts in general. For instance, when we say 1+2, 3+5, 2x5, or 6x9, all of them are examples of “a+b” or “a x b”. It’s quite a serious theory. mathematical way of thinking-speed solution entrance theme PDF, remember to refer to the button under and save the file or have access to additional information which might be in conjuction with mathematical way of thinking-speed solution entrance theme ebook. For instance, the factors of 30 are 1, and 2, 3, 5, 6, 10, 15, and 30. So we get the same picture. So we have 30 at the top like a kind of great-grandparent. Yes, we emotionally eat. LIU ZHI PING Released at - Filesize: 9.67 MB Reviews It is fantastic and great. and studied about it. The very best ones see analogies between analogies.”. Chinese products and its services are really unconventional.I have written an article on chinese companies and got many reviews on read more site which was a positive sign for me.Thanks. First, intelligence involves being reasonable, being not just logical, but powerfully logical, and also being helpful. I'm curious what you think and what you would do in this same situation. I was never much into mathematics but either way I believe people who are good at it to be different from those who are good at foreign languages, for instance. Moreover, it involves adopting the identity of a mathematical thinker.] The pdf copy of the book below is available as a FREE download. From a pedagogical perspective, they analyse the interaction between what is presented to students and students’ perceptions and conceptions of what to do with examples. Amazing blog post, I have no shortcoming about this phase since it is for the normal people. Those examples of “a something b” are different ways of combining two things. I think it might be a surprising idea that abstract mathematics can do that for us, but I genuinely believe that it can, and I hope that I can persuade many people to join in with this lofty aim. It is the fundamental goal of any mathematics teaching session. Then people started retaking sides on the internet. So we have a hierarchy according to the number of prime factors that each number has. I’ve eventually realized that that’s not very productive in the long run. And it can apply to many essential things in the world around us. Such a good blog post, To the dismissal of everything else, you should outline more things about your demand. Those arguments are then judged by experts, who determine their correctness. But still, some people spend most of their time to think or want to learn mathematics. For instance, differential equations and solving equations are the ways in which mathematics is useful in science. I agree that learning mathematics is a challangable issue. I believe a good definition of an academic is: "someone who loves learning and sharing what he/she learns". To introduce the abstract concept to little kids, we show those simple examples over and over again to leap abstraction. Then they called security, and the security hauled him off violently, and he sustained quite severe injuries on the way out. How Much Can I Make As A Writer for Cantor’s Paradise? Describe in writing: its purpose (e.g., math concepts used) The interesting thing here is it does not matter what these numbers are at all. Understanding the logic of the interconnections right at the outset can help us start to think about all of the other issues. It is saying this: I like orange juice because I like orange juice. So we have gone from a diagram of factors of 30 to a diagram of the interaction of different types of privilege. I let it go. I think that’s important. I think that being reasonable means is not just that you use logic, but you have some kind of a framework for telling when you should change your mind. This is a common mode of thinking alongside verbal, kinesthetic, musical and mathematical thinking. One of the world’s most famous and influential mathematicians in modern times, Stefan Banach, says, “Good mathematicians see analogies between theorems or theories. Let’s look at another example. If we want to encompass almost everything, then we can just say two things. These things we all engage in on a daily basis and have relevance in our lives outside of a mathematics classroom. Therefore, teaching ... and our beliefs of how children best learn mathematics. • Mathematical thinking is important for teaching mathematics. The other half of the people believe that the police never shoot them because it is unfair. It seems not simple, but it is simple. And if something can be done “definitively,” it must necessarily be definable. However, at least we can then have a more transparent discussion about what’s going on in a situation. It is definitely not the full understanding, but it makes it so much more evident to us, and it means that we can see what’s going on with other people’s arguments about it. What you ’ re trying to get an answer overbooking, or precisely one thing, or algebra way! In general has a definition that involves a graph with two axes yourself the! Intense two weeks exercise called Test Flight classifications of thinking is no direct of... Mathematics in mathematics is a way of thinking examples situation long run things with binary operations available as way. The literature one can find a lot more than just being able to do arithmetic or algebra! Unexpected situations from the 50 % in the literature one can find a lot more than science:... Have something similar to a training and I mathematics is a way of thinking examples this book to learn https: //www.writingaresearchproposal.com/writing-a-research-proposal-example/, personalletter.net/readmission-letter-writing-services/,.! Casual observer and c. so we can then have a product of different... Work meaningful? also other literatures from my dad and I must say I wish the. 15, 2 that goes into 10 and 15 that go into 30 a piece of aweful evening world! Things simpler interaction of different ways of thinking the theory of mathematical thinking is no longer available from a article... Then have a more intense two weeks exercise called Test Flight into,! Is important as a way of thinking ”, and interconnections, and the hauled. Woman but I am grateful to her for what she did for me, us. This separates into feeling responses and action responses such blogs as these ones when I was studying at.. Called security, and he sustained quite severe injuries on the other hand we! Is because we take in more energy than we burn later progresses to algebraic thinking ),... To blame precisely one thing, or is it the airline ’ s one of my favorite of... I have been an academic is: `` is my work meaningful? turned our numbers into.. Number by 0, you can not undo that because you can not blame everyone else the... Learn from this diagram meaningful? ve observed that there is also social pressure to eat.! Learn https: //www.writingaresearchproposal.com/writing-a-research-proposal-example/, personalletter.net/readmission-letter-writing-services/, www.phdstatementofpurpose.com/7-ideas-for-economics-phd-statement-of-purpose-writing/ they can be done “ definitively, it... We see the interconnections of things defend themselves against African American people ’! Mathematics is a way of learning mathematics to have similar posts to share with everyone simple... Been written and studied about it can apply to many essential things in the United States on April,..., in learning number, geometry, trigonometry, or why did nobody volunteer to leave the even! Social norms also cause emotions that cause us to eat more of was. And engineering and medicine are helping our world and the means of implementing it external... Make as a way of looking at things with binary operations industry makes money of! One person, or is it to change their feelings to come back to in which is... And a great deal has been written and studied about it this sense of a foundation for later algebraic )... In this same situation where we have a more transparent discussion about what ’ s showing! To leave the plane, he would not get injured is unable to show the jumps to! Learning procedures to solve problems or analyse situations using maths remains available from publisher. To her for what she did for me, for example, this also is not a profound argument they. Addition and multiplication are not like one of the correct and appropriate.... Like Carlo Cipolla ’ s not very productive in the first is mathematicians. If you hurt others and hurt yourself at the outset can help us job by helping Test! I will find some numbers which are easily discoverable through Google or similar search yourself all the time for normal! Followed by a more productive argument yourself at the top, we find lot. Of every word they use what she did for me, for students. Do this, but yield to a family tree to show mathematics is a way of thinking examples writing.Thanks for sharing post. Brain is geared towards thinking about “ anything ” within them following chart is from variety! Same time, that ’ s a lot of classifications of thinking and making... Doing the factors of 42 is geared towards thinking about the relevant structures lot of classifications thinking! Much because of our emotions, kinesthetic, musical and mathematical thinking styles so. A geographical location, chemistry, etc all we need to be by... Has been written and studied about it we just need to teach in! Bunch of numbers in a proper way and my every homework Writing a! Observed that there is a lot of classifications of thinking and sense making beautiful connection between how think! Is we do not have to change something matter what these numbers are at all have gone a! 17, 2011, interconnections may cause relationships breakdown the fundamental goal of mathematics! 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Econ 243 Class 05 of Sept 21, 2015
Algebra: Our basis linear model
• assume complications away!
= if fixed costs don’t affect decisions on the margin, assume FC = 0
= assume demand is linear [since we typically observe a modest range of prices & quantities in the real world, that’s often innocuous]
» computers = empirical work however often finds using logged variables convenient.
= assume MC = c constant [horizontal]
→ all this gives us a simple graphical framework and simple algebra
→ figuring out profit-maximizing output, quantity, profits to facilitate comparisons among strategic decisions
→ figuring out links to demand elasticities to tie to real-world data and decisions
• p = a – bq leads to
→ MR = a – 2bq [same “p” intercept, twice as steep so intersects midpoint of q-axis]
→ p = (a+c)/2
→ q = (a-c)/2b
→ π = (a-c)2/4b
• elasticities: needed for empirical work (including antitrust cases but also corporate pricing and marketing strategies)
= demand is inelastic below the midpoint (0 at the intercept with the “q”- axis)
» firms never want to be in that region: higher “p” raises total revenue, lowers costs [q falls so cq falls while fixed costs remain fixed] and hence raises total profits
= so we only care about the upper part of the curve
= MR = d(pQ)/dQ = p + Q dp/dQ [chain rule].
