The Probability

Probability is a measure or estimation of the likeliness or likelihood that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call probability. Thus the higher the probability of an event, the more certain we are that the event will occur. A simple example would be the toss of a fair coin. Since the the 2 outcomes are deemed equiprobable, the probability of "heads" equals the probability of "tails" and each probability is 1/2 or equivalently a 50% chance of either "heads" or "tails".                              Source - Wikipedia

Probability Questions: -   Let us find the prob for each event when a coin is tossed five hundred times and the frequency of the two events is given as Head: Three Hundred, tails = two hundreds.

Solution : - We are given that the total number of trials is five hundred. Therefore, the number of times E happens, i.e., the number of time the head comes up is three hundred.  Therefore the probability of head = the number of head / Total number of the trials Or P (E) = 300/ 500 = 0.6. Similarly the probability of tail = the number of tail / Total number of the trials Or P (F) = 200/ 500 = 0.4. Hence the prob of success is zero point six and prob of failure is zero point four. The point which is to be noted down is that the total sum of both the probability is equal to one.

Example: - Let us find the prob, when the two coins are tossed simultaneously for four hundred and eighty times and we get the following events. Two heads: one hundred and twenty times (120), One head:  Two hundred and three times. (203), No head:  one hundred and fifty seven time (157)

Solution:-  (1) The probability of getting two heads = the total number of heads divided by the total number of chances. The probability of getting two heads = 120 / 480 = ¼ = 0.25. The probability of getting one heads = 203 / 480   = 0.42292

The probability of getting no heads = 157 / 480 = 0.32708

We can see that P (E₁) + P (E₂) + P (E₃) = 1, cover all the outcomes of a trial.
Or P (E₁) + P (E₂) + P (E₃) = 0. 25 + 0.42292 + 0. 32708 = 1

Example:- The record of a weather station  shows that the out of past two hundred consecutive days, its weather forecast is correct for one hundred and forty five ( 145)  times. Let us calculate the probability that on a given day it was correct and the probability that on a given day it was not correct.

Solution: - The record of a weather station is available for two hundred consecutive days. The Probability P (E₁) that on a given day the forecast was correct = the number of days when the forecast was correct/ Total number of days for which record is available  Or P (E₁) = 145 / 200 = 0.725. P (E₂) that on a given day it was not correct = 55/200 = 0.275, Notice that P (the forecast was correct + the forecast was not correct) = 0.275 + 0.725 = 1

How do you Find Probability: -The word probably, doubts most probably, chances etc are used to define the uncertainty. The uncertainty of the probably etc can be measured numerically by means of probability. Therefore the probability started with gambling, it has been used extensively in the field of physical sciences, commerce, etc. Let there are n number of trials. The probability of an event E happening, is given by P (E) =
Number of trials in which the event happened / the total number of trials.

Let there are n number of trials. The probability of an event E happening, is given by P (E) = Number of trials in which the event happened / the total number of trials.

Derivative Of Tanx?

Derivative of tan x is a tool which is used to find the differentiation of tan x. Before we understand the differentiation of tan x it is important to understand about tan x. If we will describe tan x in terms of sin x and cos x, then it can be defined as the ratio of Sin x and Cos x.

Mathematically, tan x = Sin x/ Cos x

It is important to note that tan x is the reciprocal of the Cot x and vice versa. Apart from that it is also important to note that the derivative is simple denoted as d/dx. Now the question arises what is the Derivative of Tanx? So it can be simply written as d/dx (tan x).

The value of d (tan x)/dx is equal to Sec^2 x. Now in the coming context, you will learn how the derivative of tan x that is d (tan x)/dx is equal to Sec^2 x. This can be explained by the basic concepts of trigonometry and rules of derivative. 

So to understand the derivative of tan x it is really important to have thorough knowledge of the trigonometry as well as the differentiation. The proof of the derivative of tan x can be done with the help of the rules of the differentiation and trigonometry which are as shown below:-
 
We know that tanx = Sin x/ Cosx
 
So the derivative of tan x, that is, d (tan x)/dx = d (Sin x/Cosx)/ dx
 
We know that by the divisibility rule of differentiation,
 
That is, d (u/v)/ d x = (v du/dx – u d v/dx)/ v^2
 
So d (tan x)/dx = d (Sin x/Cos x)/ dx
= (Cos x d(Sinx)/dx- Sin x d(Cosx)/dx )/ Cos ^2 x
= (Cos x. Cos x – Sin (- Sinx) ) / Cos^2 x
= ( Cos^2x + Sin^2x) / Cos ^ 2x
 
We know that Cos^2x + Sin^2x = 1
 
Therefore, d (tanx)/dx = 1/ Cos^2 x
 
We Know that 1/ Cos x = Sec x (This is as per the trigonometric rules)
 
So d (tan x)/dx = 1/ Cos^2 x = Sec^2 x (This is in line with the above equation)
 
Hence we have seen the proof that d/dx of tanx is equal to Sec^2 x, that is, d (tanx)/dx = 1/ Cos^2 x = Sec^2 x. But from above proof, we have learnt that without knowing the differentiation rules as well as trigonometric rules, it is impossible to prove the differentiation result of the trigonometric function.

Differentiation of one function can also help in finding the differentiation of other functions. So it is very important that we should know the differentiation of each and every function. For example sometimes the question requirement is to find the differentiation of the trigonometric as well as exponential, so in that case to find the differentiation we need to know the rules along with the individual derivative of the functions.
 
We can make use of Second Derivative Calculator to find out further derivatives.

Normal Distributions

Normal Distributions

Normal distribution is also known as Gaussian distribution. It is one of the most commonly found probability distribution when studying theory of probability; more specifically when studying continuous probability distribution. Normal distribution function is a function in which the output is the probability of the occurrence of an event is such that it lies between two real numbered values. This can be illustrated with the help of the following example.

Consider a group of students that have taken a test. The distribution of marks of the students would be a normal distribution. The corresponding probability distribution of the probability of marks obtained by a particular student would also be normally distributed.

This type of distribution (abbreviated as dist’n) is very important in statistics. When the variable in question is real valued as well as random, then this type of dist’n is used when otherwise the dist’n is now known. It finds application in the fields of social sciences and natural sciences.

