20 Common Mathematics Topics ExplainedSuggested by SMS
The following twenty mathematics topics are discussed in terms of the areas that each might study and the nature of the mathematics involved. As mathematics is a constantly developing and expanding area of study, it is unrealistic to consider that this collection covers every significant area of mathematics and application of mathematics.
It is also less than ideal to list these topics in progressive difficulty as individual talent and interest will make some studies easier for one person than another. However, the hope is that they will develop in understanding and knowledge and that they will provide a guide to the breadth and brilliance of mathematics.
In preparation for a study of Algebra, it is important to have a solid grounding in concepts and applications that will provide the basis for further study. This includes a range of relatively simple concepts and conventions including, Absolute Value, Base Number System, Fractions and the Real Numbers.
Our first experience of Mathematics is usually through Arithmetic, but that’s not to say that they are synonymous. In fact, the ability to perform arithmetic has little bearing on the ability to understand and apply mathematical concepts. It is handy and certainly speeds up the process, but there are numerous examples of brilliant mathematicians and scientists that struggled with Arithmetic.
Arithmetic is largely centered on the ability to use the basic operations of addition, subtraction, multiplication and division. Much of this is done through memory rather than a process, but as the numbers become larger, most of us need to revert to pencil and paper or at least taking time to think through the task.
Furthermore, the place of Arithmetic Laws, such as the Commutative Law, Associative Law and Distributive Law, plays a significant role in developing an understanding of processes including factorization and simple algebraic operations.
Similarly, a working knowledge of the Cartesian Plane is essential to further study. The Cartesian Plane is formed by the intersection of two axes, the horizontal labeled “x” and the vertical labeled “y”. The point of intersection is called the origin and has a value of zero on each axis. From the origin the axis are graduated with positive numbers to the right and up and negative numbers to the left and down.
Points on the plane are located at the intersection of grid lines drawn from each axis. The point (2,3) is found vertically in line with the number 2 on the x-axis and horizontally in line with the number 3 on the y-axis.
This is the basis for visual representation of functions, vectors and the manipulation of figures through transformations.
Within a Pre-Algebra study there should also be a reference to the investigation of Pythagoras Theorem. Pythagoras Theorem works through the agency of right angle triangles. To apply it fully, we need to identify that the side opposite the right angle, the longest side, is the hypotenuse. With this understood the Theorem states that the sum of the squares of the two shorter sides will equal the square of the hypotenuse.
By manipulating the Theorem, it is possible to determine the value of the third side of the triangle when provided with any two other sides. When given the hypotenuse and one other side the Theorem can become:
c2 – a2 = b2
Within the study of Pythagoras Theorem, a number of triads can be found. These include, (3,4,5), (5,12,13) and (7,24,25) and they are sets of side lengths that will always form a right angle triangle. Even when they are increased by scale factors, they maintain their form, such as, (6, 8, and 10) and (15, 36, 39).
Mathematics is basically about recognizing patterns. To represent these patterns and the ideas that grow from them a system of numbers, pronumerals and conventions was devised, which is known commonly as Algebra.
This includes applications that focus on functions of degrees greater than one. The degree of a function is determined by the highest index of a variable in the function. For example, a quadratic function is a second degree function as it is expressed as f(x) = ax2 + bx + c. The exploration of these functions includes their factorization into roots.
Algebra also addresses the concepts and conventions of Set Theory. These investigate the principles of universality – all possible events, subset – sets within sets, intersection and union.
Other relevant concepts include infinity and the places geometric and arithmetic mean can play in modeling real-life events.
Geometry is the study of shapes, angles and where things really are in relation to other things. When a two-year-old is working out which block goes through which hole in the shape bucket, they have begun the study of geometry.
Through a process of tests and definitions, we are then able to identify and classify the variety of shapes through the lengths of their sides and the angles of their vertices. Such investigation leads us to recognize patterns in the angles that lines make when they intersect and how we can apply these in areas such as architecture, navigation and art.
Geometry also provides a means for discussing spatial awareness. The use of direction and distance allows us to locate items and, more importantly, it allows us to discuss their position and movement in an explicit and measurable way.
The study of transformations allows us to discuss other movements of objects as well, including reflection and rotation. We can also distinguish changes in shape through dilation, that is, enlargement or reduction.
Geometry can be as concrete as the square peg having to go into the square hole or as abstract as an imaginary plane where objects are manipulated through space and time.
Within the study of Space, a major element is the topic of Polygons. A Polygon is a two-dimensional closed figure consisting of straight lines only. Traditionally we think of hexagons, pentagons and even squares as Polygons, but the definition allows for more.
The neat evenly drawn shapes that are often examples of Polygons are actually regular Polygons. These are shapes with the same length sides and the same size angles. However, irregular Polygons are very common in real-world applications.
Beyond the concrete examples of Geometry, there is also a theoretical, more abstract, approach that allows for the study of hypercubes and hyperplanes, where traditionally geometric figures are given the scope to grow beyond their traditional dimensional conventions.
Trigonometry works from the relationships between the sides of a right-angle triangle.
The side of the triangle that isn’t connected to the right angle is called the Hypotenuse. This is the longest side of the triangle.
With that in mind, decide which vertex, corner, you are looking from. This is your orientation.
The side of the triangle that joins this vertex is called the Adjacent side.
The side that doesn’t join this vertex is called the Opposite side.
Once these sides are identified, they can be used to find unknown lengths and angles of the triangle. This is because, regardless of the length of each side, the ratio of the side lengths will always be the same for a given angle.
For example, if the vertex forms a 30° then the ratio of the Opposite side and the Hypotenuse will always be 1:2. Whether the sides are 3 and 6 inches, respectively, or 300 and 600 miles, at that angle the ratio will always be the same. Using the principles of Similar Triangles these properties can be applied to any size right-angle triangle.
The basic Trigonometric relationships are
Opposite Adjacent Opposite Sine θ° = ---------- Cosine θ° = ------------ Tangent θ° = ----------- Hypotenuse Hypotenuse Adjacent
The origin of the values attributed to each relationship is founded on the construction of a Unit Circle, that is, a circle with a radius of 1 and is centered on the origin of a Cartesian plane. As this radius is rotated around the circle, its height above the x-axis can be read as the y-coordinate and in the Sine relationship this draws the y-coordinate into ratio with the hypotenuse, the radius, of 1.
Similarly, the horizontal distance from the origin to the intersecting point of the circumference of the unit circle and the radius is the x-coordinate. This value placed in relationship with the radius, 1, gives the Cosine value.
The same values are also able to be used to determine the Tangent relationship.
Further application of the geometric properties of circles and triangles allows for greater exploration of Double and Half-Angle formulae and Addition formulae.
In preparation for the study of Calculus, there a number of elementary concepts and operations that must be understood. These provide the basis of calculations and descriptions and allow for creative and comprehensive discussions of Calculus.
Among these are Transformations of objects in space, including, Translation, Rotation and Reflection, and the use of coordinate systems other than the Cartesian, such as Polar and Spherical Coordinates. Matrix operation including the use of the Determinant, Scalar Matrices and Rotation Matrices are also important.
Furthermore, the ability to use and manipulate complex numbers, exponents and logarithms should be studied and placed in the context of functions. Exploration should look further at the way these functions are handled in calculus, as well as investigating their representation through the study of Conic Sections.
Conic Sections are the shapes created by taking cross-sections at varying angles through a solid cone. They include parabolas, through a vertical cross-section, ellipses, through an angled cross-section, and circles, through a horizontal cross-section.
Investigations into the gradient, or slope, of a graph are often drawn to the change in one variable in relation to the change in another. The subsequent discussion on the rate of change can far-reaching implications and the Mathematics behind it has a variety of applications. When such investigations move from the confines of straight line graphs to include the gradient of curves, then we enter the study of Calculus.
To determine the gradient of a curve, an approximation can be made. Using the same principle of change in one variable being placed in ratio to the change in the other, a rough idea can be gained. In order to improve the accuracy, the approximation is reduced to smaller and smaller intervals until they reach a limit that is infinitely close to, but not equal to, zero. At its smallest possible interval, the gradient is virtually a tangent to the curve, that is, a line that touches only at one point.
Being able to determine the gradient of a curve at any given point means that we are able to find the points at which the curve reaches its maximum value, minimum value or pauses, inflection points, in its progression.
The use of Calculus to model real-world motion allows us to discuss a range of “rate-of-change” scenarios. This makes it applicable in area from fluid dynamics to nanotechnology and has even been used to maximize investment portfolios.
The derivative of a formula provides us with the equation of the gradient to the curve. In its most simple form this is done by multiplying each term by the degree of the index of its pronumeral and reducing that index by 1.
y = x2 + 2x + 1 can be written with the indices y = x2 + 2×1 + 1×0
So dy = 2×1 + 1 x 2×0 + 0 x 1
dx = 2x + 2
If we allow this equation to equal 0, we can then determine the x-value of the turning point of the curve:
= 2x + 2
x = -1
Substituting this value back into the original equation provides us with the y-value:
Y = (-1)2 + 2(-1) + 1 = 1 – 2 + 1 = 0
The turning point for this curve is at (-1, 0).
More sophisticated curves and applications require more complex differentiation, but the use of Calculus to help map and model mathematical ideas is invaluable.
7. The History of Mathematics
In the words of Marcus Garvey, “A people without the knowledge of their past history, origin and culture is like a tree without roots.” While it may seem that a study of the History of Mathematics would be better suited to a Humanities program, it must be understood that such a study is deeply relevant to the understanding and future of mathematical study.
Whether it is to study the profound contributions of the Pythagoreans or to investigate the formalization of areas such Topology, the process and influences that encouraged and engendered the development of mathematics can inspire modern thinkers. Even exploring the social and political influences that brought about great advances can have a bearing on the directions and progress of mathematics today.
The origin of calculus may seem an insignificant topic against the continuing development and innovations associated with it, but the reality of the its power to describe the motion of planets holds profound significance. In the same regard, the advancement of algebra from simple roots to become the language of mathematics is fundamental to the creation of new system and understandings.
The study of history has benefits in all areas of endeavor. Perhaps it is even more relevant in the study of mathematics than most other spheres of pursuit.
8. Number Theory
The study of numbers as integers, fractions and groupings defined by their properties is a delving into the nature of the building blocks of mathematics. Scientifically, numbers can be classified and studied for their properties, similarities and individual features.
Concepts, such as prime numbers, perfect numbers and transcendental numbers, provide a structure for the continuing study of numbers beyond, real, rational and complex. This can include the development of theorems around the use and manipulation of these numbers, such as the Prime Number Theorem and the Prime Factorization Algorithm.
The study of Number Theory calls on a knowledge of concepts such as congruency and convergence, so as to discuss the characteristics of numbers and the functions through which they operate.
9. Finite Mathematics
The study of Finite Mathematics is relevant to students of psychology and sociology. Largely it is focussed on the theory of probability and matrices.
Probability furnishes the mathematical foundations for statistics. Although this is a theoretical area of mathematics, the applications used in the study of Finite Mathematics make it a more real-world specific topic. This extends into the investigation of union and intersection, while also considering mutually exclusive and disjoint events.
The use of matrices to assist in the exploration of probability includes knowledge of matrix multiplication and the applications. It also introduces the use of matrices for linear systems and the properties of inverse matrices.
These two areas of mathematics come together in the study of Finite Mathematics creating a practical and successful form of mathematics for studies that have specific needs.
10. Theory of Computation
The place of devices, mechanical and electronic, to aid calculation in mathematics is an ever developing and expanding field. Similarly, the use of software, ranging from spreadsheets to more sophisticated calculation programs, is constantly undergoing innovation and evolution.
This is probably ideally illustrated by the annual “Record-breaking” effort of university students somewhere in the world setting some poor computer to work to generate π to ten million more decimal places than the students of the year before. One can only imagine the computer sighing loudly and remarking, “If I have to” as it hacks out another list of non-recurring decimals like a cat bringing up a hairball.