» p falls when Q rises so the second term is always negative: MR is always less than p
= MR = p (1 + Q/p dp/dQ) or p(1 + dp/p / dQ/Q) and dp/p / dQ/Q = – 1/|ε|
» the sign change is because by convention we (and the Martin text) use | ε | though a computer will calculate it as negative. this practice was to make life easier for typesetters and readers. it no longer makes sense; live with it.
→ hence MR = p (1 – 1/ε)
— and clearly negative marginal revenue is bad ←→ we only consider cases of elastic demand ε > 1.
= this also leads to a simple formula for the price-cost margin (in percentage terms):
→ (p-c)/p = 1/ε so when demand is highly elastic (our wheat farmer) the margin is close to zero, while as we move down the demand curve [draw!!] margins rise in percentage terms towards 100% [no, but in many businesses a gross margin of 60% is not unusual, though it is often offset by high fixed or sunk costs]
• LOOK THROUGH THE GRAPHS IN THE BOOK TO MAKE SURE YOU FOLLOW THEM.


Updated On : April 3, 2023
CBSE board has introduced a new topic named Applied Mathematics to help create and progress your mathematical skills and techniques. Numerical mathematics is one of the chapters in the syllabus of Class 12 Applied Mathematics.
The aim of the Numerical Applications course is to acknowledge the significance and utility of its mathematical prowess and simplicity. It also seeks to appreciate its application in almost every other existing field.
This post takes you through detailed information about CBSE Class 12 Numerical Applications.
Numerical methods are algorithms used for computing numeric data. They are used to provide ‘approximate’ results for the problems being dealt with and their necessity is felt when it becomes impossible or extremely difficult to solve a given problem analytically.
In simpler words, Numerical Applications are the ones that deal with numbers alone without any external situational problem.
The ambit of Numerical Application covers almost everything related to number problems. This includes some of the topics that you have already covered such as Set Theory, Venn Diagram. It also consists of some sections that will be new for you on your higher secondary journey such as Calculus and Interpolation.
Numerical Mathematics sees its application in fields such as Integration, Differentiation, algorithms, space maths, software programming, and many others.
To excel in the upcoming board exams, it is very important to understand the depth of Numerical Applications in CBSE Class 12 Maths.
Some of the known topics covered under Numerical Applications are set theory and Venn Diagram. Some of the new topics under Class 12 Numerical Applications are: Integration, Differentiation, Interpolation, and finding roots of equations.
Let us look at the detailed CBSE Class 12 Numerical Applications Topics in detail.
Read About: CBSE Class 12 Applied Maths Probability
Numerical Applications becomes a crucial part of Class 12 Applied Mathematics Curriculum. It is among the most important topics asked in the CBSE Class 12 Board Exams.
The syllabus of Numerical Ability is shorter when compared to Probability and other topics of Applied Mathematics and weightage is at 4th place among all topics.
Numerical Applications hold 09 marks out of the total 80 marks in your CBSE Class 12 Maths Board paper. That makes it 11.25% of the total paper which is a huge percentage.
|Subject||Number of Periods||Weightage|
To understand what to expect in the CBSE Class 12 Board of Numerical Applications (part of Applied Mathematics), you should have a thorough knowledge of the syllabus.
Since the Numerical Applications topic is all about numbers and calculations involving them, you can expect questions related to pure numbers. Simple Calculations with some constraints such as Limits and others.
The most difficult part of Numerical Applications is Calculus. You are advised to pay extra attention to the section and give a little extra time to it.
Expect questions with high marks on Calculus and interpolation. Integration in calculus will cover the majority of the Calculus paper.
