Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.

One of the most important mathematical achievements of the 20th century^[1] was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

the popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups. See Rubik's Cube group.

The Königsberg Bridge problem

The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.^[19] This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy^[20] and L'Huilier,^[21] and represents the beginning of the branch of mathematics known as topology

Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.

The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable.

Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.

Algebraic K-theory

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.

K-theory was invented in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K₀, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.

The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if F is a field, then K₀(F) is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring R, K₀(R) is related to the Picard group of R, and when R is the ring of integers in a number field, this generalizes the classical construction of the class group. The group K₁(R) is closely related to the group of units R^×, and if R is a field, it is exactly the group of units. For a number field F, K₂(F) is related to class field theory, the Hilbert symbol, and the solvability of quadratic equations over completions. In contrast, finding the correct definition of the higher K-groups of rings was a difficult achievement of Daniel Quillen, and many of the basic facts about the higher K-groups of algebraic varieties were not known until the work of Robert Thomason.

Probability theory

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.^[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.^[2]

Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module. A module taking its scalars from a ring R is called an R-module.

Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication.

Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology

ring theory

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, Fermat's last theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.

Noncommutative rings are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups. It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian rings.^[1]

For the definitions of a ring and basic concepts and their properties, see ring (mathematics). The definitions of terms used throughout ring theory may be found in the glossary of ring theory.

How to Calculate the Area of a Base

In geometry, the bottom of a three-dimensional object is called a base – if the top of the solid is parallel to the bottom it is also called a base. Since bases occupy a single plane, they have only two dimensions. You can find the area of a base by using the formula for the area of that shape.

Square Bases

Cubes and square pyramids have bases that are square-shaped. The area of a square is equal to the length of one of its sides multiplied by itself, or squared. The formula is A = s2. For example, to find the area of a base of a cube with 5-inch sides: A = 5 inches x 5 inches = 25 square inches

Rectangular Bases

Some rectangular solids and pyramids have rectangular bases. The area of a rectangle is equal to its length, l, multiplied by its width, w: A = l x w. Given a pyramid whose base is 10 inches long and 15 inches wide, find area as follows: A = 10 inches x 15 inches = 150 square inches.

Circular Bases

The bases of cylinders and cones are circular. The area of a circle is equal to the circle's radius, r, squared then multiplied by a constant called pi: A = pi x r². Pi always has the same value, approximately 3.14. While pi technically has an endless number of decimal places, 3.14 is a good enough estimation for simple calculations. For example, given a cylinder with a radius of 2 inches, you can find the base's area as follows: A = 3.14 x 2 inches x 2 inches = 12.56 square inches.

Triangular Bases

A triangular prism has a triangular base. Finding a triangle's area requires two known quantities: base, labeled b, and height, labeled h. Base is the length of one of the triangle's sides, height is the distance from that side to the opposite corner of the triangle. The area of the triangle is equal to half of the base times the height: A = b x h x 1/2 You could find the area of a triangle with base length of 4 inches and height of 3 inches as follows: A = 4 inches x 3 inches x 1/2 = 6 square inches.

How to Study Times TableS

At first, learning the times tables can be challenging, but your students can learn strategies for memorizing multiplication math facts. To help your students, make sure they understand the concept of multiplication before they memorize the math facts. Find or create a multiplication grid to help with daily practice. Look for patterns, make flashcards and discover tricks that can help your students learn the times tables. Ask them to practice at home as well as in school for additional reinforcement.

Look for Visual Patterns

Studying a multiplication grid is the first step toward becoming familiar with the times tables. Find one in a math book or online, or create your own. Look for patterns in the rows and columns. For instance, every other row and column consists of even numbers. The first row and column each count up by ones and the last row all have numbers ending in a 0.

Look for reverse fact patterns. For instance 3 x 4 = 12 and 4 x 3 = 12. The more students study the multiplication grid, the more familiar the multiplication answers will become.

Use Counting Patterns

Counting patterns can help students learn their times tables rapidly. Skip counting is also helpful, such as counting by twos, fives or 10s. For example, if you are counting by fives, you would say: 5, 10, 15, 20. If a student can count by a certain number, he essentially already knows the answers to that times table. Students should start by learning to count by twos, fives and tens. As they become more advanced, they can learn to count by the other numbers.