Thankfully, there are multitudes of more constructive and purposeful applications of computation devices and principles than this. Whether through rudimentary processes or through the use of Cellular Automaton or the Turing Machine, the study of the Theory of Computation invites and encourages endeavors to improve our understanding of the nature of computational operations.
At the foundation of Topology is the concept of congruency. That is, the ability to say that a shape, argument or set of elements is identical in all ways to another. In Topology, two objects are considered the same if they can be distorted to be congruent without causing change to the properties of each. Although the basis of this branch of Mathematics had been discussed and developed for centuries, the title Topology only became the over-arching name for it near the end of the nineteenth century.
In fact, the study of Topology is often described as an investigations of the properties of an object that remain sound despite the distortion, twisting and stretching of the object. In this regard the discussion of manifolds, knots and dimensions can be held with a recognition that the fundamental topological aspects will be preserved.
Through the study of Topology and the manipulation of objects and shapes, the Mobius Strip can be formed and defined successfully. Similarly, the projective plane and the projective space, both of which involve the intersection of a set of lines through the origin are able to be discussed and explored.
Set-point Topology is more abstract and presents possibilities for further investigations into concepts such as continuity, dimension, compactness and connectedness. With these ideas in mind, the Topological Space becomes recognizable as a set with a collection of subsets that satisfy the definitions of the set.
12. Linear Algebra
Moving beyond the high school understanding of linearity being related to y = mx + c, the study of Linear Algebra invites the use of vectors, scalars and matrices to address the systems of linear functions and their transformations.
This includes the defining and application of vector spaces and the use of vector addition and scalar multiplication. Vectors are largely lines with magnitude, size, and direction. In areas such as Physics, this translates to speed and direction, producing velocity vectors. Using the fundamental operations of addition and subtraction, these vectors can be used to solve problems including exercises in the study forces.
Scalars are effectively real numbers used in the study of Linear Algebra. They can be used to multiply vectors and create new vectors within the same finite vector space.
The use of matrices as a means of representing linear transformations provides a unique representation of the transformation. Using methods of addition, subtraction and multiplication, both scalar and matrix, matrices provide a purposeful and skillful method for manipulating and problem solving in Linear Algebra.
13. Probability and Statistics
According to Benjamin Disraeli, a 19th Century Prime Minister of Britain, there are “Lies, Damned Lies and Statistics”. This is a fair assessment of the use of statistics and how they can be manipulated, but the mathematics of statistics is far less malleable.
Statistics is an area of mathematics that organizes information, commonly called data, and uses these structures to study trends. Most people will recognize terms such as average and range and these present the first ideas of “Where most of the data lies” and “How wide the data is spread”.
However, it takes only a brief consideration of the data to see that there are different types of average that we can discuss. The three most common are the:
Mean = Sum of the data / The Number of pieces of data (This is also called the Number of the data)
Mode = The most frequently occurring response.
Median = The middle number when all the data is organized from lowest to highest.
The Range of the data is the difference between the highest and lowest value of the data.
The trick to reading statistics is to look as closely at what is not being highlighted as you do at what you are being directed towards. Celebrating the success of two thirds of the class passing a test, may avoid the fact that three quarters of the class didn’t score more than 60%.
More sophisticated applications of statistics include the deriving of standard deviation and the construction of normal distribution, uniform distribution and binomial distribution. From this work, further study into poisson distribution and covariance follow as a natural progression.
Statistics are great for making a case, but need to be handled with care.
One of the most contentious areas of Mathematics is Probabilities. This largely due to the fact that the results of the theoretical calculations of Probabilities are so readily shown as inaccurate. The topic often loses credibility, its proponents lose faith in the early stages and it can take a significant amount of explanation and goodwill to pull it back into the realm of “real” mathematics.
The central rule to Probabilities is the number of favorable outcomes divided by the number of possible outcomes. In this context favorable doesn’t always mean that the outcome is pleasant or good, it simply means that it is the one being sought. No-one considers an earthquake favorable, but if you’re determining the probability of an earthquake occurring then the data supporting the event is treated as the “favorable outcome”.
More complex Probability calculations are the result of events being dependent on each other and combinations of events occurring that have no determining connection, often described as Mutually Exclusive and Conditional Probability.
14. Discrete Mathematics
The distinction between Discrete and Continuous data is important in terms of the processes and operations that can be applied. By definition, Discrete data is that which can only be discussed in whole or specific units, while continuous has the capacity to be graduate along a number line.
As examples, the number of students in a class must always be treated as Discrete data, that is you can’t have half a student. Whereas the height of the students can, if the measuring apparatus is capable, be taken to such extent that it may consist of numerous decimal places.
So, when we start to discuss Discrete Mathematics we are talking in terms of units that must stand separate from each other. Through this limitation on the values to be used, the study of algorithms, logic and binary systems can be introduced and explored.
Binary systems are limited to working with a base of 2, that is, they have only two numbers 0 and 1. Using this structure, numbers are expressed in terms 0 and 1, for example, 2 is 10. That is, 0 is 0, 1 is 1, but instead of going to a third number, the system now moves into double digits, so 2 becomes 10.
The table below shows the binary code for numbers 0 to 10.
Base 10 Base 2 Base 10 Base 2 0 0 6 110 1 1 7 111 2 10 8 1000 3 11 9 1001 4 100 10 1010 5 101 35 100011
It may be worth noting that, as binary system in base 2, powers of two become comparable powers of “ten”.
The restriction of data to Discrete forms also makes it possible to investigate areas of Combinatory, that is, the study of Permutations and Combinations. This allows us to investigate the structure and possibilities of a range of outcomes. While both are focused on the grouping of elements, the distinction between them is found in the understanding that for Permutations the order of the elements is important, while for Combinations it is not.
Calculations of Permutations and Combinations rely on the Factorial function. Indicated by the inclusion of an exclamation mark after the number, the Factorial of a number is the product of all numbers below it to 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
When calculating the Permutations of “r” elements selected from a set of “n” possible items the first thing to consider is whether or not repetition is allowed. If it is the answer is nr, that is, n x n x n … r times.
However, if repetitions are not allowed, then the number of possible items decreases with each selection. So the answer will be n x (n-1) x (n-2) … r times. This is best illustrated as n!. But to stop the process after r terms, we must also divide by the terms after r.
As an example of this consider a group of twelve students from which only three are to be chosen. The possible Permutations are given by:
12 x 11 x 10 x 9 x 8 … ------------------------ = 12 x 11 x 10 = 1320 9 x 8 x 7 …
The rule for this calculation is
n! ------ (n-r)!
When addressing Combinations the removal of the constraint of order calls for additional calculations. Largely this means reducing the formula by how many ways the objects could be ordered, so the rule for determining the Combinations possible of “r” elements selected from a set of “n” possible items without repetition is:
n! ------- r!(n-r)!
In the course of such a study, aspects of Fibonacci’s Sequence, Pascal’s Triangle and Magic Squares can be explored within the context of Discrete Mathematics.
15. Graph Theory
The area of Mathematics called Graph Theory is a significant and expansive aspect of the study of Space. It includes tangible and real-world elements, while also being rich with abstract and imaginative ideas. Although the basis of this branch of Mathematics had been discussed and developed for centuries, the first published work on the topic was only written in 1736.
This was in response to Euler’s “Seven Bridges of Königsberg”. The bridges connected two islands in the Pregel River with each other and the mainland. Euler set a challenge to discover a path that crossed all bridges only once. To simplify the problem, Euler created a topological map of the river, islands and bridges. While the map was far from an exact scale drawing of the problem, it contained all the necessary and meaningful information.
Through this exercise, Euler created the idea of Graph Theory to model real-world situations. This included, and has extended, to define the dots and lines of a diagram as vertices and edges. Other concepts were then able to be classified and discussed, such as the degree of a vertex, the number of edges attached to it, a path, the course taken connecting vertices, and a cycle, a path that ends where it began.
Graph Theory has virtually nothing to do with the display of data on a chart and graph paper has little special purpose in the study of Graph Theory. This is because Graph Theory holds a greater potential to explore the abstract and the creative side of pure mathematics.
16. Abstract Algebra
The more that we delve into the structures and nature of Algebra, the further we can move into the theoretical aspects. This is not to suggest that Abstract Algebra lacks a practical context, rather the abstraction of the ideas actually makes it more inclusive of a range of creative and diverse application.
This is evident in the exploration of a range of Algebras, including Boolean Algebra and Lie Algebra. These use the fundamentals of Algebra combined with principles of logic and additional conditions.
Abstract Algebra can also take into account the study of Group Theory. Through Group Theory we can investigate laws such as the Associative Law – a + (b + c) = (a + b) + c, the Identity property – a x 1 = a and the Inverse Law – a = √a2 . It can also present ideas regarding a variety of forms of Groups including Abelian, Cyclic, Finite and Sub Groups.
17. Calculus and Convergence
Building on the understanding and competencies attained in Calculus, a more advanced calculus program would discuss further rules for the application of calculus. These could include the Chain Rule, by which a function can be differentiated using an intermediary function, the Quotient Rule, through which functions in a fractional or divisor state can be differentiated, and the Product Rule, through which functions being multiplied together can be differentiated.
The concepts of Convergence and Divergence are necessary to the continuing study of calculus. By applying Radius of Convergence as a test, it is possible to verify that a function is indeed converging.
The divergence of a function, that is, its continuing growth toward infinity or its oscillating nature that limits its ability to converge, can be tested by using the Integral Test.
The application and study of series including Taylor Series, Power Series, Maclaurin Series and Harmonic Series is able to be carried out in terms of convergence and practical use.
Other practical applications, such as determining Arc Length and the Surface of Revolution would also be extensions on the fundamental understanding of Calculus.
18. Differential Equations
The study of Differential Equations is an introduction to the application and purpose of calculus in the mathematics. Through the exploration of derivatives, both ordinary and partial, the uses of the Differential Equations can be placed in practical and real-world scenarios.
Differential Equations contain derivatives and the process of solving them involves integration, but can also use the separation of variables method. The resulting equation will usually be of a form y = … or f(x) = … .
Through investigation of Differential Equations, methods of solving physical problems including the Fourier and Laplace transformations and Bessel Functions are explored. These solutions can involve first and second order differential equations.
The order of a Differential Equation is determined by the level of derivatives present. The highest level of highest derivative in the Differential Equation, for example, a second order Differential Equation will have at least one second derivative; it may also include first derivatives and constants.
19. Differential Geometry
The study of Differential Geometry is often seen as a study of curvature and distances, on surfaces. This can also be seen as development from the work on Surface of Revolution in Calculus.
Through Differential Geometry it is possible to investigate concepts like the Mean Curvature of a surface and the Gaussian Curvature. These study the amount of bend in a surface and curvature as a property of space.
In dealing with curvature, there is also scope to investigate the properties and characteristics of tangents. This reflects the initial intent of Calculus to define the gradient of a curve and can be extended into the development of Tangent Bundles, which are constructed from the tangent spaces of a series of points.
The use of scalars and vectors plays a role in this work as does that of matrices to represent them.
Analysis is the study of functions, real and continuous, so as to determine their features and characteristics. This can include the study of Delta Functions, Gamma Functions and Functional Analysis, which allows for the investigation of infinity vector spaces.
From generalized applications of Calculus to determine curves and stationary points, Analysis considers the use overlap of functions through Convolution and the use of measures to assist in the understanding of integration and differential equations. This can include the study of the Lebesgue Measure and the Cantor Set.
Through Analysis there is also study of Bernoulli Number and the Fourier Series. Through these the study of trigonometric functions is further explored within the realm of calculus and through the operations of Bernoulli’s Number and the Fourier Series.


