Abstract Algebra

Calculus consists of two main topics -- differentiation and integration. Differentiation is concerned with 'rates of change', for example the rate at which the position of a moving object is changing with respect to time, otherwise known as velocity. Integration is concerning with computing areas,volumes, and lenght by first approximating the region as the sum of many smaller regions which are simpler to compute, and taking the limit as the number of smaller regions increases to infinity.

At first sight, it doesn't seem like these two topics -- differentiation and integration -- have anything to do with one another. In fact, in a calculus course either one could be presented first since it wouldn't require knowledge of the other one (traditionally, differentiation is taught first, then integration, but it isn't necessary to do them in this order.)

However, the amazing fact is that these two seemingly unrelated problems are completely intertwined. The Fundamental Theorem of Calculus, one of the most amazing and profound results in all of mathematics, spells out just how the processes of differentiation and integration are related -- they are essentially reverse operations of one another, or two sides of the same coin. The Fundamental Theorem of Calculus is so named because it ties together the two main themes of the subject. F(x) = f (t)dt is a function then we have = f(t)dt but since Since dt = h we have - f (x) = f (t) - f(x)dt . and using the continuity of F(t), we have the following equality. - f (x) = 0 .Now, the punch line!The function F(x) is differentiable and F '(x) = f (x).

Many calculus books have two parts to the FTC (Fundamental Theorem of Calculus)

Part one states that the area under a section of a curve is the antiderivative evaulated at the upper limit minus the lower limit. That is:

Integral ( f(x) dx) from a to b = F(b) - F(a)

where b is the upper boundary and a is the lower boundary

and

Part two states that the derivate of integration is the integrand:

d/dx integral (f(t) dt) from 0 to x = f(x)

where x is the upper boundary and 0 is the lower boundary.

So what went into the integral that you derive is the result.

Note: it really helps to see the pictures of what is going on.

Idempotent Matrix:

An idempotent matrix, A, is the specific periodic matrix (see note) where k=1, thus having the property A2=A (we can also say A.A=A).

Inverse Matrix:

Given a square matrix, A, its inverse is B if AB=BA.

Note:

A periodic matrix, A, has the property Ak+1=A where k is a positive integer. If k is the least positive integer for which Ak+1=A, then A is said to be of period k.

ASTRONOMY

BIOLOGY

BUSINESS

Chemistry

construction

consumer

economics

education

environment

finance

geometry

government

health/life sciences

labor

Miscellaneous

Physics

Sports/ entertainment

Statistics/ demographics

Technology

Transportation

ASTRONOMY

Astronomer use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera that is attached to the telescope basically records a series of numbers - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them. Another way that astronomers use math is when they are forming and testing theories for the physical laws that govern the objects in the sky. Imagine you're on a spaceship in orbit around the moon. You have a fuel leak and are running out of power. When do you fire the ship's thrusters, and for how long and in what direction, in order to be able to return to Earth safely? Also, in addition to flying and maneuvering a spacecraft, astronauts are often involved in conducting scientific experiments aboard the spacecraft, which would involve math in other ways too.

BIOLOGY

Algebraic biology applies the algebraic methods of symbolic computation to the study of biological problems, especially in genomics, proteomics, analysis of molecular structures and study of genes. Computations in the field of biology are done in order to simulate experiments and/or calculate features of a biologic process or structure. Such as for example calculating mathematical predictions of intercellular features, cellular interaction, body reaction to chemicals and analysis of heritage. In recent years, methods from algebra, algebraic geometry, and discrete mathematics have found new and unexpected applications in systems biology as well as in statistics, leading to the emerging new fields of "algebraic biology" and "algebraic statistics." Furthermore, there are emerging applications of algebraic statistics to problems in biology. This year-long program will provide a focus for the further development and maturation of these two areas of research as well as their interconnections. The unifying theme is provided by the common mathematical tool set as well as the increasingly close interaction between biology and statistics.