It is because of the central limit theorem that this dist’n is very useful. According to this theorem, if the conditions influencing the random variable are mild then the probability distribution is normally distributed around the mean. Even of the original dist’n is not normal, it would still give us a dist’n that is centered around the mean. In general the curve of a normal distribution would look as follows:



As we can see in the above picture, the shape of the curve resembles that of a bell. Therefore a normal dist’n curve is sometimes also called a bell curve. However this is not the only dist’n that is bell shaped, there are others as well, such as: Cauchy’s, Student’s, logistic etc. The normal dist’n function can be given by the following formula:
f(x,μ,σ)=1/(σ√2π)*e^(-(x-μ)^2/(2σ^2 ))

Where,
μ is the mean of the dist’n. It is also sometimes called the expected value of the dist’n. It can also be the median or the mode of the data set.
σ is the standard deviation of the data set. That makes σ^2 the variance of the distribution.

A random variable that follows this type of Gaussian distribution is said to be normally distributed. It is also sometimes called a normal deviate. In this dist’n if we have μ=0 and σ=1, then the dist’n is called a standard normal dist’n or a unit normal dist’n and the random variable which follows this dist’n is said to be standard normal deviate.

The value of a normal dist’n is practically zero when the x value goes beyond 3 standard deviations on either side of the mean. That is why this dist’n becomes useless when there are many outliers in the data set.

The most simple case of a normal dist’n that is called the standard normal dist’n can be defined by the probability density function as follows:
φ(x)= 1/√2π e^(-1/2 x^2 )

The total area under this curve is said to be 1.

Math Phobia

Math phobia is a very common thing in children and parents. As a parent, do you like math. When you were in school, did you fail math was hard, or boring. Sometimes math just looks too difficult. If you let that mental attitudes fester, then you might end up with a phobia. At that time you should take steps to remove the dread of math, but what if your child tells you the same thing in different words – that they have a math phobia.

For Example : Find the value of (1000π)^(2/3)*(512π)^(4/3)
 
Solution : This problem looks hard for some student 
But if you observe
=>1000=2^3*5^3
=>512=2^9
=>(1000π)^(2/3)*(512π)^(4/3)
=>(2^3*5^3π)^(2/3)*(2^9)^(4/3) 
=>(2^3)^(2/3)*(5^3)^(2/3)*π^(2/3)*(2^9)^(4/3)*(π)^(4/3)
=>(2^2)*5^2*2^12*π^(2/3)*π^(4/3)
=>2^(14)*5^2*(π)^[(2/3)+(4/3)]
=>409600π^2                      

Math phobia is the one which is shown by many students, math phobia is the illogical, intense fear of not succeeding in math. It is well known that one is unable to handle the difficulty associated with learning math. Many people assume that math phobia and an inability to be successful in mathematics are inherited from one's parents. Several legitimate factors contribute to, and increase the severity of, this internal representation.

The Teacher

Math tutors play the most important role to remove the dread of math , we know the fact that many students experience math phobia in the traditional classroom, teachers should design classrooms that will make student  feel more successful. The teacher should prepare students in such a manner so that they must have the  level of success or a level of failure which they can tolerate. Therefore, incorrect responses from math tutor must be handled in a positive way to encourage student.

The student can take help of an online math tutor as well. The math tutor needs to stay eager whatever the student’s viewpoint. Like small children like numbers and games with math so they can like math if help throw online math tutor. The methods which are used by the math tutor may affect whether a student feels successful and develops mathematical self-confidence. Finally, parents and teacher attitudes can positively or negatively influence students attitudes toward mathematics, which in turn affect their levels of confidence and this method may useful to remove the dread of math phobia.

Summary:

Math  phobia is common in almost every student. Parents and teacher's right step at the right time can remove this math phobia. Parents should take help from math help online website. Tutor develops student's   mathematical self-confidence.

Educating Through Online

Idea of  Education through Online build on much of the fast development in the field of communication. Practical learning is a relatively new idea in teaching. Online tutoring is an interactive process in which students work in complete coordination with experienced tutors. The interaction between the student and the tutor is one-on-one, live and in real time. All relevant study material is sent via email, and students can engage in live chat with their tutor. For example student has a doubt with a math problem then online math tutor will send step by step solution via  email and via live chat.

Students are taught a mixture of common subjects like English, chemistry or Math .Here math is a subject many students develop a phobia about it and online math tutoring services are ones that are now being in demand. The online Math tutors can do away with the math phobia of many students.

This will because of the learning process for the student. Some of the math tutor will use the animation videos and graphics. The online Math Tutors often use whiteboards. Which gives the tutor and the student a shared screen space. In other words, it serves the purpose of a conventional blackboard that is normally used in a traditional classroom. Math helper try to understand the problems given by each student one-on-one and develop a teaching program as per their convenience.

For example if the student is in grade 3 then math tutor will develop the program as per their understanding label.

Example Which is the same as 10–3
a)4-3
b)4+3                                
c)4/3
d)4*3

Advantages of online tutoring is 24/7  availability and learning at your speed. Online tutoring sessions are customized to match the needs of each individual student. Tutor refer student friendly training methodology that makes the most difficult problems seem like an easy problem for the students.

The dedicated online math tutor can help any student who is weak in maths. With the expert guidance of the tutors a student can attempt any professional exam .The best part about online tutoring is that it operates across all over the continents and students from all over the globe.

Summary : Online tutoring is use full for all grade students and parents, tutors are available 24/7.Education through online is convenient for students, being at home student can clarify their doubts. The online Math tutors can do away with the math phobia of many students.

Online Math Tutoring

In modern life internet is one of the most powerful and far-reaching communication tool and more over online math tutor is one of the most powerful tools. Math Online is a high quality, independent online math tutoring program. For some children, math will be the most difficult thing,When a teacher imparts math knowledge to a student over the Internet, the process is known as online math tutoring.

Now a day free online math tutor is available,these math tutors will teach you maths problem step by step 

For Example Solve 2x^2+7x+6

Here we can write 7x as 4x+3x

=>2x^2+4x+3x+6

Take 2x common

And 3 common 

=>2x(x+2)+3(x+2)

Here (x+2) is common

=>(x+2)(2x+3)

=>x=-2

Or x=-3/2

Math tutor helps parents and children, some are interactive math games, some are tips and videos for studying, some online tutoring, and some others are to help parents learn to teach their kids math concepts and become children math helper.