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We were delighted to support the NSPCC by taking part in Number day on Friday 4th February.
Number day is a great way to make maths fun and bring about a positive 'can do' attitude towards the subject. Throughout the day, pupils completed a range of activities but they were also invited to Dress up for digits. Pupils were asked to wear an item of clothing with a number on it or get even more creative by dressing in a maths or number theme. Have a look at our photos to see the different things we did during the day. More information can be seen on our class Seesaw blogs.


I linear programming
I LINEAR PROGRAMMING
In a decision-making embroilment, model formulation is important because it represents the essence of business decision problem. The term formulation is used to mean the process of converting the verbal description and numerical data into mathematical expressions which represents the relevant relationship among decision factors, objectives and restrictions on the use of resources. Linear Programming (LP) is a particular type of technique used for economic allocation of ‘scarce’ or ‘limited’ resources, such as labour, material, machine, time, warehouse space, capital, energy, etc. to several competing activities, such as products, services, jobs, new equipment, projects, etc. on the basis of a given criterion of optimally. The phrase scarce resources mean resources that are not in unlimited in availability during the planning period. The criterion of optimality generally is either performance, return on investment, profit, cost, utility, time, distance, etc.
George B Dantzing while working with US Air Force during World War II, developed this technique, primarily for solving military logistics problems. But now, it is being used extensively in all functional areas of management, hospitals, airlines, agriculture, military operations, oil refining, education, energy planning, pollution control, transportation planning and scheduling, research and development, etc. Even though these applications are diverse, all I.P models consist of certain common properties and assumptions. Before applying linear programming to a real-life decision problem, the decision-maker must be aware of all these properties and assumptions.
The word linear refers to linear relationship among variables in a model. Thus, a given change in one variable will always cause a resulting proportional change in another variable. For example, doubling the investment on a certain project will exactly double the rate of the return. The word programming refers to modelling and solving a problem mathematically that involves the economic allocation of limited resources by choosing a particular course of action or strategy among various alternative strategies to achieve the desired objective.
STRUCTURE OF LINEAR PROGRAMMING
General Structure of LP Model
The general structure of LP model consists of three components.
- Decision variables (activities): We need to evaluate various alternatives (courses of action) for arriving at the optimal value of objective function. Obviously, if there are no alternatives to select from, we would not need LP. The evaluation of various alternatives is guided by the nature of objective function and availability of resources. For this, we pursue certain activities usually denoted by x1, x2…xn. The value of these activities represents the extent to which each of these is performed. For example, in a product-mix manufacturing, the management may use LP to decide how many units of each of the product to manufacture by using its limited resources such as personnel, machinery, money, material, etc.
- The objective function: The objective function of each L.P problem is a mathematical representation of the objective in terms of a measurable quantity such as profit, cost, revenue, distance, etc. In its general form, it is represented as:
- The constraints: There are always certain limitations (or constraints) on the use of resources, e.g. labour, machine, raw material, space, money, etc. that limit the degree to which objective can be achieved. Such constraints must be expressed as linear equalities or inequalities in terms of decision variables. The solution of an L.P model must satisfy these constraints. The linear programming method is a technique for choosing the best alternative from a set of feasible alternatives, in situations in which the objective function as well as the constraints can be expressed as linear mathematical functions.
These activities are also known as decision variables because they arc under the decision maker’s control. These decision variables, usually interrelated in terms of consumption of limited resources, require simultaneous solutions. All decision variables are continuous, controllable and non-negative. That is, x1>0, x2>0, ….xn>0.
Optimise (Maximise or Minimise) Z = c1x1 + c2X2. … cnxn
Where Z is the measure-of-performance variable, which is a function of x1, x2 …, xn. Quantities c1, c2…cn are parameters that represent the contribution of a unit of the respective variable x1, x2…, xn to the measure-of-performance Z. The optimal value of the given objective function is obtained by the graphical method or simplex method.
APPLICATION AREAS OF LINEAR PROGRAMMING
Linear programming is the most widely used technique of decision-making in business and Industry and in various other fields. In this section, we will discuss a few of the broad application areas of linear programming.
These applications fall into categories of farm economics and farm management. The former deals with agricultural economy of a nation or region, while the latter is concerned with the problems of the individual farm.
The study of farm economics deals with inter-regional competition and optimum allocation of crop production. Efficient production patterns can be specified by a linear programming model under regional land resources and national demand constraints.
Linear programming can be applied in agricultural planning, e.g. allocation of limited resources such as acreage, labour, water supply and working capital, etc. in a way so as to maximise net revenue.
Military applications include the problem of selecting an air weapon system against enemy so as to keep them pinned down and at the same time minimising the amount of aviation gasoline used. A variation of the transportation problem that maximises the total tonnage of bombs dropped on a set of targets and the problem of community defence against disaster, the solution of which yields the number of defence units that should be used in a given attack in order to provide the required level of protection at the lowest possible cost.
- Product mix: A company can produce several different products, each of which requires the use of limited production resources. In such cases, it is essential to determine the quantity of each product to be produced knowing its marginal contribution and amount of available resource used by it. The objective is to maximise the total contribution, subject to all constraints.
- Production planning: This deals with the determination of minimum cost production plan over planning period of an item with a fluctuating demand, considering the initial number of units in inventory, production capacity, constraints on production, manpower and all relevant cost factors. The objective is to minimise total operation costs.
- Assembly-line balancing: This problem is likely to arise when an item can be made by assembling different components. The process of assembling requires some specified sequence(s). The objective is to minimise the total elapse time.
- Blending problems: These problems arise when a product can be made from a variety of available raw materials, each of which has a particular composition and price. The objective here is to determine the minimum cost blend, subject to availability of the raw materials, and minimum and maximum constraints on certain product constituents.
- Trim loss When an item is made to a standard size (e.g. glass, paper sheet), the problem that arises is to determine which combination of requirements should be produced from standard materials in order to minimise the trim loss.
- Portfolio selection: This deals with the selection of specific investment activity among several other activities. The objective is to find the allocation which maximises the total expected return or minimises risk under certain limitations.
- Profit planning: This deal with the maximisation of the profit margin from investment in plant facilities and equipment, cash in hand and inventory.
- Media selection: Linear programming technique helps in determining the advertising media mix so as to maximise the effective exposure, subject to limitation of budget, specified exposure rates to different market segments, specified minimum and maximum number of advertisements in various media. (if) Travelling salesman problem The problem of salesman is to find the shortest route from a given city, visiting each of the specified cities and then returning to the original point of departure, provided no city shall be visited twice during the tour. Such type of problems can be solved with the help of the modified assignment technique.
- Physical distribution: Linear programming determines the most economic and efficient manner of locating manufacturing plants and distribution centres for physical distribution.
- Staffing problem: Linear programming is used to allocate optimum manpower to a particular job so as to minimise the total overtime cost or total manpower.
- Determination of equitable salaries: Linear programming technique has been used in determining equitable salaries and sales incentives.
- Job evaluation and selection: Selection of suitable person for a specified job and evaluation of job in organisations has been done with the help of linear programming technique.
Other applications of linear programming lie in the area of administration, education, fleet utilisation, awarding contracts, hospital administration and capital budgeting.
ADVANTAGES OF LINEAR PROGRAMMING
Following are certain advantages of linear programming:
- Linear programming helps in attaining the optimum use of productive resources. It also indicates how a decision-maker can employ his productive factors effectively by selecting and distributing (allocating) these resources.
- Linear programming techniques improve the quality of decisions. The decision-making approach of the user of this technique becomes more objective and less subjective.
- Linear programming techniques provide possible and practical solutions since there might be other constraints operating outside the problem which must be taken into account. Just because we can produce so many units docs not mean that they can be sold. Thus, necessary modification of its mathematical solution is required for the sake of convenience to the decision-maker.
- Highlighting of bottlenecks in the production processes is the most significant advantage of this technique. For example, when a bottleneck occurs, some machines cannot meet demand while other remains idle for some of the time.
- Linear programming also helps in re-evaluation of a basic plan for changing conditions. If conditions change when the plan is partly carried out, they can be determined so as to adjust the remainder of the plan for best results.
LIMITATIONS OF LINEAR PROGRAMMING
- There should be an objective which should be clearly identifiable and measurable in quantitative terms. It could be, for example, maximisation of sales, of profit, minimisation of cost, and so on, which is not possible in real life.
- The activities to be included should be distinctly identifiable and measurable in quantitative terms, for instance, the products included in a production planning problem and all the activities can’t be measured in quantitative terms for example if labour is sick, which will decrease his performance which can’t be measured.
- The resources of the system which arc to be allocated for the attainment of the goal should also be identifiable and measurable quantitatively. They must be in limited supply. The technique would involve allocation of these resources in a manner that would trade off the returns on the investment of the resources for the attainment of the objective.
- The relationships representing the objective as also the resource limitation considerations, represented by the objective function and the constraint equations or inequalities, respectively must be linear in nature, which is not possible.
- There should be a series of feasible alternative courses of action available to the decision makers, which are determined by the resource constraints.
- While solving an LP model, there is no guarantee that we will get integer valued solutions.
- Linear programming model does not take into consideration the effect of time and uncertainty. Thus, the LP model should be defined in such a way that any change due to internal as well as external factors can be incorporated.
- Sometimes large-scale problems can be solved with linear programming techniques even when assistance of computer is available. For it, the main problem can be fragmented into several small problems and solving each one separately.
- Parameters appearing in the model are assumed to be constant but in real-life situations, they are frequently neither known nor constant.
- Parameters like human behaviour, weather conditions, stress of employees, demotivated employee can’t be taken into account which can adversely effect any organisation
- Only one single objective is dealt with while in real life situations, problems come with multi-objectives.
When these stated conditions are satisfied in a given situation, the problem can be expressed in algebraic form, called the Linear Programming Problem (LPP) and then solved for optimal decision.
For example, in finding out how many men and machines would be required lo perform a particular job, a non-integer valued solution will be meaningless. Rounding off the solution to the nearest integer will not yield an optimal solution. In such cases, integer programming is used to ensure integer value to the decision variables.
II SITUATION ANALYSIS
Phang furniture system Inc. (Fursys) manufactures two models of stools, Potty which is basic model and a better model called Hardy.
Maximum of 350 pounds plastic per day at the rate of $1.5 per pound by Keow supplies Up to 30 boxes of legs per day at the rate of $7.5 per box. Each box has 10 sets of legs by Yuen supplies Using linear programming the optimal production should be determined for maximum profit.
The production units are in terms of number on daily basis. Therefore the decision variables are:
Let, X1 = No. of Potty’s production daily
X2 = No. of Hardy’s production daily
The objective in the problem is to attain maximum profit. We have selling price for Potty and Hardy as $12.75 and $18. We need to calculate the unit profit gained by selling Potty and Hardy.
Cost of production for 1 Potty = one pound plastic + one set of leg
= ($1.5*1) + $0.75(1)
Profit made by selling = $12.75 – $2.25 = $10.5
Cost of production for 1 Hardy = 1.5 pound of plastic + one set of leg
= ($1.5*1.5) + ($0.75*1)
Unit profit made by selling Hardy = $18 – $3 = $15
Potty requires one pound of plastic and Hardy requires 1.5 pound plastic. So the total plastic used daily is:
(1)X1 + (1.5)X2
This plastic supply can’t exceed the limit of 350 pounds daily, so constraint is
(1)X1 + (1.5)X2 <= 350
Both the model require one set of each legs each for its production. So the sets of legs used daily is
(1)X1 + (1)X2
The no of set of legs can’t exceed the limit of 300, so the constraint is
(1)X1 + (1)X2 <= 300
Potty can be manufactured in 15 minutes and Hardy can be manufactured in 24minutes. So the total time taken for manufacturing both stools in order to achieve maximum profit is:
(15)X1 + (24)X2
The production time can’t exceed 80 hours(4800 minutes) on daily basis. Therefore the constraint is,
(15)X1 + (24)X2 <=4800
Negative production of Potty and Hardy stool is not possible. Therefore,
Maximize, 10.5X1 + 15X2 (total daily profit)
Subject to constraints, X1 + 1.5X2 <= 350(plastic in pound)
X1 + X2 <= 300(sets of legs)
15X1 + 24X2 <= 4800(production time in minutes)
Solution from winqsb
Assignment WINQSB output.bmp
According to WINQSB, when Potty produced(X1) = 266.67 and Hardy produced(X2) = 33.33, Fursys can get a maximum profit of 3,300. Therefore the optimum solution is
X1 = 266.67
X2 = 33.33
Fursys makes a maximum profit of $3300 per day. Its fixed cost, namely for overheads and family labour is about $2800 per day. Therefore,
Net income of Fursys is = Profit- Fixed cost
= $3300-$2800 = $500
Range of optimality
After achieving optimal solution, Fursys will be concerned about how the solution may be affected if any one of the objective function co-efficient is changed. Depending on the value of the objective function co-efficient the optimal solution may vary.
Assignment WINQSB output.bmp
From above table we can conclude that:
9.375 <= C1 (UNIT COST OF ONE POTTY) >= 15
10.500 <=C2 (UNIT COST OF ONE HARDY) >=16.800
The amount, the optimal profit will change per unit increase in the variable from its lower bound, while assuming there are no changes in the input parameters is called reduced costs. Reduces costs are usually zero.
Shadow price is the premium value above the existing unit value for the resource if the need arises to purchase more resources, which means slack or surplus is zero.
When there is a slack or surplus of resources there is no need to purchase more. Hence the shadow price is zero. In the above problem after one day of production, there is a surplus of 33.333 pounds of plastic, therefore there is no shadow price. But all sets of legs were used to manufacture stools and therefore the slack or surplus for sets of legs is zero. So it has a shadow price of $3. Now if Fursys wants to buy more sets of legs it has a perceives value to pay is
original price + shadow price = $0.75 + $3 = $3.75 per set of legs
Range of feasibility:
The range of feasibility is the range of values for which the shadow prices of resources remain unchanged, however optimal solution will change. When the amount or number of resources goes beyond the range, a new shadow price arises.
In this problem, when the number of legs go beyond 320, the value of the shadow price changes. Therefore, for the same shadow price, only 20 more sets of legs can be purchased. Likewise when we see for the plastics to be bought, the upper limit is infinity and since there’s already surplus of plastic, there’s no need to buy anymore.
Analysis of available solutions
Option1: Seeking additional source of plastic
As shown in figure from winqsb output that at the end of a day’s production there is a surplus of plastic 33.333 pounds. Since there is surplus of plastic, there is no need to look for additional sources of plastic.
However if the demand of the products increase and surplus is finished then Fursys can purchase additional plastic. From the range of feasibility we can see that the upper limit of the amount of plastic is infinity, therefore any amount of plastic can be purchased.
Option2: Taking up Yuen Supplies offer to deliver an extra cost of 10 sets of legs
From the WINQSB solution we can see that the maximum sets of legs the maximum no of set of legs can be purchased per day is 320. The current number of legs used per day is 300, so we can conclude that Fursys can buy 10 extra set of legs from Yuen supplies as it is under feasibility.
Option3: Adding a part time worker (4 hours a day) for $50 per day
Fursys considers its labour cost as sunk for business. By adding up an extra worker, the cost of worker will be considered as sunk cost only.
Adding up a worker will increase production time by 240 minutes per day, this lies within the limit of range of feasibility. Hence option of extra worker can be taken into account.
Application of 100% rule to evaluate option 2 and 3 can be implemented at the same time
100% rule is used to evaluate whether different options available for a company are feasible or not. It involves calculations of increase or decrease in an objective function coefficient to the maximum possible increase or decrease as determined by the limits of the range of optimality. Here we will consider option 2 and 3 for Fursys and will see if both options are feasible at the same time.
Option 2 considered first
When Fursys buy 10 extra set of legs then:
Total cost of legs = 300*0.75(for 300 legs) + 25(for extra 10 legs)
Cost of 310 legs = $250
Cost of 1 leg = 250/310 = $.80
When Fursys buys 300 legs then cost of each set of leg =$ 0.75
So there is an increase in price of legs by $.05 by buying 10 extra set of legs, since profit is inversely proportional to increase in cost price so profit decrease by $0.05 for both Potty and Hardy
Applying 100% rule
New cost price for Potty’s is = $10.45
Now we will calculate % change in profit
Formula is: percentage change = (change/maximum change) * 100
= (0.05/1.125) * 100
New cost price for Hardy’s is = $14.95
Similarly like potty we will calculate % change in profit for hardy
Formula is: percentage change = (change/maximum change) * 100
The additional worker works for 4 hours i.e. 240 minutes and will receive $50 as wages
Since Fursys considers labour cost as sunk cost so there will be no effect on cost of product
Now calculating % change for time constraint. Maximum permissible production time is 600 minutes. We will calculate % change in time
Formula is: percentage change = (change/maximum change) * 100
According to 100% rule, adding all percentage change we get 45.5% which is less then 100%, so option 2 and option 3 can be used together.