Business

Business in mathematics involves a lot of arithmetic, algebra ang geometry. But major portion is of mathematics that is found in business in algebra. Getting off to a good start is the goal of understanding this topic. You may be looking for the answers to some deep and dark mathematical secrets. This topic helps you light the way toward realizing that the basic math algebra involved in business was never meant to be a secret. We might not see the relevance in some mathematical processes. Most of the math in business is not compartmentalized into one section or another. Fractions and decimals are found in all application. Proportions and percentages are rampant. Measurements are necessary for many different business processes. In other words, the math in business involves computation shared by all different aspects. The main trick in doing the math is to know when to apply what.

Chemistry

First, algebra is applied in everything we do. Algebra can be applied to chemistry in many ways: to manipulate equations and solve for a problem. For example, here is a gas equation from chemistry PV=nRT. P is the pressure (in atm), V is the volume (in L), n is the moles, R is a constant (.082059 L*atm mol-1 K-1), and T is the temperature (in K). In recent years computer algebra techniques and symbolic computation systems have found increasing use for solving problems in chemistry and for chemistry education.Let's say you are given all the information and need to find the temperature, and this is where algebra comes into play: T= PV/nR . You can complete General Chemistry as well as Organic Chemistry with only algebra under your belt.

Economics

Algebra includes examples that demonstrate the foundation operations of matrix algebra and illustrations of using the algebra for a variety of economic problems.

The authors present the scope and basic definitions of matrices, their arithmetic and simple operations, and describe special matrices and their properties, including the analog of division. They provide in-depth coverage of necessary theory and deal with concepts and operations for using matrices in real-life situations. They discuss linear dependence and independence, as well as rank, canonical forms, generalized inverses, eigenroots, and vectors. Topics of prime interest to economists are shown to be simplified using matrix algebra in linear equations, regression, linear models, linear programming, and Markov chains.

Construction

We use algebra in construction to figure square footage, cubic footage, and angles when building, you can use it to tell how many feet, sg units, sg feet, the perimeter, the area, all of them. Its is important, for example, in deciding how much material they need they will have to do some rough calculations. Mathematics in algebra is used by construction workers in many ways. When setting out a site, mathematics is used to get the dimensions correct. It is also used calculating the amount of materials to order, and when cutting materials to size. Very few tasks do not involve some use of mathematics.

Consumer

Consumer Math provides a basic understanding of the fundamental math life skills needed after graduation. Course content includes the following topics: pay (earning money, gross pay, net pay, deductions), banking (checking and savings accounts), taxes, budgeting, food purchase, clothing purchase, buying a car, use of credit cards, public transportation, renting an apartment, buying a home, insurance, investing (retirement, school expenses, emergencies) and the use of leisure time. Calculators are an integral part of instruction and are used during assessment. Various teaching methods are employed at the discretion of the instructor and I.E.P. to meet the needs of the student.

Education

Mathematics is every thing. No matter what you want to be in life mathematics is important as your breakfast, lunch and dinner. God used mathematics when creating the world if not it won't still be here after all this millions of years. Mathematics is the essentials of life. Take a good look at every thing you do each day, mathematics is involved. As all the students who had gone to grade school they already done discussing the basic math algebra. They encountered the simplest form until now. Math algebra is taught by teachers so students may be aware of the things around them. Like in buying, measuring and counting.

Environment We see a diversity of waves in our everyday experience. Electromagnetic waves carry television and radio to our homes, ultrasound waves are used to monitor the growth of a baby in the mother's womb, and a variety of waves on the surfaces of rivers, lakes and oceans affect the coastal environment. Mathematical models help us understand these disparate phenomena.Until recently, critical questions about the mathematical theory for the existence of solutions for the equation were unresolved, and solution of this equation strained the resources of the most powerful completers. However, mathematical advances have now made its solution routine, allowing accurate predictions of wave evolution. Early numerical techniques to solve the equation were slow and cumbersome. But now, several efficient techniques exist which can yield reliable results.Not only has the mathematical theory of water waves helped us to understand and protect our environment, but its insights have also had a significant impact on technological development. Although the solitary wave is now well understood, other water waves still have mysterious effects on our environment and remain objects of active mathematical research.