Online math tutor, you get 24/7 availability of classrooms with highly qualified instructors. You do not need to step out of your house for math tutor and can study at a time of your own choice, with the help of  math tutors you can even schedule your class according to your convenience and interact with the subject matter maths expert as and when required. 

With the help of free online math tutor you can clarify your doubt at any time and any where.

Parallelogram Properties

Properties of a parallelogram

A quadrilateral is a plane geometric figure that has four sides and four vertices. If in such a quadrilateral, both the pairs of the two opposite sides are parallel and they are also congruent to each other, then such a quadrilateral is called a parallelogram. (Abbreviated as ||gm). The figure below shows a sample parallelogram.



In the above picture, ABCD is a parallelogram. The two sides AB and DC are parallel to each other. This is indicated by the single arrows. The other two sides AD and BC are also parallel to each other. This is indicated by double arrows on both the sides. Also the length of the side AB is same as that of DC and the length of the side AD is same as that of BC. The four angles of the parallelogram are: Properties of parallelogram:

1. The first property we already stated in the definition of the parallelogram, that the opposite sides are parallel to each other.

2. The opposite sides are also congruent to each other. This we saw in the figure above and its description.

3. The opposite angles are congruent. Thus in the above figure,
4. The adjacent angles are supplementary. Thus in the above figure:
mmmm
5. The diagonal of a || gm divide the || gm into two congruent triangles. This can be shown with the help of the following figure:

In the above figure, ABCD is a || gm. AC is its diagonal. Now if we consider the triangles DAC and BAC, we see that one of the sides AC is common to both the triangles. We already established as the property number 2 of the || gm that the opposite sides are congruent. Therefore the side DC is congruent to AB and DA is congruent to CB. Therefore by the SSS congruency theorem, the two triangles DAC and BAC are congruent. Hence the property 5 of the || gm stands proved.

6. Area of a parallelogram: Since we just established that the diagonal of the parallelogram divides the || gm into two congruent triangles, if we know the area of one of these triangles we can find the area of the || gm by doubling the area of the triangle. Let us now see how to find the area of the || gm.

Consider the || gm ABCD shown in the figure below:

The triangle BDC has the length of base = DC = b and the altitude = h. The area of this triangle is therefore given by the formula:
∆ =(1/2)*base*height
∆ =(1/2)* b*h

Now we already established that the area of the || gm is twice the area of this triangle. Thus  the area of the | | gm ABCD would be:
A=2* ∆
A=2*(1/2)* b*  h
A=b*h

 Properties of normal distribution is a topic of statistics and therefore shall be tackled under a separate article.

The Stem And Leaf Plots

Stem and Leaf Plot Definition:- The stem and leaf plot is used for the presentation of the quantitative data in the graphical formats. It is similar to the histogram by which the shape of the distribution of the data can be found. It is the useful tool which can be used in exploratory data analysis. This plot was popular during the type writer time. In modern computer the machine language is zero and one.

So this technique of layout of the data is obsolete in modern computers. The stem and leaf display is also called as the stem plot. The stem and leaf displays retain the data to at least two significant digits. This display contains two columns which are separated by a vertical line. The left column contains the stem and the right column contains the leaf of the data.

For example in the nub thirty two the stem is left most digits which are three and the leaf is the rightmost digit which is two is the leaf. In numb ten one is the stem and zero is the leaf of the number. In number twenty nine the leftmost digit is two and rightmost digit is nine. The number two is called the stem and the number nine is called the leaf.

To construct the stem-leaf displays the data numbs are arranged in the ascending orders.  The data value may be rounded to a particular place value that can be used as the leaf. The remaining digits to the left of the rounding digit will be used as the stem. Stem and leaf plots can be used to find the range, median, mean and media. Other statistics parameters can also be calculated with this available data.

Irregular Polygon Definition:- Polygon is the plane figure which is bounded by the finite chain of straight line segments and closed in a loop to form a closed chain or circuit. These straight line segments are known as edges or sides. The junction of the two sides is called the vertex or corner. Polygon means a shape which has many sides and angles.

A regular polygon means the shapes which have many sides of equal lengths and many angles which are equal in measurements. Irregular polygons are those in which length of each side is not equal and the measurement of angles is also not equal.  The polygon in which one or more interior angles are greater than one hundred and eighty degrees is called as concave polygon.

The polygon which has only three sides cannot be concave. The convex polygon has opposite properties to the concave. It means one or more angles of the polygon are less than one hundred and eighty degrees.

A line which is drawn through the concave polygon can intersect it more than two places. It is also possible that some of the diagonals lie outside of the polygons.  In convex polygon all diagonal lie inside the polygon.  The area of the concave polygon can be found by assuming it as other irregular polygon.

Spherical Geometry

Spherical Geometry :- The geometrical symmetry is associated with the two dimensional surface of the sphere. It is not Euclidean. The main application of the spherical geometry is in navigation and astronomy. The plane geometry is associated with the point and lines. On the sphere, points are defined as usual sense. The straight lines are not defined as the usual sense. In spherical geometry angles can be defined between great circles, resulting a spherical trigonometry which is differ from the ordinary trigonometry in many respect; the sum of the interior angles of a triangle exceeds one hundred and eighty degrees.

Let r is the radius of the sphere. The volume of the sphere is 4/3 pi r3 cubic meter.  To find the volume of the sphere we can divide it into number of infinitesimally small circular disk of the thickness dx. The calculation of the volume of the sphere can be done as below. The surface area of the disk is equal to pi r².

Now the volume of the sphere can be found by finding the integral of the area within the limits of minus r to plus r. The formula can be derived more quickly by finding the surface area and then by integrating it within the limit of zero to r. The spherical geometry is not the elliptical geometry but shares with the geometry the property that a line has no parallel through a given point. The real projected plane is closely associated with the spherical geometry.

Midpoint Formula Geometry : - The point which is exactly at the centre of the two points is known as the middle point. This point divides a given line segment between the two equal halves. The middle point is called the midpoint in the geometry. The midpoint formula has been taken from the section formula. We have to find a point which divides a line in a particular ratio in the section formula.