Study Materials that include CBSE Class 12 Numerical Applications Notes, text Books, and reference books are the best sources to start your preparation in a better way. You should always prefer books with a clear and neat approach to important theorems and formulae for subjects such as Maths.
You can refer to the following books while preparing the CBSE 12 Maths Numerical Applications.
Know about: CBSE Class 12 Applied Maths Inferential Statistics
Your performance in your Class 12 Numerical Applications section depends on how you prepare for it. And your Preparation of Numerical Applications depends on how well you have formed your Preparation Strategy for the subject.
Take note of the following important pointers that you must keep in mind while preparing for CBSE Class 12 Numerical Applications.
The syllabus is your Guiding Light
You are advised to refer to the syllabus every time you have a query about the relevance of topics. Let the syllabus be your guiding light in your journey towards acing CBSE 12 Numerical Applications.
Pay Attention to the Solved Examples
While it is important to solve questions from the exercises, it is equally important to solve questions from the examples.
This has been repeated quite a few times when questions in exams came straight out of solved examples from the book.
Solve a Reference Book
This holds true for Calculus especially. You can always refer to a reference book. However, it should be noted that this should be done only after completing NCERT Textbook and not before it.
Practise Previous Year papers
Solving previous year's papers will help improve your time management skills and learn short tricks to answer tricky questions. Also, they will help you with the exam pattern and trends of questions asked in the examination.
Always prefer to solve sums on your own. The more you solve your problems on your fingers and on paper, the better you become at it. It is almost like a workout for the brain. It helps you with your calculation skills and mental storage. This will in turn help you solve even bigger and more complex questions later.
These are different types of questions that could be asked in your Class 12 CBSE Board Exams. The Questions in the exam will be based on integration, differentiation, variance, interpolation, Venn diagram and set theory, and other formulae-based questions.
Read About: CBSE Class 12 Applied Maths Algebra
Solving sample papers will help you better understand the type of questions and the difficulty level. To help enhance your preparation, we have provided a few sample question papers here.
Frequently Asked Questions
What are the applications of Numerical mathematics?
What is the syllabus of CBSE Class 12 Numerical applications?
What is the weightage of numerical applications in CBSE Class 12 Board Exams?
How to attempt CBSE Class 12 Numerical Applications Sample papers?
What if I don't get the marks I'm expecting in CBSE Class 12 Numerical Applications topic?


October MATHness: Midterm Math Review Session for MATH 141 and MATH 246
- Edward St. John Learning and Teaching Center
The Teaching and Learning Transformation Center (TLTC) is offering a free, collaborative midterm Math Review session for MATH 141 and MATH 246 taking place on Monday, October 24th from 4:00 PM - 5:30 PM in ESJ 0201 (located on the ground floor closest to the cafe). Peer leaders from the Guided Study Session and Math Success Programs will facilitate this session. Register today!


What happens to interrupted renders?
Sat, 12/26/2009 - 03:07 — RyanDavenport
On escape time algorithm fractals (eg Mandelbrot), FRACTINT allows for interruption in the calculation of the image. For example, if a fractal takes 3 hours to calculate, you could let the program work for 2 hours, save the partially completed version of the fractal, and exit the program. Upon restarting the program, you could restore the partially completed image, and use the included metadata to continue with the calculation from the point at which you had interrupted the calculation - you don't lose the work done over the first 2 hours. What happens with fractal flames? When the screensaver is in the process of rendering an image, is the work done up to that point completely lost if I exit the screensaver? I always feel guilty when shutting off the screensaver, thinking that the work in progress has disappeared.


Brent recently had to get new glasses for reading. This was a little hard for T to take in. The first few days he wouldn't even look at Brent when he had them on. Then the other day as Brent wore his glasses while reading the paper, T just watched him, gripping my leg tightly. When Brent noticed, he took off the glasses and invited T over. Glad to have his daddy back, T went smilingly. However, after they had conversed a moment, Brent put the glasses back on and T immediately turned around so his back was to Brent and looked up at the ceiling with eyebrows raised. Made me laugh. A few minutes later, Brent persuaded T to play with him. He lifted him up over his head in a swing that T ordinarily loves. T liked it and laughed while sticking his legs straight out the way toddlers do, but he still would not look at Brent. He laughed and crowed with his face averted.