Finance

It deals with money and what happens when you borrow money, open a savings account to earn interest, or retire. When it comes to money, as you may have learned, there are many people who want to take your money in various clever ways. There is a saying "a fool and his money are soon parted". Knowing financial theory would keep you with your money throughout your life. So do not skimp on this section!

You can find here a collection of finance solvers related to middle school algebra. Of particular interest are the present value solver, mortgage duration calculator, mortgage payment calculator. There are many others to choose from, as well. You can also check comparing simple interest vs. compound interest, basics of mortgages, and explanation of present value vs. future value, and many more!

Geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this leads into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.

Physics

It is important that I emphasize here that physics cannot be derived from mathematics alone. Let me back up a bit and fill in the holes. In order to understand fully a physical process, physicists try to derive the process from other more fundamental concepts. For example, in the early 1600's Johannes Kepler constructed a model of the solar system which he then used to predict the exact locations of planets with hitherto unheard-of precision. This is usually the first step in trying to understand a physical system--figure out HOW it works and then try to figure out WHY it works that way. It took Sir Isaac Newton's formulation of gravity to explain why Kepler's model works. And it took Newton's discovery of three fundamental ways that matter interacts to derive his theory of gravity. So, in the end, starting with Newton's three "rules," you can derive Kepler's model: The planets move the way they do because of gravity, and gravity works the way it does because it follows three basic rules or "laws" for forces. This is what we mean by deriving a complicated physical concept from more simple ones.

Technology

Define technology, From the perspective of the classroom, technology can mean calculators, iPods, cell phones, Active Boards, computers, the internet, and on and on. Since the advent of machines that can start doing the things that kids are supposed to learn (spelling or addition) there has been a struggle with what to do by hand, and for how long to do it.

As far as Algebra goes, the biggest topic of discussion is the graphing calculator. Those of us that were working on Head First Algebra all learned Algebra before graphing calculators existed, so when we sat down to write the book, there was a discussion about how much to include them. We decided (as a team, editors, authors and all) that the best way to go was to assume that students would and could use a basic calculator to do division and multiplication but NOT solving equations. After all, the point of studying Algebra is to learn how to do that yourself. Here's the problem. Just knowing that a calculator that exists that can solve an equation presents a giant motivational challenge. "Why do I need to know how to do that, if the calculator can?" Ugh. That is a perfectly reasonable and typical question out of anyone learning Algebra. Especially if they think that Algebra is just about solving for X. Because if that's all it is a calculator can do that.

Health/life sciences

We also use algebra in life sciences in looking for the exact nutrients we need in our body. Like in knowing how many milliliters is equivalent to 8 glass of water that we need in our body. Good health is one of those things that we don't really notice until we get sick or injured, and then we really miss it. Like mathematics, health consists of many components; we are going to explore a few of them.

These mathematics activities focus on 1) assessing the nutritional value of fast food, 2) analyzing the numbers associated with our heart, and 3) looking at how medicines affect our bodies over time. Our heart plays an important role in your health. The heart moves oxygen and other nutrients to all the different parts of the body and helps carry away the waste products. Here are a few activities to get us thinking about our hearts. 1. Do you think your heart has beaten a billion times? 2. One's heart rate is usually reported in beats per minute. Take your pulse and figure out how many times your heart beats in 10 or 15 seconds. Use this to figure out your resting heart rate in beats per minute. 3. At this rate, how many times does your heart beat in one day? In these statements, it talks about the number of beats. Number there is referred to math.