In the section formula a point which divides the line in the m and n ratio is to find out. If the value of the two ratios which are represented by m and n is equal or m= n then we get the mid points at the given line. 
Let AB is a line and P is a point anywhere in between A and B. Let the coordinates of point A are (x₁, y₁) and coordinate of point B are (x₂, y₂). P is the point which lies in between points A and B has the coordinates (x, y). The coordinates on the mid-point of the line segment joining the points (x₁, y₁) and B(x₂, y₂) are given by  {(x₁+x₂)/2  ,(y₁+y₂)/2}. these are obtained by replacing l and m in the above formula. If a point P divides the given line segment  joining the points (x₁, y₁) and B(x₂, y₂) , then the coordinate of point P are given by  {(x₁+kx₂)/(k+1)  ,(y₁+ky₂)/(k+1)} These are obtained by dividing the numerator and denominator in the above expression the  replacing l/m the by k.

Obtuse Angle

                                                                                                          
In this article we shall learn about What is an Obtuse Angle? And also about Obtuse Scalene Triangle. Before studying about obtuse angle let us learn the definition of ‘angle’. In terms of geometry, an angle is defined as a measure between the circular arc and its radius and it’s shape is formed by two rays called as the sides (arms) of the angle and it shares an end point which is common to the two rays i.e., vertex of the angle. The units to measure an angle are degrees in sexagesimal system, radians in circular system and grades in centismal system.

We have various types of angles such like acute angles, right angles, obtuse angles, supplementary angles, straight angle, reflex angles etc. There are few properties which are listed below:

•    The measure of it can be positive or negative.
•    The measure of it can exceed 360°.
•    When we have two angles such as 30° and 390° where 390° = 360° + 30° where it means that the two terminal sides 30° and 390° belongs to same plane. Hence these two angles are called Coterminal angles.
•    It is expressed in terms of degrees and radians.

So far, we have learnt about definition of angle and its properties, now let us define obtuse angle?
It is defined as an angle whose measure will be greater than 90 degrees but less than 180 degrees. If the measure of this is not between 90 degrees to 180 degrees is not considered as an obtuse angle. Even if the measure is exactly 90 degrees is called as a right angle but not an obtuse. In other words, it is defined as the swipe between the quarter and the half rotation of a circle whose measure varies from 90 degrees to 180 degrees.

Now let us see about Obtuse Scalene Triangle. Firstly, Scalene Triangle is defined as a triangle, whose all sides are unequal and all angles are unequal. Where as an obtuse scalene triangle definition is similar to a scalene triangle where one of its angle is greater than 90°. An obtuse scalene triangle has one obtuse angle and two acute angles, where the two acute angles may be equal or unequal. If the two acute angles are equal then it is known as an obtuse isosceles triangle. There are few facts about scalene triangle such as

•    All interior angles are different.
•    The side which is opposite to the smallest angle will be the shortest side.
•    Similarly, the side which is opposite to the largest angle will be the longest side.

In order to find the area of an obtuse triangle the best formula to use is “heron’s formula”.

According to heron’s formula, the area A of a triangle whose sides are a, b, c is as follows:

A = √s(s-a)(s-b)(s-c), where a, b, c are sides of triangle and ‘s’ is the semi perimeter of the triangle. i.e., S = (a + b + c) / 2.

Polynomials

An expression with a single term is called as monomial, with two terms as binomial and with three terms as trinomial. If the number of terms is more, then such expressions are given a general name as polynomials, the word ‘poly’ means ‘many’. So it can also be written as poly-nomial to emphasize the meaning. Therefore, a poly-nomial is an expression that contains a number of terms consisting variables with constant coefficients. The terms of expression are usually arranged in descending order of the variable powers. Since a constant can also be expressed with the variable power as 0, a constant term can also be a part of a poly-nomial.
But as per convention in algebra, the definition of a polynomial includes certain restrictions. A polynomial can be built up with variables using all operations except division. For example, x³ – (2x + 3)/(x) + 7 cannot be called as a polynomial. However, this restriction applies only division by a variable and not for division by any constant, because such divisions can be considered as equivalent to fractional coefficients. The other restriction is that the exponents of the variables can only be non-negative integers.

A polynomial is generally an expression but acts as part both in equations and functions and such equations and functions are named with prefix ‘poly’ in general. In fact we convert a polynomial function to an equation while attempting to find its zeroes. Therefore, it is imperative to know how to factor a polynomial so that the zero product property can be used to determine the solutions. Using the zero product way is the easiest way to find the solutions of the variables.

Polynomial equations with a single variable with degrees 1 and 2 can easily be solved and the latter type is more popularly known as ‘quadratic’ equation. Mostly quadratic equations are possible to solve by factoring but even otherwise it can be done by using the quadratic formula. But equations with higher degrees are not all that easy to solve. But thanks to the great work by the mathematicians, there are ways to do that. Let us see some of the helpful concepts enunciated by the mathematicians.

As per fundamental theorem of algebra, the number of roots (the number of solutions of variables when equated to 0) of a polynomial is same as the degree of the same. This concept guides us to do the complete solution. We must also be aware that in some cases the solutions or some of the solutions may be imaginary. To get an idea on this, Descartes’s rule of sign changes helps. As per this rules we can figure out the number of real solutions, both positive and negative. Subsequently we can figure out the imaginary solutions with the help of fundamental theorem of algebra.

The rational zero theorem helps us to know what are the possible zeroes of a function. By a few trials, we can figure out a few zeroes and can reduce the polynomial to the level of a quadratic. Thereafter, the remaining zeroes can easily be figured out.

In addition, these days there are many websites advertising as ‘Factor Completely Calculator’ to help us in finding the solutions of a polynomial.

Correlations

Zero Order Correlation :- Zero order correlation means there is no correlation between the two quantities. They vary independently.  If classes in one variable are associated by the classes in the other, then the variables are called correlated. The correlations is said to be perfect if the ratio of two variable deviations is constant. Numerical measure of correlation is called co- efficient of correlations. A group of n individuals may be arranged in the order of merit with respect to some characteristics.

The same group would give different order for different characteristics. Consider order corresponding to two characteristics A and B. The correlations between these n pairs of ranks is called rank correlation in characteristics A and B for the group of individuals.  When the variation of the value of one variable becomes the cause of the variation of the value of other variable then it is called the correlation between the two variables. When the variation between two quantities is directly proportional then it is called positive correlations. It means when one variable increases other also increases or when one variable decreases other variable also decreases.

