Algebra is thought a fundamental branch of mathematics which puts the light on how to deal with all situations involving numbers and variables. By Nature and historically, there is so much to articulate about teaching and studying of Algebra as a generalized arithmetic which goes through systematic mathematical operations such as induction, generalization and proof. So, gradually pupils get several means to enhance their Algebra level, for example by getting the information from tutors or software programs, which offer bit by bit solutions. Packages designed for algebra studying offer all the available methods for resolving particular problems with a technological touch. Many students don’t even know how very usable Algebra is! They complain about its impracticality neglecting that Algebra, generally math, instructs their mind how to think logically and correctly. The school is the most orthodox way of learning algebra, from being a kid till becoming an adult students get their lessons from the teacher. With the wide growth of technology, new techniques have been institutionalized to learn Algebra, such as using software packages which is a more handy way to learn Algebra. These packages deliver information in a step-by-step approach in to student’s minds.

Algebra’s Covered Area

Same as any other branch of science, Algebra addresses a lot of fields and includes many theories and constructs. Gcf, or Greatest Common Factor , is one such concepts. Gcf means to rewrite the polynomial as a product of simpler polynomials or of polynomials and monomials. Solving fractions is one of the principal parts of algebra which essentially gives pupils the chance to apply it to the real world. non-linear function represents any function which is a solution of a quadratic polynomial . Among other significant factors of algebra , multiplying and dividing radicals is also one of the primary ones. An individual can multiply and divide with radicals only if the index, or root, is the same. Other related areas are Adding and Subtracting Radicals; an individual can add or subtract radical terms only if both the index and the radicand are the same. Matrix operations, another primary areas of algebra which has a wide applicability when it comes to the real life, includes operations such as adding, subtracting, multiplying and dividing. Other primary areas are finding x-intercept of a line and y-intercept of a line – to get the x-intercept of a line, substitute zero for y in the equation and vice versa for finding y-intercept of a line.

Although a lot of students are not really sure of what is included in algebra but it is not terrifying as a lot of people think. Some of the fundamental functions of algebra include finding least common multiple , factoring difference of cubes and reducing fractions.

When mathematics is leaned by pupils, particularly at college stage it is very significant for them to apply some planning and also some persistence in learning mathematical concepts from easy to hard ones. There are a lot of college mathematics courses of study at first stage that are built-up around constructing a solid path for the road into higher mathematics as they enable learning the core skills that are required.

Who offers aid?

For helping you in mastering algebra, there is a number of computer software packages and these include programs such as algebra tutors. A tutor for algebra is also a surefire way to enhancing your algebra skills. The software products are very good and will help you with all aspects of algebra including quadratic inequalities, graphing rational inequalities, simplifying complex fractions with different denominators and on master the basic and advanced topics of algebra.

If you start learning mathematics from the beginning, it is a good idea to do one of the first stage courses of study as they will teach you the very basics and build on that so that you are able to get up to the more sophisticated stage of maths. The beginning courses also teach you the science of algebra through the use of a significant number of different models and methods so this is a perfect starting point for the novice to algebra.

Why Virtual Tutors are Not Perfect?

The only trouble with maths courses of study is that they sometimes do not follow things in a very logical order and this especially is the case when you are a pupil at grade school. A lot of the students at school are actually very ill-prepared for learning algebra and they are not knowledgeable of the fundamental principles of this valuable subdivision of mathematics.

Algebra is a large component of mathematics that uses generalization by replacing alphabetic characters for numbers. It is this generalization that often frightens many and shudders some. topics of algebra range from working with inequalities to factoring binomials all the way to finding the inverse of a matrix. Most people start to study Algebra by adding, subtracting, reducing and simplifying algebraic expressions. later on, they would move on to more advanced stages of solving algebraic equations using the least common multiples and converting fractions in to decimals.

Exponents and Radicals and Graphing Oh My!