Blackjack Math - How To Use CVCX (Blackjack) (2018)
Blackjack Math � Understanding the Importance of Math in Blackjack Blackjack Math � Understanding the Importance of Math in Blackjack An Introduction to Blackjack Math If you think your winning at blackjack has all to do with luck and skill and nothing to do with math, think again; in blackjack math plays a vital role in determining whether you win or lose.
There is a strong connection between the concept of probability and math, and that becomes very apparent when you begin to use math when playing a game of blackjack.
To win at blackjack on a regular basis, besides knowing the rules in detail you also need to know some strategy.
The math blackjack card counting of math in blackjack becomes clear from one simple fact � the strategies in blackjack are all math-based.
More about Math in Blackjack � Basic Strategy To understand clearly the role of math in blackjack you will need to take a look at the strategies involved in blackjack.
Professional blackjack players, researchers, and people with a head for numbers have spent a lot of time researching for workable blackjack strategies.
There are basic strategies and more advanced strategies like card counting, both of which are dependent on math.
The first of the basic strategies came about more than half a century ago, in the previous millennium, thanks to three years worth of effort by Roger R.
Cantey, James McDermott, and Herbert Maisel, four officers of the army.
In all, there are calories pizza blackjack jackpot different possible combinations that can translate to blackjack decisions, all of which come under 30 different rules.
However, while this may seem math blackjack card counting, you must also know that learning just basic blackjack strategy is not enough always; it is a good idea to know other strategies like card counting as well.
Blackjack Math and Card Counting Blackjack math and card counting have a strong connection, because the latter would not make sense without the former.
Card source is an advanced technique that demonstrates truly the usage of math in blackjack.
Using this technique and a computer, experts including computers have been able to devise a card counting technique that works.
While the usage of math in blackjack math blackjack card counting crucial to winning, it is here the only math blackjack card counting that is important.
Even if you are an expert card counter, you may still lose money if you do not keep in mind the other seemingly lesser but equally important concepts in blackjack.
These include simple things like having the right-sized bankroll, the ratio between bets to bankroll, and others.
Can You Count Cards At Online Blackjack?
Card counting in Blackjack is a strategy which allows players to predict to some. card counting can be learned only by people with good mathematical skills is�...
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Here is a breakdown of the expected vocabulary for math that
we are teaching and that you can support at home.
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Probability distributions as models for mortalityAuthor(s): M.A. Buford; W.L. Hafley
Source: Forest Science
Publication Series: Scientific Journal (JRNL)
Station: Southern Forest Experiment Station
PDF: Download Publication (1.0 MB)
DescriptionThe necessary attributes for a mortality model for an even-aged forest stand are stated. The Weibull distribution, the gamma distribution, and the negative binomial distribution are proposed based on their previous use in failure research and as mortality models. A distribution derived from the Richards generalization of the von Bertalanffy growth equation is proposed. The four functions are examined mathematically and empirically using data from a loblolly pine spacing study to determine their usefulness as mortality models. The negative binomial distribution and its continuous analog, the gamma distribution, show instability under right-censoring and are computationally difficult. The Weibull distribution shows extreme instability under rightcensoring due to constraints on the location of the inflection points of its probability density function, limiting its value as a mortality model. The distribution derived from the Richards generalization of the von Bertalanffy function is stable under right-censoring, shows no constraints on assumable shapes, and is computationally simple.
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- We recommend that you also print this page and attach it to the printout of the article, to retain the full citation information.
- This article was written and prepared by U.S. Government employees on official time, and is therefore in the public domain.
CitationBuford, M.A.; Hafley, W.L. 1985. Probability distributions as models for mortality. Forest Science: 31(2) 331-341. 11 p.
Keywordsnegative binomial distribution, gamma distribution, weibull distribution, Richards function
- The Chapman-Richards Distribution and its Relationship to the Generalized Beta
- Individual tree basal-area growth parameter estimates for four models
- Modelling diameter distributions of two-cohort forest stands with various proportions of dominant species: a two-component mixture model approach.
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My husband and I recently rediscovered Yahtzee. I have always loved this game. Lots of strategy and math.
When it came time to add up my score, I got up to grab my cell phone to use the calculator. It dawned on me that I was missing an easy authentic math experience – for me. Rather than allow my brain to become rusty – I added up my score myself!
Authentic learning does not always have to involve a big project. The point of authentic learning is to make it real and make it count. Sometimes this involves a big project. Sometimes a short and sweet project. And sometimes just an authentic experience.
There are many wonderful math games to play with children that improve math skills. Just don’t forget that keeping track of and adding up scores are also fantastic authentic math experiences. This is also a great authentic way to teach estimation – before actually adding up the scores, do a quick estimation to see who probably won. And a great authentic way to teach checking your work – add of the score two times to make sure you get the same number both times.
By not grabbing that cell phone or calculator at the end of the game to add up the scores, you can double or triple the authentic math skills practiced and reinforced. And in case you are wondering – I won that round of Yahtzee! This doesn’t usually happen, so I think it is important to note this on my blog.
One thought on “Authentic Teaching – Math”
Very interesting to read. I am a Director of a school in western Kenya. The country is in the process of reforming its education. We have competency based curricular up to grade 3 this year. I am interested in project based learning. I’m learning and trying to get my teachers interested in project based learning. Any materials up to grade four in mathematics, English and science will go a long way to assist me.