When the variation between two quantities is indirectly proportional then it is called negative correlations. It means when one variable increases other decreases or when one variable decreases other variable increases. For example the production of any quantity is indirectly proportional to the cost of that quantity. When the temperature increases the charge of the electricity bill also increases. This is the example of positive correlations.

On the other hand during the summer season the length of night hours decreases. It means the day hours are increasing and night hours are decreasing. During the cold season the day hours are decreasing whereas the night hours are increasing. There is a negative correlation between the day and the night hours. The reception of the radio signals is indirectly proportional to the distance from the transmitter. The reception near the transmitter is better than the reception at a distance. As the distance between the transmitter and the receiver increases the signal strength in the receiver decreases. It means the signal strength is indirectly proportional to the distance from the transmitter.

This is an example of the negative correlation. Zero order correlation means there is no correlation and all the quantities have their own graph.

Correlation Equation: - Correlation equations are written to find the order of correlations. We can find the perfect positive correlation, negative correlation or zero order correlations.  The numerical measure of correlation is called co- efficient of correlation and is defined as
r = (∑XY)/(n σx σy ) =(∑XY)/√(∑X² ∑Y²), where X and Y are the deviation from the mean positions.

Deviations are used to find the lower and upper bound of that quantity. Let a resistance value is written as 200 ± 5%.

The lower value of the resistance is equal to 190 and upper value is 210 ohm.
σ²x=(∑x²)/n  ,   σ²y=(∑y²)/n

Where, X = deviation from mean, ¯x=x-¯x

Y = deviation from mean, ¯y=y-¯y
σ_(x =  Standard deviation of x series )
σ_(y =  Standard deviation of y series )
n = number of values in two variables

We can also use the direct Method by substituting the value of σ_(x ) and  σ_(y ) in the above mentioned formula.

r = (∑XY)/(n σx σy ) = (∑XY)/√(∑X² ∑Y²)          or (n∑xy- ∑x∑y)/√(n∑x²-(∑x )²4 ×{n∑y²-∑x²}

Where r is the coefficient of correlation which can be used to find the rank of two quantities.

Vectors

In this section we are going to understand the basic concept of vectors and we will see a very important operation that can be done on vectors which is cross product. 

Let us see what are vectors first.Vectors are used to represent quantities where magnitude and direction both are essential to define a quantity properly. A vector hence represents magnitude and direction of a quantity. For example quantities like velocity and acceleration needs to be defined with magnitude and direction as well. A velocity of 5 km/hr due east represents that the object is moving with a speed of 5 km/hr in east direction. This means that velocity is a vector quantity. 

A vector is denoted graphically by an arrow where the head of the arrow represents the direction of the vector and its length represents the magnitude of the vector. The length of a vector is used for comparing magnitude of 2 or more vectors. A vector with shorter length will have smaller magnitude than other vectors. 

A vector a can be represented in 3 dimensional space as:
 a= a₁i+a₂j+a₃k

Here i, j and k are unit vectors in the x, y and z direction respectively. 

Let us see how cross product can be performed on the vector  Cross product is also known as vector product. This is due to the fact that the result after the product is also a vector. It is a binary operation on 2 vectors in 3 D space. The resulting vector is perpendicular to both the vectors and thus it is perpendicular to the plane in which initial vectors lie. 

If the direction of the two vectors is same or they have zero magnitude or length, then the cross product of the two vectors results zero. The magnitude of the cross product vector is equal to the area of the parallelogram which has the two vectors as its sides. If the two vectors are perpendicular and form a rectangle, then the resulting magnitude will be the area of rectangle i.e. product of lengths of the vectors. 

The operation of cross product is denoted by ×. If the two vectors are ‘a’ and ‘b’ then the cross product of the two vectors is denoted as a×b. Note that a×b is not equal to b×a. The resulting vector of this cross product denoted by c is a vector that is perpendicular to both the vectors. Its direction can be given by using right hand rule. Formula for Cross Product of Two Vectors is given as:
a×b=|a||b|sinӨ n

Here Ө is the smaller angle between the vectors a and b. |a| and |b| denotes the magnitude of the vectors a and b respectively. n here represents the unit vector perpendicular to the plane of a and b. Its direction can be given by right hand rule. The right hand rule says that if a×b is the cross product that you are calculating then role your fingers of right hand from vector a to vector b, the direction of your thumb will give the direction of resulting vector as shown below:

If the vectors a and b are parallel to each other, then the cross product of then is a zero vector as sin0 = 0
Cross product is considered as anti commutative. This means that a×b = -(b×a) 

Algebra Word Problems

In the field of mathematics where many forms are present algebra is one such form that deals with the basic concepts of Mathematics. We all know that Maths is a game of mathematical operations, so to deal with addition, subtraction, multiplication, division with the known and unknown values of the variables is known as algebra. In a simple way to say is that when we play computer games then we deal with jumping, running or finding secret things and in Maths we play with letters, numbers and symbols to extract the secret things. Now let’s see what is included in algebra.

Get easy steps to solve Algebra Problems (click here)


WHAT IS THERE IN ALGEBRA?
1. Algebraic terms – Algebraic terms are those terms which are infused with the values of variables. For example 3a.
In this example 3 is the numerical coefficient and a is the variable. The main work of numerical coefficient is also to deal with + 3 or – 3 of the variable.

2. Algebraic expressions – Algebraic expressions deal with a logical collection of numbers, variables, positive or negative numbers in the mathematical operations. For example we can write an expression – 3a + 2b.

3. Algebraic equations – Algebraic equations refer to the equivalency of expressions considering both the left hand side and the right hand side. For example an equation is x – 35 = 56k2 + 3. Over here both left hand side and right hand side are dealing with expression. If both the expressions are equal then that is said to be an equation.

ALGEBRA WORD PROBLEMS

The main part in Algebra is to solve the word problems, which we also did in Pre Algebra Word Problems.  It can be harder if you don’t think and do it but it can be easy if you think and do it. If we say in English how are you and the simple answer to the question is I am fine which can be simply understood but to convert the English language in to the language of maths requires certain techniques. You need to practice a lot with proper understanding of the translations. Let’s go step by step.

1. First of all read the entire word problem carefully. Make it a point that we should not start solving the problem by just reading half of a sentence. Read it properly so that you can gather the information that what you have and what you still need to have.