Practice With Flashcards

Study the times tables one at a time. Make flashcards for that specific times table. Put the factors on the front of the flashcard and the answers on the back. For instance, if you are studying the 2 times table, one card might have 2 x 2 on the front and a 4 on the back. You can use the flashcards to test yourself or to test someone else. The more you use flashcards, the faster you will be able to memorize the times tables.

After moving on to a new times table, go back and review sets of flashcards you have already mastered to retain the information. Parents and teachers can use incentive programs for encouragement. A child could earn a sticker on a chart after mastering each set of flashcards -- and once the chart is filled with stickers -- he could receive a prize or privilege. Flashcards can also be a useful method to study pop quizzes or tests.

Learn the Tricks

Teach your students tricks to help them learn specific times tables faster.

In the 9 times table, the numbers in the answer column add up to nine. For example, 2 x 9 = 18 (1 + 8 = 9), 3 x 9 = 27 (2 + 7 = 9) and 4 x 9 = 36 (3 + 6 = 9).

In the 11 times table, the answers all consist of duplicate digits. For example, 2 x 11 = 22 and 3 x 11 = 33.

In the 10 times table, 10 times a number equals that number with a 0 after it. For example, (10 x 1 = 10, 10 x 2 = 20, 10 x 3 = 30).

How to Teach Mathematics in the Primary School

According to the National Association for the Education of Young Children, children's knowledge of math skills at the primary level "predicts their math achievement for later years." Using different activities that allow children to use and develop math skills develops strong logic and reasoning skills in children. Teaching math skills to primary level students should be done using multiple teaching strategies to optimize student learning.

Engage the students in math activities such as sorting, organizing, patterning, mapping and making pictures or drawing to find the answers to math problems.
Provide materials to enhance math discoveries. Math manipulatives, number lines, the hundreds chart and play money give students tangible items they can use to make connections to their math skills.
Introduce one math concept in several different ways, demonstrate it to the class, allow the children to work in pairs on problems, and have them engage in math games or activities related to the concept.
Ask children to explain their thinking process. Have them explain in their own words how they came to the answer, or they can show you using manipulatives or drawings.
Encourage children to make connections between math they know and new concepts. Ask questions guiding children to make their discoveries about mathematical concepts. Have the children predict the answer based on what they know, then have them work out the problem to find out if they were right. For example, in a subtraction problem, they can predict the answer will be lower than the top number.
Support the students in building math skills by encouraging them to ask questions and use reasoning skills.

Easy Ways for My Child to Learn Multiplication

Multiplication tables are often taught by rote and sometimes are difficult for students to grasp. Certain techniques, however, turn multiplication into a trick or a game that might reel in reluctant learners and encourage them to find the fun in math.

Larger Numeral Multiplication Trick

Students who struggle with multiplication might appreciate this quick trick for multiplying any two numbers between 11 and 19 in their heads. Not only is the result impressive, but performing it may encourage reluctant children to practice multiplying smaller numbers so that they can do it. Start with any two numbers between 11 and 19, for instance 12 times 15. The larger number goes on top, making the equation 15 times 12. Ask your student to add the top number to the right hand number from the bottom number. In this case 15 plus two, making 17, then add a zero, for 170. Now they multiply the two righthand numbers, five times two in this example, making 10. The last step is to add the two numbers, 170 plus 10, and they have their answer. Multiplying 15 times 12 equals 180.

Rules

Some multiplication tables have rules. These are often the easiest for students to learn. Teach them that any number times zero equals zero and any number times one equals itself. Once they know that multiplying by 10 involves adding a zero to the end of any number, they'll have three of the tables down. Elevens are easy to learn once students know that it doubles a number, so that two times 11 is 22 and three times 11 is 33, and so on. Learning these rules gives confidence to students who are learning the multiplication tables because they are easy to master.

Counting

Students can add two more multiplication tables when they know how to count by twos and fives. Once they have these ways of counting down, they'll be able to figure out the two and five times tables even up to large numbers. Now that students have mastered times tables for zero, one, two, five and 10, they'll have the tools for figuring out less formulaic multiplication tables. Teach them that the four times tables is simply twice the twos or the fives minus the number. The sevens are the fives plus the twos. Eventually students should be able to memorize the times tables to 12, but having tools for figuring out difficult answers on their own may reduce stress during that process.

