Compound Interest

What is compound interest?

When you borrow money from a bank, you pay interest. Interest is really a fee charged for borrowing the money, it is a percentage charged on the principle amount for a period of a year - usually.

If you want to know how much interest you will earn on your investment or if you want to know how much you will pay above the cost of the principal amount on a loan or mortgage, you will need to understand how compound interest works.

* Compound interest is paid on the original principal and on the accumulated past interest.

Formula:

P is the principal (the initial amount you borrow or deposit)

r is the annual rate of interest (percentage)

n is the number of years the amount is deposited or borrowed for.

A is the amount of money accumulated after n years, including interest.

When the interest is compounded once a year:

A = P(1 + r)n

However, if you borrow for 5 years the formula will look like:

A = P(1 + r)⁵

This formula applies to both money invested and money borrowed.

Frequent Compounding of Interest:

What if interest is paid more frequently?
Here are a few examples of the formula:

Annually = P × (1 + r) = (annual compounding)

Quarterly = P (1 + r/4)⁴ = (quarterly compounding)

Monthly = P (1 + r/12)¹² = (monthly compounding)

Introduction

A simple linear equation is a statement of equality between two algebraic expressions involving an unknown quantity called the variable. In a linear equation the power of the variable is always equal to 1. The two sides of an equation are called Left-Hand Side (LHS) and Right Hand Side (RHS).

Equations of condition

The two expressions (LHS and RHS) are equal only for a particular value of the variable (x). These equations are called equations of condition. An equation of condition is generally referred to as an equation.

Identical equations or identities

The two expressions (LHS and RHS) are always equal for any value we give to the variable. Equations that are true for any value of the variable are called identical equations or briefly an identity. Solving linear equations The process of finding the value of the unknown quantity for which the equation is true, is called solving the equation. The value so found is called the root or solution of the equation. The process of solving a simple equation depends upon the following axioms: Addition property, Subtraction property, Multiplication and division property.

Ratio and Proportions

CONTENTS

Ratio
Comparing ratios
Proportion
Rate
Converting rates

Ratio

A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.

Examples:

Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.

1) What is the ratio of books to marbles?
Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4.
Two other ways of writing the ratio are 7 to 4, and 7:4.

2) What is the ratio of videocassettes to the total number of items in the bag?
There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total.
The answer can be expressed as 3/15, 3 to 15, or 3:15.

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Comparing Ratios

To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions.

Example:

Are the ratios 3 to 4 and 6:8 equal?
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal 24, the answer is yes, the ratios are equal.

Remember to be careful! Order matters!
A ratio of 1:7 is not the same as a ratio of 7:1.

Examples:

Are the ratios 7:1 and 4:81 equal? No!
7/1 > 1, but 4/81 <>

Are 7:14 and 36:72 equal?
Notice that 7/14 and 36/72 are both equal to 1/2, so the two ratios are equal.

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Proportion

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.
3/4 = 6/8 is an example of a proportion.

When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Question marks or letters are frequently used in place of the unknown number.

Example:

Solve for n: 1/2 = n/4.
Using cross products we see that 2 × n = 1 × 4 =4, so 2 × n = 4. Dividing both sides by 2, n = 4 ÷ 2 so that n = 2.

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Rate

A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h. The fraction expressing a rate has units of distance in the numerator and units of time in the denominator.
Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.

Example:

Juan runs 4 km in 30 minutes. At that rate, how far could he run in 45 minutes?
Give the unknown quantity the name n. In this case, n is the number of km Juan could run in 45 minutes at the given rate. We know that running 4 km in 30 minutes is the same as running n km in 45 minutes; that is, the rates are the same. So we have the proportion
4km/30min = n km/45min, or 4/30 = n/45.
Finding the cross products and setting them equal, we get 30 × n = 4 × 45, or 30 × n = 180. Dividing both sides by 30, we find that n = 180 ÷ 30 = 6 and the answer is 6 km.

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Converting rates

We compare rates just as we compare ratios, by cross multiplying. When comparing rates, always check to see which units of measurement are being used. For instance, 3 kilometers per hour is very different from 3 meters per hour!
3 kilometers/hour = 3 kilometers/hour × 1000 meters/1 kilometer = 3000 meters/hour
because 1 kilometer equals 1000 meters; we "cancel" the kilometers in converting to the units of meters.

Important:

One of the most useful tips in solving any math or science problem is to always write out the units when multiplying, dividing, or converting from one unit to another.