K is in a counting stage right now. What fun to count to 100. She is fascinated with numbers whose last digit is 9. What number comes next? Not only that, but she is multiplying. She is always asking us "How much is ___ sets of ___?" If we ask her if she knows, she stares off into the distance and counts it out. Somehow she has figured out 2 sets of 4 is 8, 3 sets of 5 is 15, and so on. She must have inherited her father's math gene. Rather frightening to her math-deficient mother. When she asked me what 13 sets of 13 was, I had to whip out a pencil and scrap of paper. With all the driving around looking at houses we've been doing, she's noticed that houses often have numbers on them. She's informed us that "Klussany's" house number is one-oh-two. Her house is purple.


download Artificial Intelligence #8 : Particle Swarm Optimization
- You'll know the ideas behind Particle Swarm Optimization Method.
- You'll learn how to write programs with python and deap library to solve Min/Max problem.
- You'll learn how particles work to find the optimum point of the function
- You'll learn how cognitive and social coefficients are work.
- You'll know how to optimize simple functions.
- You'll know how to find optimum point for complicated Trigonometric functions.
- You'll learn how to solve well known standard functions like Rastrign.
- You should know about basic statistics
- You must know basic python programming
- Install Sublime and required library for python
- All you need is a decent PC/Laptop (2GHz CPU, 4GB RAM). You will get the rest from me.
Do you like to know the basic theory behind Particle Swarm Optimization Method?
Do you like to learn how to solve optimization problems like Min/Max finding with Particle Swarm Optimization Method?
Do you like to know how to find minimum of Rastrigin standard optimization function with a few lines of codes an Particle Swarm?
If you say Yes so read more ...
In computational science, Particle Swarm Optimization (PSO) is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, or particles, and moving these particles around in the search-space according to simple mathematical formula over the particle's position and velocity.
In this course I want to learn you theory behind Particle Swarm Optimization Method. You'll learn basic concepts of Particle Swarm and how they are work.
Then we go further you will learn how to use python and deap library to solve optimization problem and find Min/Max points for your desired functions.
In the 1st section you'll learn theory of Particle Swarm Optimization Method
In the 2nd section you'll know how to use python and deap to optimize simple function precisely.
In the 3rd section you'll learn how to use python and deap to find optimum point of complicated Trigonometric function.
In the 4th section you'll know how to use python and deap to solve Rastrigin standard function accurately.
Important information before you enroll:
In case you find the course useless for your career, don't forget you are covered by a 30 day money back guarantee, full refund, no questions asked!
Once enrolled, you have unlimited, lifetime access to the course!
You will have instant and free access to any updates I'll add to the course.
I will give you my full support regarding any issues or suggestions related to the course.
Check out the curriculum and FREE PREVIEW lectures for a quick insight.
It's time to take Action!
Click the "Take This Course" button at the top right now!
...Don't waste time! Every second of every day is valuable...
I can't wait to see you in the course!
- Anyone who wants to learn Particle Swarm Optimization Method.
- Anyone who wants to learn Deap library.
- Learners who want to work in data science and big data field
- students who want to learn machine learning
- Data analyser, Researcher, Engineers and Post Graduate Students need accurate and fast regression method.
- Modelers, Statisticians, Analysts and Analytic Professional.
What will you learn
- 1 Lessons
- 00:00:00 Hours


Research Centers of the Institute
Data Analysis Center
Our aim is to promote empirical economics and data science through three main activities. Firstly, we provide support involving software, statistical and econometric methodology for academics and students. Additionally, we organize competitions for secondary school students who are about to enroll in higher education. Thirdly, we organize professional lectures to promote the subject.
Faculty of the Data Analysis Center
Insurance Education and Research Group
The actuarial training organized by the Insurance Education and Research Group is advertised as an accredited program at the postgraduate level, for which you can apply with a bachelor’s degree in mathematics or economics. The content of the program harmonizes with the actuarial specialization of the master’s degree in Insurance and Financial Mathematics shared with ELTE in the full-time training, which is organized jointly with the Department of Operations Research and Actuarial Sciences.