Calculator Hands

Ask your students if they know that their hands are quick and efficient nine times table calculators. This trick is impressive enough that it might encourage an interest in math in even reluctant students. Have them lay their hands on their desks in front of them. Starting from the left, tuck under one finger at a time. The left pinkie reflects nine times one. There are no fingers on the left and nine on the right, making the answer nine. Now unfold the pinkie and tuck in the ring finger. One finger on the left and eight on the right represent 18, or the answer to nine times two. This works through 10, when students tuck in their right pinkie, leaving nine fingers on the left and zero on the right, or the number 90.

How to Use TouchMath

TouchMath is a multisensory math program designed for pre-K through third grade. The program helps make math concepts easier and more accessible for students with different learning styles or learning difficulties. The approach uses auditory, visual and tactile strategies for understanding numbers and operations. You can use the program to help students prepare for new math principles, to supplement grade-level programs, or for enrichment activities.

Every number from one through nine has physical points on the actual number that the user will touch. These are the “TouchPoints.” Numbers one through five each have single points the user touches. Numbers six through nine have double points or a combination of double and single points that the user taps. These points are represented by dots. The student touches the pencil to the number while counting aloud. For example, the number one has a single dot. Number two has two dots. Number three has three dots -- one at the top where the number begins, one in the middle after the first curve and one at the bottom, where the number ends. As the students touch each dot, they count.

Teaching the System

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Teach the students how to use the program by showing the numbers with the actual dots on the numbers. Explain that the number of dots on an arithmetic number helps them know the name and value of that number. Show the students how to use the program, pointing at each dot as you count aloud by ones for each given number. Then, have the class count aloud with you as you repeat the process for each number one through five. For numbers six through nine, explain that some dots are now double dots. Not enough room exists on the numbers to use single dots exclusively, so you count some dots twice. You can show this to the students by counting dots on the device. For the number six, count "one, two" on the first dot, "three, four" on the second dot and so on. For numbers seven and nine, you will see single and double dots.

Teaching Arithmetic Operations

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Once the students have learned how the program works for individual numbers, you can use the system to teach addition, subtraction, multiplication and division. For addition, students touch the dots while counting forward. For subtraction, students touch the dots while counting backward. For multiplication and division, the students will count in sequences. Offer plenty of opportunities for the students to practice using the device for math operations. The program will help students visualize and eventually memorize the value of the numbers or number sentences.

The Aims & Objectives of Primary School Mathematics

Numbers and Basic Functions
At the most basic level, mathematics involves counting, recognizing numbers and performing simple operations like addition and subtraction. In the primary grades, students should be provided with numerous opportunities to master these skills. Upon exiting primary school, children should be comfortable with writing and identifying numbers, rote counting forward and backward and comparing numbers and quantities. Primary students should have knowledge of number facts and families. They should also be able to add, subtract, multiply and divide numbers.

Measurement and Estimation
In the primary grades, students should be taught about measurement of length, weight and capacity. Children should be introduced to comparative language, such as "shorter," "heavier" and "longer," and should be able to apply these concepts when considering different objects and units of measurement. Primary students should also learn about money and time and be able to measure time in terms of hours, days, months and years. In addition to measurement, children should be taught about estimation of quantities and capacities.

Geometry
Learning about shapes, symmetry, position and direction is a key objective for primary school mathematics. Students should be exposed to two- and three-dimensional shapes and be able to identify, name and draw them. Children in the primary grades should also have an understanding of line and rotational symmetry, as well as the manipulation of objects in space. Additional spatial concepts of position, like "above," "under," "next to" and "beyond," should be addressed in primary mathematics education as well.

Data Collection and Interpretation
Being able to collect, organize and interpret data is an important skill that is taught in the primary grades. Students should be given opportunities to answer questions through sorting and organizing data by using graphs, charts, tables and Venn diagrams. They should also learn to compare objects and data based on a given criteria.

Critical Thinking and Problem Solving
Mathematics should be used to develop critical thinking and problem solving. Presented with a problem or situation, primary students should be able to identify the proper strategies needed to come to conclusions and carry out calculations. Throughout primary school, students should progress from using concrete objects and written calculations to carrying out operations mentally. Children in primary school should also have the ability to identify and continue patterns, provide examples and non-examples of mathematical statements, and form and test hypotheses.