Statistics/ demographics

Non-technical explanations of methods, with worked examples, enable students without a background in algebra, calculus or statistics to learn demographic methods, for interpreting demographic data and indices. The attributes of people in a particular geographic area. Used for marketing purposes, population, ethnic origins, religion, spoken language, income and age range are examples of demographic data.These uses numbers and equations in solving those numbers.Includes techniques for analysis of population at regional and local, as well as national, scales - of particular interest to geographers and planners but usually omitted from demographic texts

Sports/ entertainment

to determine the amount of supplies needed to run the concession stand (based on prior attendance stats) to graph data and track trends collected from the competitions to calculate budgets and determine whether a team is operating in the red (debt) or in the black (profit)

I can't really add to that save to point out two things:

1) Algebra is not an isolated branch of maths, but the building-blocks for mathematics, because it abbreviates and encodes the relationships between quantities or values, and the arithmetical steps needed to solve the problem.

2) All the above describe largely professional uses, or things like personal finance. It can also crop up in your hobbies, particularly crafts and obviously amateur science.

The Definition of an Anti-Symmetric Matrix:

If a square matrix, A, is equal to its negative transpose, -A', then A is an anti-symmetric matrix.

Notes:

1. All diagonal elements of A must be zero.

2. The cross elements of A must have the same magnitude, but opposite sign.

More precisely, I think you're asking whether the set of n X n matrices forms an abelian group under multiplication. The answer is no (assuming n>1). For example

(1 0)(0 1) = (0 1)

(0 0)(0 0) (0 0),

but

(0 1)(1 0) = (0 0)

(0 0)(0 0) (0 0). However, the set of n x n diagonalmatrices does form an Abelian set. This is true regardless of the direction of the diagonality, right-to-left or left-to-right. Note that the resulting matrix will also be diagonal, but always right-to-left.

Say there's a relation ~ between the two objects a and b such that a ~ b. We call ~ an equivalence relation if:

i) a ~ a.

ii) If a ~ b. then b ~ a.

iii) If a ~ b and b ~ c, then a ~ c.

Where c is another object.

The three properties above are called the reflexive, symmetric, and transitive properties. Were those three properties all that was needed to define the equalityrelation, we could safely call them axioms. However, one more property is needed first. To show you why, I'll give an example.

Consider the relation, "is parallel to," represented by . We'll check the properties above to see if is an equivalence relation.

i) a a.

Believe it or not, whether this statement is true is an ongoing debate. Many people feel that the parallel relation isn't defined for just one line, because it's a comparison. Well, if that were true, then you would have to say the same thing for everybinary equivalence relation; e.g., a triangle couldn't be similar to itself, or, even more preposterously, the statement a = a would have to be tossed out the window too. But, just to be formal, we'll use the following definition for parallel lines:

Two lines are not parallel if they have exactlyone point in common; otherwise they are parallel.

So, with that definition in hand, i) holds for .

ii) If a b, then b a. True.

iii) If a b and b c, then a c. True.

Thus, the relation is an equivalence relation, but two parallel lines certainly don't have to be equal! So, we need an additional property to describe an equality relation:

iv) If a ~ b and b ~ a, then a = b.

Let's check iv) and see if this works for our relation :

If a b and b a, then a = b. False. But, does it hold for the equality relation?

If a = b and b = a then a = b. True. This is what's known as the antisymmetricproperty, and is what distinguishes equality from equivalence.

But wait, we have a problem. We used the relation = in one of our "axioms" of equality. That doesn't work, because equality wasn't part of the signature of the formal languagewe're using here. By the way, the signature of the formal language that we are using is ~. So, any other non-logical symbol we use has to either be defined, or derived from axioms.

Well, we have three possible ways out of this. We can either:

1) Figure out a way to axiomize the = relation through the use of the ~ relation.

2) Define the = relation.

3) Add = to our language's signature.

Well, 1) is not possible without the use of sets, and since the existence of sets isn't part of our signature either, we'd have to define a set, or add it to our language. This isn't very hard to do, but I'm not going to bother, because the result is what we're going to obtain from 2).