Example:

If Juan runs 4 km in 30 minutes, how many hours will it take him to run 1 km?
Be careful not to confuse the units of measurement. While Juan's rate of speed is given in terms of minutes, the question is posed in terms of hours. Only one of these units may be used in setting up a proportion. To convert to hours, multiply
4 km/30 minutes × 60 minutes/1 hour = 8 km/1 hour
Now, let n be the number of hours it takes Juan to run 1 km. Then running 8 km in 1 hour is the same as running 1 km in n hours. Solving the proportion,
8 km/1 hour = 1 km/n hours, we have 8 × n = 1, so n = 1/8.

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Average Rate of Speed

The average rate of speed for a trip is the total distance traveled divided by the total time of the trip.

Example:

A dog walks 8 km at 4 km per hour, then chases a rabbit for 2 km at 20 km per hour. What is the dog's average rate of speed for the distance he traveled?
The total distance traveled is 8 + 2 = 10 km.
Now we must figure the total time he was traveling.
For the first part of the trip, he walked for 8 ÷ 4 = 2 hours. He chased the rabbit for 2 ÷ 20 = 0.1 hour. The total time for the trip is 2 + 0.1 = 2.1 hours.
The average rate of speed for his trip is 10/2.1 = 100/21 kilometers per hour.

Average rate of speed

Polynomials

A polynomial is a mathematical expression involving a sum of Powers in one or more variables multiplied by coefficient. A polynomial in one variable with constant coefficient is given by

(1)

The individual summands with the coefficient (usually) included are called monomials (Becker and Weispfenning 1993, p. 191), whereas the products of the form in the multivariate case, i.e., with the coefficients omitted, are called terms (Becker and Weispfenning 1993, p. 188). The highest power in a univariate polynomial is called its order, or sometimes its degree.

Any polynomial with can be expressed as

(2)

where the product runs over the roots of and it is understood that multiple roots are counted with multiplicity.

A polynomial in two variables (i.e., a bivariate polynomial) with constant coefficient is given by

(3)

The sum of two polynomials is obtained by adding together the coefficient sharing the same powers of variables (i.e., the same terms) so, for example,

(4)

and has order less than (in the case of cancellation of leading terms) or equal to the maximum order of the original two polynomials. Similarly, the product of two polynomials is obtained by multiplying term by term and combining the results, for example

			(5)
			(6)

and has order equal to the sum of the orders of the two original polynomials.

A polynomial quotient

(7)

of two polynomials and is known as a rational funtion. The process of performing such a division is called longdivision, with synthetic division being a simplified method of recording the division.

For any polynomial , divides , meaning that the polynomial quotient is a rational polynomial or, in the case of an integer polynomial, another integer polynomial (pers. comm., N. Sato, Nov. 23, 2004).

Exchanging the coefficient of a univariate polynomial end-to-end produces a polynomial

(8)

whose roots are reciprocal of the original roots.

horner's root provides a computationally efficient method of forming a polynomial from a list of its coefficients, and can be implemented in Mathematica as follows.

  Polynomial[l_List, x_] := Fold[x #1 + #2&, 0, l]

The following table gives special names given to polynomials of low orders.

polynomial order	polynomial name
2	quadratic polynomial
3	cubic polynomial
4	quartic
5	quintic
6	sextic

Polynomials of fourth degree may be computed using three multiplications and five additions if a few quantities are calculated first (Press et al. 1989):

(9)

where

			(10)
			(11)
			(12)
			(13)
			(14)

Similarly, a polynomial of fifth degree may be computed with four multiplications and five additions, and a polynomial of sixth degree may be computed with four multiplications and seven additions.

Polynomials of orders one to four are solvable using only rational operations and finite root extractions. A first-order equation is trivially solvable. A second-order equation is soluble using the quadratic equation. A third-order equation is solvable using the cubic equation. A fourth-order equation is solvable using the quartic equation. It was proved by Abel and Galois using group theory that general equations of fifth and higher order cannot be solved rationally with finite root extractions (Abel's impossibility theorem).

However, solutions of the general quintic equation may be given in terms of Jacobi theta functions or hypergeometric functions in one variable. Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein's method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or "Siegel functions" must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the th order polynomial equation in finite form. These functions turned out to be "natural" generalizations of the elliptic functions.

Irrational Numbers

Famous Irrational Numbers

	PI is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...)
	The number e another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...)
	PI is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...)
	The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820... (and more ...)
	Many square roots, cube roots, etc are also irrational numbers. Examples: √3 1.7320508075688772935274463415059 (etc) √99 9.9498743710661995473447982100121 (etc) But √4 = 2, and √9 = 3, so not all roots are irrational.

History of Irrational Numbers

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and it was was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!