Appreciation and Uses
One of the most important aims of for students at this level is to develop a positive attitude toward mathematics. Students should understand and appreciate the functionality of mathematics. In addition to valuing math, primary school students need to be taught how to use mathematics in their everyday lives. They should be exposed to all the uses of mathematics, from counting out change or telling time to using angles in architecture or art.

How to Learn Pre Algebra Step-by-Step

Whether you are anticipating taking a pre-algebra class in the future, are struggling with a current pre-algebra class, or need to master the basics to enter a beginning algebra class, learning pre-algebra step-by-step can help you understand the material that you will build on in later courses. Trying to go too fast and skimming over the basics can hurt your understanding of more complex problems later on. Therefore, working methodically through pre-algebra material will help you progress in a more productive way.

Study numbers and their properties. Though students who are ready for pre-algebra will already be familiar with basic functions and operations, including addition, subtraction, multiplication and division, a good knowledge of more complex numerical operations and properties, such as decimals, square roots, negative numbers, and integer properties, will prove to be invaluable in algebra studies later on.
Work with ratios and proportions. Students may already be familiar with basic ratios, which describe the relationship of one amount to another, and proportions, which compare ratios, but may need to practice these concepts to work with them at a more advanced level. Problem sets, online practice, and diligent corrections will help prepare students for the more complex problems they will soon encounter.
Study factoring. Factoring will prove to be extremely useful in algebra, for problems involving exponents, complicated expressions that need to be simplified, and other topics. Begin by approaching basic factors, breaking down numbers like 4 into factors of 2 and 2 or 4 and 1. Take your knowledge to the next level by studying more complex factoring topics, like finding the greatest common factor of two numbers, or performing prime factorizations of a number.
Develop your understanding of fractions. Though you may already have worked with fractions in a variety of capacities, develop this knowledge further by working through problem sets that require you to manipulate fractions by adding, subtracting, multiplying, and dividing fractions, as well as problems that require you to convert from decimals to fractions, and vice versa.

What Is Infinity in Math?

In math, infinity is a concept that refers to an endless quantity that's larger than every real number. The symbol for infinity resembles a sideways number eight. Students are introduced to the concept of infinity during or before middle school, but they usually don’t use infinity much until calculus.

What Infinity Is

Although infinity is larger than any number in existence, it is not a real number. Unlike real numbers in which you add two numbers to produce a larger number such as 2+5 = 7, if you add infinity + 1, you get infinity. If you add infinity to infinity, you will see that infinity + infinity = infinity. Infinity is not only enormous, it is also endless. You cannot measure infinity; add any quantity to infinity, and you will always get infinity.

Mathematical Examples

Although infinity isn’t widely applied before calculus, math has many examples of infinity. For example, the sequence of numbers -- 1, 2, 3 and so on -- extends infinitely. When you write certain fractions in decimal form, they will repeat infinitely. For example, a calculator will show that 2/3 equals 0.6666, but the row of sixes in the number 0.6666 doesn’t end after four digits. The sixes in the number 0.6666 continue as far as a calculator screen will allow; in theory, the number 0.6666 extends forever -- infinitely. In geometry, a line segment has two distinct endpoints – points A and B. A line, however, will extend infinitely in either direction.

How to Use Counters in Math

A knowledgeable teacher recognizes that young children learn best when engaged in hands-on activities that allow them to further explore abstract ideas or concepts, especially in math. Counters are an excellent tool that children can use in their attempts to master math skills including counting, adding, subtracting, making patterns and comparing numbers. Although there are commercially made counters that commonly are small round plastic discs or squares, incorporating counters such as dried beans, blocks, buttons or counting bears into math activities can prove to be a simple yet effective teaching strategy.

Early Math Concepts

Counters are helpful in teaching children basic math skills such as counting, sorting and patterning. Provide children with a variety of counters to use for different activities to promote participation and keep them engaged. Give children specific tasks to complete using counters. For example, you can ask children to show you a certain number of counters, to count a group or set of counters, or to group counters by size or color.

Basic Functions

Once children have mastered counting and identifying numbers, you can use counters to help teach skills such as addition, subtraction, multiplication and division. Start by asking children to find the sum or difference of two numbers using counters to represent the problems. Counters can also be used along with a line of numbers to engage children in "jumping" their counters to specific numbers in the line to solve basic addition and subtraction problems. In the upper primary grades, children can use counters to represent multiplication and division problems by combining and separating groups of different objects.