Anyways, speaking of 2), let's define =.

For all predicates (also called properties) P, and for all a and b, P(a) if and only if P(b) implies that a = b.

In other words, for a to be equal to b, anyproperty that either of them have must also be a property of the other. In this case, the term propertymeans exactly what you think it means; e.g. red, even, tall, Hungarian, etc.

So, the million dollar question is, by defining =, are our properties now officially axioms? For three of the properties, the answer is no. In fact, because we just defined =, we've turned properties ii), iii), and iv) from above into theorems, not axioms. Why? Because, property iv)still has that = relation in it, which we had to define. So, iv) is a true statement, but we had to use another statement to prove it. That's the definition of a theorem! And, since the qualifier for iv)'s truth was that a ~ b and b ~ a, we can now freely replace b with a in ii), giving us "If a ~ a, then a ~ a." Well, now ii)'s proven as well, but we had to use iv) to do it. Thus, both ii) and iv) are now theorems. Finally, iii) can be proven in a similar was as ii) was, so it, too, is a theorem.

However, our definition of = only related a to b, it never related a to itself. Thus, we need to include i), from above, as an axiom.

Just for kicks, let's try plan 3) too.

The idea here is to make = a part of our signature, which means now we don't need to define it. In fact, we can't define it if we put it in our signature; because by placing it there, we're assuming that it's understood without definition. Therefore, iv) must now be assumed to be true, because we have no means to prove it; that sounds like an axiom to me! However, just like before, we can prove both ii) and iii)through the use of iv), so they get relegated back to the land of theorems and properties. Interestingly though, iv)makes no mention of reflexivity, and since our formal definition of = is gone, we have no way to prove i). Once again, we have to assume that it's true. Thus i) is an axiom as well.

So, to paraphrase our two separate situations:

In order for the relation ~ to be considered an equality relation between the objects a and b, oneaxiom must be satisfied if we define =:

1) For all a, a ~ a,

as well as three theorems:

1) If a ~ b, then b ~ a

2) If a ~ b and b ~ c, then a~ c, where c is another object

3) For all a and b, if a ~ b and b ~ a, then a = b.

Additionally,

In order for the relation ~ to be considered an equality relation between the objects a and b, twoaxioms must be satisfied if we put = into our signature:

1) For all a, a ~ a

2) For all a and b, if a ~ b and b ~ a, then a = b,

as well as two properties:

1) If a ~ b, then b ~ a

2) If a ~ b and b ~ c, then a~ c,

where c is another object.

What the one right above did is include "=" into our formal language, but "=" is equality, so he actually came up with a fairly well axiom before he finishes with the circular looking one.

His axiom: We say ~ is an equality relation means whenever x ~ y, for any condition P, P(x) iff P(y)

The axiom is the bolded part.

After discussion with my Math prof. this morning, that axiom becomes a properties follows from this more formal definition. It does not need to include any more things then what we already have for the formal language.

We say ~ is an equality relation on a set A if (a set is something that satisfies the set axioms)

For any element in A, a ~ a.

If follows that P(a) is true and y ~ a, then P(y) is also true, vice versa. Because in this case, y has to be a for it to work.

You might argue well the definition for an equivalence relation have this statement in it too, does that mean equivalence IS equality?

No! It's the other way around, equality is equivalence. Equality is the most special case for any relation, say *, where a * a.

Take an equivalence relation, say isomorphisms for instance (don't know what that word mean? Google or as it on this website), we know any linear transformation T is isomorphic to T, in particular this isomorphism IS equality. Of course it would be boring if isomorphism is JUST equality, so it's MORE.

The other axioms in a definition of a relation are to differ THEM from equality, because equality is the most basic. Equality must always be assumed, it always exist, any other relation is built upon it. It is the most powerful relation, because ALL relations have it. (I mean all relations, say *, such that a * a for all a must at least be equality)