Welcome to our in-depth guide on how to calculate an amortized loan. In the world of finance, understanding the intricacies of amortization is crucial, whether you’re a seasoned professional or a beginner navigating the complexities of loans.
In this comprehensive article, we’ll delve into the nitty-gritty details of amortized loans, providing you with the knowledge and tools to master this financial concept.
What is an Amortized Loan?
An amortized loan is a financial arrangement where regular, fixed payments are made to gradually reduce the principal amount and pay off both the interest and the principal over a specified period. Unlike simple interest loans, which only charge interest on the principal amount, amortized loans ensure a systematic repayment of both principal and interest, making them a popular choice for mortgages and long-term loans.
Calculating an Amortized Loan
The Basic Formula
To calculate an amortized loan, you can use the following formula:
- �P is the monthly payment,
- �r is the monthly interest rate,
- ��PV is the present value or the loan amount,
- �n is the total number of payments.
Breaking Down the Formula
Monthly Interest Rate (�r)
The monthly interest rate is a crucial factor in the formula. It is calculated by dividing the annual interest rate by 12. For example, if the annual interest rate is 6%, the monthly interest rate would be 0.0612=0.005120.06=0.005.
Total Number of Payments (�n)
The total number of payments represents the number of months over which the loan will be repaid. For a 30-year mortgage, �n would be 360 (30 years multiplied by 12 months).
Let’s consider a practical example to illustrate the calculation of an amortized loan. Suppose you have a loan amount (��PV) of $100,000, an annual interest rate of 5%, and a loan term of 20 years. Using the formula, you can determine the monthly payment required to amortize the loan.
After plugging in the values and performing the calculations, you’ll find the monthly payment amount.
Advantages of Amortized Loans
Predictable Monthly Payments
One of the significant advantages of amortized loans is the predictability of monthly payments. Borrowers can plan their budgets more effectively, knowing that the payment amount remains constant throughout the loan term.
Unlike some interest-only loans, amortized loans ensure a steady reduction of the principal amount with each payment. This gradual reduction leads to a quicker accumulation of equity in assets such as homes.
Tips for Effective Amortization
Make Extra Payments
Accelerate your loan repayment by making extra payments when possible. This not only reduces the overall interest paid but also shortens the loan term.
Consider refinancing your loan if market conditions offer lower interest rates. Refinancing can result in significant savings over the life of the loan.
In conclusion, mastering the calculation of an amortized loan is an essential skill for anyone navigating the world of finance.
By understanding the formula, breaking down its components, and exploring the advantages of amortized loans, you’re equipped to make informed financial decisions.