2.  Then, pick the information so that it can be changed into certain variables. Some might be known and some might be unknown. If required then draw pictures and symbols in the rough work so that you can deeply understand the problem without leaving any loopholes.

3. Note the key words as for different mathematical operations there are different key words –

a) Addition – increased by, total of , sum, more than, plus
b) Subtraction – decreased by, minus, less than, fewer, difference between
c) Multiplication – of, times. Multiplied by, product of, increased or decreased by the factor of
d) Division – per, out of, quotient
e) Equals -  is, gives, yields, will be


4. After going though these expressions carefully you can solve various word problems logically. Limit Problems are part of Calculus.

Rational Inequalities

Rational Inequalities : - The rational expressions are written in the P(x)/q(x) form where p(x) and q (x) are the polynomials i.e p(x), q(x)∈z(x), and q(x)≠0 are known as rational functions. the set of rational function is denoted by Q (x).

Therefore Q(x) = (p(x))/(q(x)) (where p(x), q(x) ∈z(x) and q(x) ≠0. for, if any a(x)∈z(x), then we can write a(x) as  ( a(x))/1, so that a(x)∈z(x)  and therefore,(a(x))/1  ∈Q(x);i.e  a(x)∈Q(x). The rational function example is given below.   

(2x+4)/(x²+5x+8)>0
(ax+b)/(cx+d)<0 p="">
Rational inequalities means the left hand side is not equal to the right hand side of the equation. The inequalities are associated with the linear programming.  The step by step procedure to solve the inequalities is given here. 

The very first step to solve the inequality problems is to write the equation in the correct form. There are two sides in the equation, left hand side and the right hand side. All the variables are written in the left hand side and zero on the right hand side. The second step is to find the critical values or key. To find this we have to keep the denominator and the numerator equal to zero.

The sign analysis chart is prepared by using the critical values. In this step the number line is divided into the sections. The sign analysis is carried out by assuming left hand side values of x and plugs them in the equation and marks the signs. Now using the sign analysis chart find the section which satisfies the inequality equations.

To write the answer the interval notation is used. Let us solve an example (x + 2) / (x² – 9) < 0. Now put the numerator equal to zero, we get x = -2. By putting the denominator equal to zero we get x = +3 and x = -3.  Mark these points on the number line. We get minus three, minus two and plus three on the number line.
Now try minus four, plug the value of x equal to minus four we get – 0.286.

Plug x equal to minus 2.5 we get + 0.1818. Now plug 2 in the equation we get minus zero point eight.
Now we try x equal to plus four which is at the right hand side of the number line. We get plus 0.86. The sign is minus, plus and plus sign. The equation is less than zero. To get less than zero we need all the negative values of the equation.

For x equals to plus three and minus three the equation is undefined. For x equals to minus two the equation is zero. The equation is satisfied by the value of x which is less than minus three. Therefore the answer which is the solution of the equation is minus infinite to minus three.

Rational Expressions Applications : - The equations which do not have equal sign are called the expression. The expressions are used to simplify the equations. The expressions are written to form the conditions of the problems.

According to the direction of the question the expressions are formed to solve a problem.

Set Operations

Set Operations: - The collection of the distinct objects is called the sets. For example the number 5, 7 and 11 are the distinct object when they are considered separately. Collectively they can form a set which can be written as [5, 7, 11]. The most fundamental concept of the mathematics is the sets.  The objects which are used to make the sets are called the elements. The elements of the sets may be anything, the letters of the alphabet, people or number. The set are denoted by the capital letters.  The two sets are equal when they have equal elements. The set can be represented by two methods, roster or tabular form and the sets builder form. In the roster form the elements are separated by commas. For example a sets of positive and odd integers less than or equal to seven is represented by {1, 3, 5, 7}.

The set of all the vowels of the alphabet is {a, e, I, o, and u}. The sets of the even numbers are {2, 4, 6,}. The dots indicate that the positive even integers are going to infinite. While writing the set in the roster form the element must be distinct. For example the set of all the letters in the word “SCHOOL” is {s, c, h, o, and l}. In the set builder form all the elements of the sets have the common property which is not possessed by any element which is outside of the set. For example vowels have the same property and can be written as {a, e, I, o, and u}. The set in the set builder form can be written as V = {x; x is a vowel in the English alphabet}.  A = {x; x is the natural number and 5 < x < 11}. C = {z; z is an odd natural number}.

Operations on Sets: - Like the mathematical operations, there are number of operations which are carried out by the sets. We will study them one by one. Union of sets is the union of the all the elements of a set and all the elements of the other  taking one element only once. Let A = {2, 3, 4, 5} and B = (4, 5, 6}. Therefore the union  of the two sets is written as A ∪B  = {2, 3, 4, 5, and 6}. Intersection of two sets A and B is the set of all elements which are common in both the sets. If the intersections of the two sets are equal to null then the set is called the disjoint set.

The difference of sets A and B can be defined as the A – B. It is equal to the elements which belong to set a and the element which does not belong to the set B. The operation on the sets such as union and intersection satisfy the various laws of the algebra such as the associative laws, the commutative laws, idempotent laws, identity laws, distributive laws, De Morgan’s laws, complement laws and involution law. These all the laws can be verified by the Venn diagram. 

Dispersion Statistics

Measures of Dispersion : - The dispersion is also known as the variability is the set of constant which would in a concise way explain variability or spread in a data. The four measures of dispersion or variability are the range, quartile deviations, average deviation and the standard deviation. The difference between two extreme observations in the given data is known as the range. It is denoted by R. In frequency distribution, R = (largest value –smallest value). It is used in statistical quality control studies rather widely. Median bisects the distribution. If we divide the distribution into four parts, we get what are called quartiles, Q(1 ),Q2 (median) and Q(3.)

The first quartile Q(1,) would have 25 % of the value below it and the rest above it; the third quartile would have 75% of values below it. Quartile deviation is defined as, Q. D.  = 1/2  ( Q3-Q1). If the average is chosen a, then the average deviation about A is defined as A.D. A.D. (A) = 1/n ∑|(xi- A)|  for discrete data. The Standard deviation is also called as the Root mean square deviation. The formula for the standard deviation is given as Standard deviation,σ=√(1/n ∑(xi- x ̅)^2 ) for discrete data.