There are many another higher level themes. Firstly there are exponents. An exponent is the small number placed to the right and slightly above a number or algebraic expression. In this example (x + y)³ where the 3 is the exponent and denotes the power to which that number is raised. The above algebraic expression is expressed as, x plus y to the third power. While working with powers you will often see exercises that require you to add, subtract, multiply and divide. You can work with rational and negative exponents . If that isnt enough to make your head spin, then you can move on to radicals. A radical, in simple terms, is the reversing of an power. The V beside 4 denote a radical expression which means, the square root of 4, which equals 2. The inverse of “V” symbol is “^” which denotes that the number it refers to is a multiple of the principle number. Therefore, 2^2 which is read as 2 to the 2nd power, equals 4. Equally with exponents, radical expressions can be added, subtracted, multiplied and divided. Radicals can be switched into powers and exponents back into roots. If you find that exponents and roots don’t really shake your interest, you could move on to graphing. The best way to start with graphing is to draw lines and try to figure out if they are horizontal, vertical or neither. Another question to ask is whether the line has an x-intercept or a y-intercept Can you find the slope of the line? After you master graphing lines, a whole world of circles, parabolas and hyperbolas awaits you!

Help is Available

If you are learning algebra and this all seems a little too much, dont sweat. Did you know that there are many resources out there that can help you master fractional equations, quadratic formulas and polynomials. The custom is to decide on a math tutor, but the cutting-edge algebraic software are no different. In fact, they are as good as math tutors. Know that frustration is not your only friend because Any of these instruments can help you become an algebra expert in no time.

That banks are very risky – is proven by the inordinate number of regulatory institutions which supervise banks and their activities. The USA sports a few organizations which insure depositors against the seemingly inevitable vicissitudes of the banking system.

The FDIC (Federal Deposit Insurance Corporations) insures against the loss of every deposit of less than 100,000 USD. The HLSIC insures depositors in saving houses in a similar manner. Other regulatory agencies supervise banks, audit them, or regulate them. It seems that you cannot be too cautious where banks are concerned.

The word “BANK” is derived from the old Italian word “BANCA” – bench or counter. Italian bankers used to conduct their business on benches. Nothing much changed ever since – maybe with the exception of the scenery. Banks hide their fragility and vulnerability – or worse – behinds marble walls. The American President, Andrew Jackson, was so set against banks – that he dismantled the nascent central bank – the Second Bank of the United States.

A series of bank scandals is sweeping through much of the developing world – Eastern and Central Europe to the fore. “Alfa S.”, “Makedonija Reklam” and TAT have become notorious household names.

What is wrong with the banking systems in Central Eastern Europe (CEE) in general – and in Macedonia in particular? In a nutshell, almost everything. It is mainly a crisis of trust and adverse psychology. Financial experts know that Markets work on expectations and evaluations, fear and greed. The fuel of the financial markets is emotional – not rational.

Banks operate through credit multipliers. When Depositor A places 100,000 USD with Bank A, the Bank puts aside about 20% of the money. This is labelled a reserve and is intended to serve as an insurance policy cum a liquidity cushion. The implicit assumption is that no more than 20% of the total number of depositors will claim their money at any given moment.

In times of panic, when ALL the depositors want their money back – the bank is rendered illiquid having locked away in its reserves only 20% of the funds. Commercial banks hold their reserves with the Central Bank or with a third party institution, explicitly and exclusively set up for this purpose.

What does the bank do with the other 80% of Depositor A’s money ($80,000)? It lends it to Borrower B. The Borrower pays Bank A interest on the loan. The difference between the interest that Bank A pays to Depositor A on his deposit – and the interest that he charges Borrower B – is the bank’s income from these operations.

In the meantime, Borrower B deposits the money that he received from Bank A (as a loan) in his own bank, Bank B. Bank B puts aside, as a reserve, 20% of this money – and lends 80% (=$64,000) to Borrower C, who promptly deposits it in Bank C.

At this stage, Depositor A’s money ($100,000) has multiplied and become $244,000. Depositor A has $100,000 in his account with Bank A, Borrower B has $80,000 in his account in Bank B, and Borrower C has $64,000 in his account in Bank C. This process is called credit multiplication. The Western Credit multiplier is 9. This means that every $100,000 deposited with Bank A could, theoretically, become $900,000: $400,000 in credits and $500,000 in deposits.

For every $900,000 in the banks’ books – there are only 100,000 in physical dollars. Banks are the most heavily leveraged businesses in the world.

But this is only part of the problem. Another part is that the profit margins of banks are limited. The hemorrhaging consumers of bank services would probably beg to differ – but banking profits are mostly optical illusions. We can safely say that banks are losing money throughout most of their existence.