Warning! Math-related post! Don’t let that scare you off! Math can actually be fun, and will help your reasoning abilities.
As I was wandering around the blogosphere, I stumbled upon an interesting article over at Michellalianna that has nothing to do with what I am going to write about. Well, nothing but the image that was used for the post. The image is in fact the featured image for this post. This picture bothers me. Not as an androsexual transwoman, but instead as a mathematician—in training. As a math-loving person, the notation bothers me. I think that the usage of the equals sign doesn’t make sense. So, I am going to argue that the proper subset notation should be used.
Some preliminary stuff first: I don’t disagree with the message of the picture; I simply think the notation is off. Clearly this post will have some math in it, but I will do my best to keep it simple, easy to understand, and informal.
Before I dive into why I think the proper subset notation should be used, let me give you some mathematical details on the two symbols. I am thinking in terms of set theory. A set is simply a collection of elements—like an array in programming or simply an unordered list. Now, a set A is a subset of the set B if and only if every element in A is an element of B, i.e. A is contained in B. In set theory we would denote this by , and read it as A is a subset of B. It’s important to note here that every set is subset of itself because every set contains itself. Now, two sets are equal if and only if , i.e. they both contain each other thus A=B. Lastly, A is a proper subset of B if and only if A is contained in B, but A != B. This is denoted in set theory as , and read as A is a proper subset of B. Easy enough, right?
Well, I am going to state my case mathematically, then I will use words to describe what I am talking about. Let P be the set of all human beings. Let G be the set of all sets of people that share a common trait, and let GP be in G and be equal to P. Let Ra denote a unary relation that produces the set of all civil rights related specifically to that set—how this looks, I have no idea.
Now, lets —this constrains X to only being groups in G—then . We want X to be equal to GP because then RX would contain everyone’s civil rights. In the case of the picture, X is the set of all trans people—as broadly speaking as possible—so, —while X is in G, it is not equal to GP which implies that RX isn’t equal to RGP. So, in order for the equals sign to make sense, the set on the left must be the human species, i.e. . Now, RGP contains everyone’s civil rights, thus —for every group g in G that is not equal to GP, g is a proper subset of GP which implies that Rg is a proper subset of RGP.
Did you skip that? It might make more sense after I give some explanation. In order for X rights—the rights of the set we are talking about—to be equal to civil rights, it needs to contain everyone. In the picture, X doesn’t contain everyone; it only contains trans people. Thus the equals sign should be a proper subset sign.
Realistically speaking, this isn’t important at all. The message of the picture is clear. It’s really about the perspective you are using. If you look at the image mathematically, then it’s invalid. If you instead look at it sociopolitically, then everything is in order. The math perspective is useful for skeptics because math improves one’s reasoning abilities. Skepticism and Math both require attention to detail, and biasless—okay, minimized bias—approaches to finding the truth.
P.S. While wordpress is supposed to support LaTeX out of the box, it’s not parsing it. If you would like a copy of the article, email me.


Early Professional Success is only the part of the story……….
While swimming laps, Pavel Etingof thinks about math. The crowds and noise on a city bus do little to distract Allen Knutson from the equations he scribbles on a notepad he keeps handy. Francis E. Su gave up his songwriting hobby to spend more time on his proofs.
The constant devotion of these and other mathematicians to their work has allowed them to produce seminal proofs and impressive results that have won them high praise early in their careers -- all are under 35.
Abel died of tuberculosis at age 26 after solving a 300-year-old problem and discovering what are now known as Abelian functions. Although death cut short the careers of those two men, Albert Einstein lived for 50 years after formulating his most famous equation, E=mc2, when he was 26.
Early professional success is only part of the story, even there are more failures than sccess however. Many researchers in other fields show early promise but typically take more time to make important contributions because of the nature of their work.
But starting at a young age doesn't necessarily mean one's career will end early or that later contributions will pale in importance -- the second half of the legend. In fact, Mr. Simonton found that mathematicians make their best research contributions (which he defined as the ones mentioned most often by historians and biographers in reference books) at what many might consider doddering old age: 38.8. That age is very similar to those he found in other sciences: 40.5 in biology, 38.2 in physics, and 38.0 in chemistry.
"Life takes a lot of time and effort," Mr. Fefferman says. "I think the big jump there came with taking care of babies, taking night shifts. There's nothing like sleep deprivation to make one less than brilliant."
"Doing the great mathematical work requires a hell of a lot of energy," says Mr. Etingof, of M.I.T. and
Many of these forms of public recognition are given only once to a researcher, so "there's an impression that [older mathematicians] have run out of steam," says Mr. Simonton, even if their work continues at the same level.
Doing significant work late in one's career involves seeking out problems that require more knowledge than young mathematicians can have accumulated, according to George W. Mackey, 84, an emeritus professor of mathematics at Harvard. That often means learning about several different areas of math and looking for ways to tie them together, he says.
"In mathematics, it's not a game where the fastest wins," says Edward V. Frenkel, a 32-year-old professor at
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.
So if we re-phrase the above lines what Frenkel said it ought comes out like a philosophy “In Life, it’s not a game where the fastest wins, but rather it’s more who can see farther, who can see deeper. That’s the one who achieves more.”At this point, tell me how many times we've heard this story in our childhood " The Hare & the Tortoise", isn't it sounding the same?