The Square of the standard deviation is known as the variance. It is denoted by the square of sigma. Out of these measures, the last σ is widely used as a companion to x ̅ on who is based, when dealing with dispersion or scatter. Measure of dispersion is calculated for the data scattering. Deviation means how a value is deviated from it mean or average value. The mean of the two groups of the data may be same but their deviation may be high.

Central Tendency Measures : - The central tendency measures are also called the statistics central tendency. The clustering of data about some central value is known as the frequency distribution. The measure of central tendency is the averages or mean. The commonly used measures of central values are mean, Mode and median. The mean is the most important for it can be computed easily. The median, though more easily calculated, cannot be applied with case to theoretical analysis. Median is of advantage when there are exceptionally large and small values at the end of the distribution. The mode though easily calculated, has the least significance. It is particularly misleading in distributions which are small in numbers or highly unsymmetrical. In symmetrical distribution, the mean, median and mode coincide.

For other distributions, they are different and are known to be connected by empirical relationship. Mean – Mode = 3 (mean – median). The sum of the values of all the observations divided by the total number of observations is called the mean or average of a number of observations.  The value of the middle most observations is called the median. Therefore to calculate the median of the data, it is arranged in ascending (or descending) order. The observation which is found most frequently is known as mode.

The central tendency measures and the variability or dispersion are used in the statistical analysis of the data.

Interpolation

Interpolation is a concept that is used in numerical analysis. It means finding an intermediate value from the given data. That is, for a set of function values, sometimes a situation arises to know what the value of the function is, for some intermediate value of the variable.

Let us explain the concept with a simple example. Let the ordered pairs of the data of a function be (0, -1), (1, 7), (2, 22), (3, 40), (4, 69), (5, 98) and we need to find the function value of 3.8 of the variable. The attempt to find that value is called interpolation and there are several methods. Let us describe those one by one.

The first method is to round the given value of the variable to the nearest value in the data and take the corresponding value of the function. So for the given data, the value 3.8 of the variable can be rounded to 4 and assume the value of the function approximately as 40. But as one can easily see it is a crude method and far from accurate. This method can only be used just for guidance.

The next method is called as linear interpolation. That is, the function is assumed to be linear in the interval that contains the required value of the variable and accordingly the value of the function is determined. In the given example 3.8 falls in the interval [3, 4]. Considering the function to be linear in this interval, the slope of the function in this interval is (69 – 40)/(1) = 29. So the value of the function at x = 3.8 is, 29(0.8) + 40 = 63.2. Though this method also is not very accurate still the accuracy is much better than that in method 1. This method, even at the cost of sacrificing some accuracy, is preferred because it is easy to work with. In general the linear interpolation y at a point x  is given by the formula y = [(y_b – y_a)/(x_b – x_a)](x – x_a) + y_a, where [a, b] is the interval in which the required point occurs. This is the algorithm used in any linear interpolation calculator.

It is always possible a curve rather than a straight line could better cover the points plotted from a data. In other words a polynomial function can give a better interpolation. Finding a suitable polynomial function for a given data is called as polynomial regression.  Suppose we consider the same data, the three degree polynomial function f(x) = 3x² + 5x – 1 will be very close to the desired results. In such a case evaluating the function for x = 3.8, the value of the function is will be a very accurate interpolation. The accuracy can further be improved when a polynomial function of degree same as the number of data points is determined.

A high level interpolation polynomial can be derived by extensive methods and such an interpolation is called as Lagrange interpolation introduces by the famous Italian mathematician.

Analytical Geometry

Analytical Geometry:

Another term for analytical geometry is Cartesian geometry or co ordinate geometry. It refers to the study of relationships between points, lines, planes etc against a back drop of the co ordinate system, be it in three dimensions or two dimensions. In our lower grade normal geometry, we knew of geometric shapes such as triangles, angles, squares, rectangles etc. The same shapes in co ordinate geometry are described by the co ordinates of their vertices, or by the equations and slopes of the lines joining these vertices.

The basic of analytical geometry deals with the concept of lines. The topics covered are parallel lines, perpendicular lines and inclination of lines with respect to the coordinate systems or with respect to each other. A line is described by its slope or gradient. This slope can be found using the co ordinates of two points that lie on the line. If two lines have the same gradient then they are said to be parallel to each other. If the product of the slopes of two lines is -1, then the two lines are said to be perpendicular to each other. The slope can also be defined by the tangent of the angle that the line makes with the positive x axis direction. Thus if we know the slope we can use the arctan function to find the inclination of the line with respect to the positive x direction.

All the other geometric shapes are studies with the help of this concept of straight lines. For example consider a quadrilateral. It is a closed shape formed by four line segments that are parts of four lines. If both the pairs of opposite sides are parallel and congruent to each other, (that means if both the pairs of opposite sides have the same slope and same length) then such a quadrilateral is called a parallelogram. In a parallelogram if all the adjacent sides are perpendicular to each other, (in other words, if the product of the slopes of adjacent sides is -1), then the parallelogram becomes a rectangle. Similarly other geometric shapes can also be studied in this way.

Analytical geometry is most useful in studying three dimensional objects. This is sometimes also called analytical solid geometry. In three dimnesional space, there are three coordinate systems that can be used. Besides the Cartesian co ordinate system, there can also be the cylindrical coordinate system and the spherical co ordinate system. All these three systems can be interchangeably used to study various types of curves, surfaces or solids in space. Just like how a point in the Cartesian co ordinate system is defined by its x y and z co ordinate, in the cylindrical co ordinate system it is defined by three parameters. They are, (a) its distance from the origin (r), (b) the angle of the line joining the point and the origin with the positive x axis (θ) and (c) its perpendicular distance from the x-y plane (z co ordinate).

Also check out the video streaming of Analytical Geometry

Variables

                                                                                                               
What are Variables?

A variable is something which always varies. That is it does not have a fixed value. In any situation if we are not sure of a value for a quantity then we represent such things using alphabets like x, y, z, a, b, c etc known as variables.

Let’s assume that Peter is working in a hotel and earns $12 per hour and on top of his hourly wage he also get tips for each hour. This expression $(12 + x) would give how much peter earn in a given hour. Here x denotes the tips earned. Now, you might also realize that number of tips or the amount of tips Peter make per might change dramatically from hour to hour. It varies consistently. Since, we are not sure of the tips Peter make each hour exactly, this value is called a variable.