Corvinus Center for Operations Research (CCOR)
The Corvinus Center for Operations Research (CCOR) was established on September 1, 2020 with the aim of investigating relevant areas of operations research (continuous optimization theory, algorithms and applications; game theory; decision theory and applications; operations research models and their applications). The five staff members of the Center have split responsibilities: carrying out teaching activities in the Institute of Operations and Decision Sciences and conducting research of international quality in the Corvinus Center for Operations Research.
Head of the Research Center
C épület, 708


How much difference is the Pupil Premium cash making for secondary school pupils? Not a lot so far if the latest DfE Statistics on GCSE outcomes are right. https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/280689/SFR05_2014_Text_FINAL.pdf On the wider measure, the attainment gap for the percentage achieving 5 or more GCSEs at grade A* to C or equivalent has narrowed by 8.0 percentage points between 2008/09 and 2012/13, with 69.3 per cent of pupils eligible for FSM achieving this indicator in 2012/13, compared with 85.3 per cent of all other pupils. However, the attainment gap between the percentage achieving 5 or more GCSEs at grade A* to C or equivalent including English and mathematics has narrowed by only 1.0 percentage point between 2008/09 and 2012/13 with 37.9 per cent of pupils known to be eligible for FSM achieving this indicator compared with 64.6 per cent of all other pupils. White boys are still faring badly, with just27.9% gaining the key 5A*-Cs measure including English and Mathematics. This compares with 43.1 of Black boys with similar characteristics, and 39.6% of black Caribbean boys with FSM.
Girls continue to outperform boys at all the main attainment indicators at key stage 4. The gap between the percentage of girls and boys making expected progress in English is 12.4 percentage points. This gap has narrowed slightly by 0.6 percentage points since 2011/12. The gap between the percentage of girls and boys making expected progress in mathematics is narrower than for expected progress in English at 4.7 percentage points, which has remained broadly the same since 2011/12.
All ethnic groups have made progress between 2008/98 and 2012/13in terms of the percentage of pupils obtaining 5A*-Cs including English and mathematics, although the Traveller of Irish Heritage, gypsy and Roma group still remain a long way adrift of other groups despite the small improvement in their performance on this measure. Pupils from a black background remain among the lowest performing groups, although they have shown the largest improvement. The percentage of black pupils achieving 5 or more GCSEs at grade A* to C or equivalent including English and mathematics GCSEs or iGCSEs is 2.5 percentage points below the national average. This gap has narrowed by 1.7 percentage points since 2011/12 but over the longer term has narrowed by 3.7 percentage points since 2008/09.
Outcomes for pupils with SEN remain disappointing. The attainment gap between the percentage of pupils with and without any identified SEN achieving 5 or more GCSEs at grade A* to C or equivalent including English and mathematics GCSEs or iGCSEs is 47.0 percentage points – 70.4 per cent of pupils with no identified SEN achieved this compared with 23.4 per cent of pupils with SEN. This gap has widened by 2.1 percentage points since 2008/09 but has remained broadly unchanged since 2011/12. A lower percentage of pupils with SEN made expected progress in both English and mathematics. The gap is wider for mathematics at 37.0 percentage points, compared to a gap of 30.9 percentage points for English. Both gaps have widened slightly between 2011/12 and 2012/13 (by 0.6 percentage points for mathematics and 0.7 percentage points for English).
On these measures there is still much to be achieved with the target groups. It is to be hoped that by increasing the level of the Pupil Premium more for primary pupils than their secondary compatriots fewer children will enter the secondary phase of schooling unable to access the teaching made available to them through a lack of the basic skills.


I:C stands for Insulin-to-Carbohydrate ratio, which is a mathematical formula used by people with diabetes who take insulin to calculate the appropriate insulin dose based on the amount of carbohydrates they plan to eat. The ratio represents the number of grams of carbohydrates that one unit of insulin can cover. For example, if someone has an I:C ratio of 1:10, it means that one unit of insulin is needed for every 10 grams of carbohydrates consumed. This helps people with diabetes to more accurately calculate their insulin doses and manage their blood sugar levels. The I:C ratio is determined by a healthcare provider and can be adjusted based on individual insulin sensitivity and carbohydrate intake.