Follow these steps to ensure you have the correct calculations:
- Begin by entering the longest service out of the pedestal that is farthest from the transformer. This row will always have a quantity of one for the number of houses.
- The calculator adds the number of houses serviced from the rows above, so on the second row enter the number of houses served out of the farthest pedestal from the transformer, less the one service entered on the first row.
- Continue by entering the cable distance and number of houses fed out of each pedestal, working from the farthest back to the transformer.
- Enter only the distance from house to pedestal, pedestal to pedestal, and pedestal to transformer. The calculator will add length to the cable for terminations.
Note: The maximum total voltage drop from the first transformer to the farthest customer is five per cent. This calculation is based on 240-volts.
Voltage Drop Calculator
The calculator factors in internal wiring.


Limitations of Dimensional Analysis
Limitations of Dimensional Analysis.
(1) Dimensions are not unique. Two Physical Quantities can have same dimensions. For example torque and Work Done.
(2) Dimensions of [K] such as (1/2), 1 or 2p etc. cannot be deduced by dimensional analysis.
(3) The method of dimensions can not be used to derive relations other than product of power functions. For example,
s=ut+(1/2)at2 or y = a sinwt
cannot be derived by using dimensional analysis.
(4) The method of dimensions cannot be applied to derive formula if in mechanics a physical quantity depends on more than 3 physical quantities as then there will be less number (= 3) of equations than the unknowns (>3).
(5) Even if a physical quantity depends on 3 physical quantities, out of which two have same dimensions, the formula cannot be derived by theory of dimensions


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Taking into account ALL of the forces acting on a particular body is not always an easy task. The angle used as argument of the cosine function is always determined by going on an arc from the positive x-axis in counter-clockwise direction towards the force of which you want to determine the x-component. A particle remains at rest if the sum of the x-components and the sum of the y-components and the sum of the z-components of ALL forces acting on it, are zero. Here are my calculations for the x-components of the four forces and then the x-component of the resultant: In mathematics we know this also as the absolute value. So, here is the problem as depicted in Figure 3. If the labels F 1 , F 2 , F 3 , and F 4 are not given, you must label them. Provide a wide range of slots. Two of the forces have negative x-components cos is negative as is cos Use the graphical output to visualize magnitude and direction of the resultant or equilibrium force changing one of the given forces. There are different structures to these types of deals; some require you pay back the free bets you were given out of your winnings should you come out with profits , while others will require playthrough wagering requirements that prevent you from withdrawing your winnings until a set number of real money bets is met, and these can only be met with your real money deposit wagers. My recommendation to you is to use these programs in three ways: Have been independently verified for fairness and withdrawal speed by our staff writers. Like Us on Facebook. Now the three components are evaluated according to: Up to or more free spins may be offered to players at various sites. Formally we write this as: I furthermore entered already an x-y coordinate system. Given is the magnitude of the force and the values of the angle beta and gamma.
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Free Online Casino Game Two programs are available for you. Alonzo Mourning Shows Support for Operation No Deposit Bonus Codes - The list of casino codes available online is seemingly endless. And the magnitude of a force like the resultant is calculated according to: This is a popular type of casino promotion because it provides new players with real money credits that can be used to try out various online gambling sites for free. Resultant of 2 Forces Problem 3. We interpret each of the three forces in Figure 3. All you have to do is sign up to the casino to claim your free chips. It allows you to ascertain your knowledge of the definition of terms and your understanding of important golden nugget casino online sizzling hot online kostenlos spielen ohne anmeldung. You've Won a Free Spin. I furthermore entered already an x-y coordinate system. Believe it or not, players can actually win big when using these offers, but it should be noted that winnings are often subject to withdrawal requirements. Two computer programs Using the concept of x- y- and z-components is also a great way to write computer programs which can tackle generalized problems. And the magnitude of a force like the resultant is calculated according to:


Throughout this help file:
Hexadecimal numbers are prefixed with a dollar sign (e.g. $A5).
"Word" refers to a two-byte integer.
"Long word" refers to a four-byte integer.
Multi-byte integers are always encoded in little-endian notation (e.g. the hex word $1234 would be represented in memory by $34, $12 and the long word 123456789 would be encoded as $15, $CD, $5B, $07).
All numeric fields are unsigned integers unless stated otherwise. In numeric fields that support them, negative numbers are represented in two's-compliment form.
mod indicates modulo. So if an entry says "n mod 4 = 0," this indicates that n is a multiple of 4 (i.e. 0, 4, 8...).


Chinese Name 柳万民
Email wanminliu (at) gmail.com
Postal Address Box 480, 751 06 Uppsala, Sweden
Office Room 14144, Ångströmlaboratoriet, Plan 4, Hus 1, (Lägerhyddsvägen 1), Uppsala University.
I am a Researcher at the Department of Mathematics of Uppsala University. I was a research fellow at the IBS Center for Geometry and Physics during 2015 Dec – 2018 Nov. I received my PhD from the Hong Kong University of Science and Technology (HKUST) in 2015, under the guidance of Prof Wei-Ping Li.
Curriculum Vitae (2020 Feb) / PDF
My research area is Algebraic Geometry. My research interests include Bridgeland stability conditions, moduli space, derived category, birational geometry, counting invariants and wall-crossing.
- Bridgeland stability condition – new aspects to symmetry and classification
- Bridgelands stabilitetsvillkor – nya aspekter av symmetri och klassificering
- 2020 Fall, Linear Algebra II (1MA024), Timetable of Programme E3 and Timetable of Programme X2.A. Zoom Links of lectures are available via login Studium.
- 2020 Spring, supervise a undergraduate student: Patrik Deigård, Liouville’s equation on simply connected domains, Bachelor Thesis (Kandidatarbete), 57 pages, Uppsala University, 2020 June, DiVA, id: diva2:1466472.
- 2019 Fall, Differential Geometry (1MA011) for undergraduate students in Uppsala University.
- ——, Song Yang, Xun Yu, Classification of full exceptional collections of line bundles on three blow-ups of P^3, Journal of the Korean Mathematical Society, Vol. 56, No. 2 (2019), 387-419, arXiv:1810.06367 [math.AG]. (A previous version, titled by On exceptional collections of line bundles of maximal length on the blow-ups of P^3, is available at CGP17025, 2017 July.)
- ——, Bayer-Macrì decomposition on Bridgeland moduli spaces over surfaces, Kyoto Journal of Mathematics, Vol. 58, No. 3 (2018), 595-621, arXiv:1501.06397 [math.AG] or CGP16027.
- ——, Bayer-Macrì decomposition on Bridgeland moduli spaces over surfaces, PhD thesis, (see the PDF for an updated version of the library version).
Geometry and Mathematical Physics Seminar of USTC, June 4, 2020, Contractibility of space of stability conditions on the projective plane via global dimension function / PDF
- Online Seminars;
- UU Geometry and Topology Seminar, Wed 14:15–16:15;
- UU Algebra and Geometry Seminar, Tue 15:15–17:00;
- UU Geometry and Physics Seminar;
- KTH & SU Algebra and Geometry Seminar;
- Chalmers/GU Algebraic Geometry and Number Theory Seminar
- Classification of full exceptianl collection of line bundles (of FINITE type) on the blow-up of P^7 at a point
- Notes on Maps in the Homotopy Category of dg-Categories / PDF, CGP Derived seminar on Bertrand Toën’s Lecture on DG-categories, 2017.
- THE RISING SEA: Foundations of Algebraic Geometry – an HTML version of Ravi Vakil’s book
The purpose of the HTML version (via the tool pdf2htmlEX) is for a quick preview of the book without downloading the whole PDF. The PDF is available at http://math.stanford.edu/~vakil/216blog/index.html under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
- pdf2htmlEX – Convert PDF to HTML without losing text or format.
- Talk on TUG2013: Online publishing via pdf2htmlEX HTML / PDF
(Last modified: Aug 31, 2020)


Number of nonzero digits in representation of n in primorial base.
While working on a hobby project of mine that searched for prime generating polynomials, I was testing out different data structures for efficient primality testing. One such data-structure was a hash set based on Wheel Factorization of polynomial coefficients.
Ultimately I decided to go with a simpler approach of binary searching a prime sieve. But, in implementing and evaluating complexity of the wheel factorization scheme, I ran across an interesting function: the number of nonzero digits in the representation of a number in the primorial base. This measurement is useful for determining the complexity of a number, going through levels of wheel factorization.
Being the curious person I am, I decided to look up this function in the OEIS, and found that it was not there. So, I decided to add it myself: oeis.org/A267263. That page contains some sample code, and connections to other integer sequences already on OEIS.
It’s not much, but when it was first approved I felt extremely proud that I had contributed to the OEIS. I hope to add more sequences in the future.