The SPREAD is the difference between interest paid to depositors and interest collected on credits. The spread in Macedonia is 8 to 10%. This spread is supposed to cover all the bank’s expenses and leave its shareholders with a profit. But this is a shakey proposition. To understand why, we have to analyse the very concept of interest rates.

Virtually every major religion forbids the charging of interest on credits and loans. To charge interest is considered to be part usury and part blackmail. People who lent money and charged interest for it were ill-regarded – remember Shakespeare’s “The Merchant of Venice”?

Originally, interest was charged on money lent was meant to compensate for the risks associated with the provision of credit in a specific market. There were four such hazards:

First, there are the operational costs of money lending itself. Money lenders are engaged in arbitrage and the brokering of funds. In other words, they borrow the money that they then lend on. There are costs of transportation and communications as well as business overhead.

The second risk is that of inflation. It erodes the value of money used to repay credits. In quotidian terms: as time passes, the Lender can buy progressively less with the money repaid by the Borrower. The purchasing power of the money diminishes. The measure of this erosion is called inflation.

And there is a risk of scarcity. Money is a rare and valued object. Once lent it is out of the Lender’s hands, exchanged for mere promises and oft-illiquid collateral. If, for instance, a Bank lends money at a fixed interest rate – it gives up the opportunity to lend it anew, at higher rates.

The last – and most obvious risk is default: when the Borrower cannot or would not pay back the credit that he has taken.

All these risks have to be offset by the bank’s relatively minor profit margin. Hence the bank’s much decried propensity to pay their depositors as symbolically as they can – and charge their borrowers the highest interest rates they can get away with.

But banks face a few problems in adopting this seemingly straightforward business strategy.

Interest rates are an instrument of monetary policy. As such, they are centrally dictated. They are used to control the money supply and the monetary aggregates and through them to fine tune economic activity.

Governors of Central Banks (where central banks are autonomous) and Ministers of Finance (where central banks are more subservient) raise interest rates in order to contain economic activity and its inflationary effects. They cut interest rates to prevent an economic slowdown and to facilitate the soft landing of a booming economy. Despite the fact that banks (and credit card companies, which are really banks) print their own money (remember the multiplier) – they do not control the money supply or the interest rates that they charge their clients.

This creates paradoxes.

The higher the interest rates – the higher the costs of financing payable by businesses and households. They, in turn, increase the prices of their products and services to reflect the new cost of money. We can say that, to some extent, rather than prevent it, higher interest rates contribute to inflation – i.e., to the readjustment of the general price level.

Also, the higher the interest rates, the more money earned by the banks. They lend this extra money to Borrowers and multiply it through the credit multiplier.

High interest rates encourage inflation from another angle altogether:

They sustain an unrealistic exchange rate between the domestic and foreign currencies. People would rather hold the currency which yields higher interest (=the domestic one). They buy it and sell all other currencies.

Conversions of foreign exchange into local currency are net contributors to inflation. On the other hand, a high exchange rate also increases the prices of imported products. Still, all in all, higher interest rates contribute to the very inflation that are intended to suppress.

Another interesting phenomenon:

High interest rates are supposed to ameliorate the effects of soaring default rates. In a country like Macedonia – where the payments morale is low and default rates are stratospheric – the banks charge incredibly high interest rates to compensate for this specific risk.

But high interest rates make it difficult to repay one’s loans and may tip certain obligations from performing to non-performing. Even debtors who pay small amounts of interest in a timely fashion – often find it impossible to defray larger interest charges.

Thus, high interest rates increase the risk of default rather than reduce it. Not only are interest rates a blunt and inefficient instrument – but they are also not set by the banks, nor do they reflect the micro-economic realities with which they are forced to cope.

Should interest rates be determined by each bank separately (perhaps according to the composition and risk profile of its portfolio)? Should banks have the authority to print money notes (as they did throughout the 18th and 19th centuries)? The advent of virtual cash and electronic banking may bring about these outcomes even without the complicity of the state.

Sam Vaknin ( samvak.tripod.com ) is the author of Malignant Self Love – Narcissism Revisited and After the Rain – How the West Lost the East. He served as a columnist for Global Politician, Central Europe Review, PopMatters, Bellaonline, and eBookWeb, a United Press International (UPI) Senior Business Correspondent, and the editor of mental health and Central East Europe categories in The Open Directory and Suite101.

Until recently, he served as the Economic Advisor to the Government of Macedonia.

Visit Sam’s Web site at samvak.tripod.com