Random Variables

In this section i will introduce you to the concept of a random variable. For me this is something I had lot of trouble for getting my head around. That’s really because it called variable. Generally, variable are unknowns used in algebraic equations/ expressions.

For example: x + 2 = 15. We can find the value of x, by subtracting 2 from both sides.
Or if we have an equation in two unknowns that is y = 2x + 5, here x is an independent quantity and y is a dependent quantity. So, we can assume any x value and find respective value for y. In this case we will have infinite combinations of x and y satisfying the equation.

A random variable is kind of same thing that it can take multiple values but it is not something which you really ever solve for. A random variable is usually denoted by a capital letters say X. It can take bunch of different values but we are never solving for it. In fact it a function that maps from you from the world of random processes to the actual numbers. Let us see one example.

Let the random process be it is going to rain tomorrow or not. Now, how are we going to quantify this, let say X = 1 if it rains tomorrow and X = 0 if does not rain. It is not compulsory that we have to use 1 and 0. Here we can assign to any number. So, we need to keep in mind is a random variable is not a traditional variable.

It is defined as a numerical value to each outcome of a particular experiment. For every element of an experiment’s sample space, it can take only one value. When a random variable can take finite (countable) finite set of outcomes then it is known as discrete random variable. Individual outcomes for a random variable are denoted by lower case letters. A continuous random variable would take on any of the countless number of values in the given line interval.

Using Matrices

In this article we shall study about one of the methods used to solve system of linear equations using matrices. Before we study about the method let us first see few definitions

Consider the following system of simultaneous non – homogeneous linear equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂

Expressing the above equations in matrices, we get

These equations can be represented as a matrix equation as AX = D, where







Here A is called the coefficient matrix.X is called the variable matrix.
D is called the constant matrix.

Augmented Matrices

The coefficient matrix augmented with constant column matrix, is called the augmented matrix, generally denoted by [AD]. Hence the augmented matrix of the above system of simultaneous linear equations is 

Sub Matrices

A matrix obtained by deleting some rows or columns (or both) of a matrix is called a sub matrix.

Definition (Rank of a matrix)

Let A be a non – zero matrices. The rank of A is defined as the maximum of the orders of the non – singular square matrices of A. The rank of a null matrix is defined as zero. The rank of A is denoted by rank (A).

It is to be noted that :-

If A is a non zero matrix of order 3 then rank of A is
(i)    1 if every 2 x 2 sub matrices is singular
(ii)    2 if A is singular and atleast one of its 2 x 2 sub matrices is non – singular.
(iii)    3 if A is non – singular.

Consistent and Inconsistent systems

A system of linear equations is said to be
(i)    Consistent if it has a solution
(ii)    Inconsistent if it has no solution.

A system of three simultaneous equations in three unknowns whose matrix form is AX = D has
(i)    A unique solution if rank (A) = rank ([AD]) = 3
(ii)    Infinitely many solutions if rank (A) = rank([AD]) < 3
(iii)    No solution if rank (A) is not equal to rank ([AD])

It is to be noted that the system is consistent if and only if rank (A) = rank ([AD])

The different ways of solving non homogenous systems of equations are
(i)    Cramer’s Rule
(ii)    Matrix inversion method
(iii)    Gauss – Jordan method

In Gauss – Jordan method we try to transform the augmented matrix by using elementary row transformations. So that the solution is completely visible that is x = α, y = β and z = γ. We may get infinitely many solutions or no solution also.

For solving a system of three equations in three unknowns by Gauss – Jordan method, elementary row operations are performed on the augmented matrix as indicated below.

(i)    Transform the first element of 1^st row and 1^st column to 1 and transform the other non zero elements if any in of 1^st row and 1^st column to zero.
(ii)    Transform the second element of 2^nd row and 2^nd column to 1 and transform the other non zero elements if any in of 2^nd row and 2^nd column to zero.
(iii)    Transform the third element of 3^rd row and 3^rd column to 1 and transform the other non zero elements if any in of 3^rd row and 3^rd column to zero. 

Matrices Calculator
We shall study about Matrices Calculator in some other article.

Right Circular Cylinder

Right Circular Cylinder :- If the number of circular sheets is arranged in a stack, if the stack is kept vertically up then it is called the right circular cyl. If the base is circular and the other sheets are kept at ninety degrees at the base, then it is called the right circular cyl.  The right circular cyl can be made by a rectangular sheet by folding it in round shape. The area of the sheet gives us the curved surface area of the cyl because the length of the sheet is equal to the circumference of the circular base which is equal to 2 π r . Therefore the curved surface area of the cyl is equal to the area of the rectangular sheet. The area of the rectangular sheet is equal to the product of the length and the breadth. The curved surface area of the cylinders is also equal to the product of the perimeter of the base and the height of the cylinders.  If the top and bottom of the cylinders is also consider because it is needed to make the circular tin, If the area of the bottom and top of the cylinders is included in the curved surface area we get the total surface area. It means total surface area of the right circular cylinder is equal to the area of the base plus the area of the top of the cylinder plus the curved surface area of the cylinder. The cylinder has two circular base at the bottom and at the top.The diameter of the base of the cylinder can be measured directly with the help of the scale. 

What is the Formula for Volume of a Cylinder :- As we know that the volume of cuboids is equal to the product of the length, breadth and the height. The volume is three dimensional. The cuboids are built up with the rectangle of the same size sheets. In the same way the right circular cylinders can be made by using the same principle. So, by using the same argument as for cuboids, we can see that the volume of a cylinders can be obtained by the product of the base and height of the cylinders.  Therefore, the volume of a cylinder = the base area of the cylinder X height of the cylinder. As we know that the base of the cylinder is the circle. The area of the circle is equal toπ r². The height of the cylinder can be assumed as h. Therefore the volume of the cylinder = π r²h, where r is the base radius and h is the height of the cylinder.

Example :- Let us find the volume of the cylinder which has the radius of the base is twenty one centimeter and its height is thirty five centimeter.

Solution :- The formula for the volume of the cylinder, V = π r²h
Radius r = 21 centimeter, height h = 35 centimeter
Therefore the volume of the cylinder V = π ×21² ×35 = 48490.48 cubic centimeters.