Ralph Gootee graduated from SU in 2005 with a BS degree in mathematics and computer science. During his time at SU, Ralph impressed his professors and peers with his enthusiasm for research, strong work ethic, and academic drive. Ralph went on to become co-founder and CTO of PlanGrid, which was funded by YC in 2012 and acquired by Autodesk in 2018 for $875m. Prior to PlanGrid, Ralph worked for Pixar as a rendering engineer, helped Sony launch its first Android smartphone, and researched missile defense for Johns Hopkins University where he earned his master’s degree in mathematics. Ralph spends his free time programming, reading literature, drawing, and rock climbing.


Whether they are heavy or occasional, many gamblers are interested in lottery. The main lottery’s charm is that it’s really easy money if the odds work in your favor, but of course, it can enable you to get bankrupt if you obtain too addicted.
Online Lottery Results: If You Really Want To Win
If you’re like every other gambler, then you definitely are probably wondering tips on how to better your online lottery results by understanding the math behind probabilities. ruay Many lottery experts are wont to provide this little bit of advice: brush up on your math and you will continue to work the odds in your favor. So return to your math books, or better yet, ask your math teacher.
Online Lottery Results: How To Increase Your Chances
What you can do to enhance your online lottery results would be to play a system entry, or consistently betting on some numbers. If you’re playing a six-number lottery, then choose four numbers to be constants in your combination with both other numbers being variable.
Better yet, if you will find a lottery that enables you to play a 15-number system, you’ve over 5,000 better chances of winning than the usual six-number combination. A 15-number system lotto is usually higher priced than ordinary lotto, so anticipate to spend money.
Online Lottery Results: The Damper On Number Systems
The sole downside to number systems is that they are expensive, although you are able to always share betting costs with friends with similar interests. If you’re unable to find willing partners, what you can do would be to bet on lottery games with an eight-number winning combination. If you switch from the six-number set to an eight-number set, you really boost your odds of winning by 33%. For best online lottery Results, make six of eight numbers constant, and both other numbers variable.
Getting Good Online Lottery Results Or Spotting A Good Lotto Website
The Internet doesn’t lack are lottery websites. You will find two types of lottery websites: one which generates online lottery results from its system, and one which generates online lottery results from lottery draws in actual brick-and-mortar locations.
In finding a great website that generates good online lottery results, it always pays to do your homework. Since there are so many scam sites on the market, your job is to find reputable and legitimate sites which can be proven to provide out honest online lottery results. In general, most of reliable sites give you a 100% cash back guarantee. A good website would also give you a free testing system. If they’re not being provided by the website you are looking at, then you definitely may want to reconsider and find another site that provides better online lottery results.


Board Games Could Be a Win for Your Kid's Math Skills
MONDAY, July 10, 2023 (HealthDay News) -- Family game night can be more than just a fun time: New research suggests it may even help build some early math skills in young children.
While past research has pointed to games as a way to enhance reading development and literacy, a new comprehensive review finds that number games like Monopoly, Othello, and Chutes and Ladders may help children with math.
“Board games enhance mathematical abilities for young children,” said lead author Dr. Jaime Balladares, from Pontificia Universidad Católica de Chile, in Santiago. “Using board games can be considered a strategy with potential effects on basic and complex math skills. Board games can easily be adapted to include learning objectives related to mathematical skills or other domains.”
These number-based games help improve counting, addition and the ability to recognize if a number is higher or lower than another for young children, according to the study.
Children benefit when they play board games a few times a week supervised by a teacher or another trained adult, the study authors said.
The researchers reviewed 19 studies published from 2000 onward involving children aged 3 to 9. Most of the studies focused on the relationship between board games and mathematical skills.
The children each received special board game sessions about twice a week for 20 minutes over about six weeks.
Some of the studies grouped children into either the number board game or to a board game that did not focus on numeracy skills, while in other studies all the children participated in number board games but were assigned different games.
All children were assessed on their skills before and after the interventions, with the authors rating success in four categories, including the ability to name numbers; basic number comprehension (such as nine is greater than three); the ability to add and subtract; and interest in mathematics.
In some cases, parents were provided training on using math in games.
The results showed that children had significantly improved math skills for more than 52% of tasks analyzed. In 32% of cases, children in the intervention groups gained better results than those who did not take part in the board game intervention.
The findings were published online July 6 in the journal Early Years.
“Future studies should be designed to explore the effects that these games could have on other cognitive and developmental skills,” Balladares said in a journal news release. “An interesting space for the development of intervention and assessment of board games should open up in the next few years, given the complexity of games and the need to design more and better games for educational purposes.”
The American Academy of Pediatrics has more on early childhood development.
SOURCE: Early Years, news release, July 7, 2023


This app helps you to convert the commonly used units of measurements for Length / Distance, Mass / Weight, Area, Volume, Temperature and Time.
You can convert a value from the chosen unit to other units. e.g. kg to grams, kg to pound, miles to kilometers, miles to yards, etc.
The conversion results are shown for two chosen units. You have also got an option to view the conversion results in a list for all units of that category.
You can also share conversion results with others.
This app can work completely offline.


Thredded Plugin: TeX math in Markdown via KaTex
A Thredded markdown plugin that renders TeX math server-side via KaTeX.
This is what it looks like:
Add this line to your application's Gemfile:
And then execute:
bundle rails g thredded:markdown_katex:install
Wrap math in
See the Kramdown Math documentation for more information.
After checking out the repo, run
bin/setup to install dependencies.
rake spec to run the tests.
You can also run
bin/console for an interactive prompt that will allow you to
To install this gem onto your local machine, run
bundle exec rake install.
To release a new version, update the version number in
and then run
bundle exec rake release, which will create a git tag for
the version, push git commits and tags, and push the
.gem file to
Bug reports and pull requests are welcome on GitHub at https://github.com/glebm/thredded-markdown_katex.
This project is intended to be a safe, welcoming space for collaboration, and contributors are expected to adhere to the Contributor Covenant code of conduct.
The gem is available as open source under the terms of the MIT License.


It appears that there has been an order-of-magnitude change in the number of known viruses.
To add further empirical proof to the theory that evil never rests, we have assembled a chart delineating the increase in the number of viruses known to Bitdefender throughout the past four years.
Unsurprisingly, the numbers go up, up, up, 16-fold in all. The data does not quite fit an exponential increase, although there’s an interesting inflexion point a couple of years in that we can’t quite explain – it seems the virus writers got even more productive around that time, for some reason.
So, here’s the chart. The numbers on the vertical axis represent the ratio by which the number of known viruses has increased as compared to the first data point (represented as having value=1), while the horizontal axis represents days elapsed:


NJSLA tests are administered to public school students in grades 3-9 and measure how well students meet the NJ Student Learning Standards. NJSLA consists of 2 units of English Language Arts and 2-3 units of Mathematics and are administered entirely on a computer. Although schools have over a month in which to complete testing, most will complete all testing within one or two weeks. Total testing time ranges from 5.5 hours in 3rd grade to 6 hours in grades 4-9.
NJSLA ELA and math results are reported in five separate categories that run from Level 1, "did not yet meeting expectations" to Level 5, "exceeded expectations". Each category has a scaled score, or a cut score, connected to it. The lowest NJSLA score possible is 650, and the highest is 850. Levels 4 and 5 are considered passing scores.
It is possible that some schools will use NJSLA scores to help with course placement (i.e. advanced/honors classes).


The Discover Dairy Lesson Series meets Pennsylvania state education standards in math, science and reading, as well as Common CORE standards in Literacy and Math. The series also incorporates standards aligned with STEM. Standards are listed with each individual lesson and are included in a summarizing chart below.
To view a matrix of educational standards addressed in the upper elementary school lessons, click here.
To view a matrix of educational standards addressed in the middle school lessons, click here.
For more information about the lesson series, email firstname.lastname@example.org.


Here are some examples of fitting simple statistical models in rjags. They are useful for beginners learning Bayesian inference using the rjags library. Notice in each case all that is required is specification of the probability density function for the observed data (i.e. the likelihood function) and the prior distribution functions for the unknown parameters.
Beyond that the user needs to be familiar with the rjags syntax for compiling the model, running the MCMC algorithm, sampling from the posterior distribution and examining the sample. The mcmcplots library is necessary for the last step.