Inequalities:

Inequalities:

Introduction to inequalities:

In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not.

* The notation a < b means that a is less than b.

* The notation a > b means that a is greater than b.

* The notation a ≠ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.

In each statement above, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".

In contrast to strict inequalities, there are two types of inequality statements that are not strict.

* The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b)

* The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b).

This was a very general view about the meaning of Inequalities,now let us understand the look at a more definite explanation,An inequalities is a statement, that relates the size or order of two objects or about whether they are the same or not.Solving linear inequalities is very much similar to solving linear equations, except for one small but important change: you flip the inequality sign whenever you multiply or divide the inequality by a negative number. In linear equation we don’t do that.

Procedure for Solving Inequalities:


As in the case of solving inequalities equations, there are certain rules for the inequality problems which do not change the solutions. Here is a list of "permissible'' manipulations:

Step1: Adding/subtracting the same number on both sides.

Step2: Switching sides and changing the orientation of the inequality sign.

Step3: Multiplying/dividing by the same positive number on both sides.

Step4: Multiplying/dividing by the same negative number on both sides and changing the orientation of the inequality sign.

* The notation a < b means a is less than b.
* The notation a > b means a is greater than b.
* The notation a ≠ b means a is not equal to b,
* The notation a ≤ b means a is less than or equal to b (or, equivalently, not greater than b)
* The notation a ≥ b means a is greater than or equal to b

Inequalities - Example Problems:

Solving inequalities for Addition and Subtraction:

1. 4x+2 < -2x+14

Solution:

4x+2-2 < -2x+14-2

4x < -2x+12

4x+2x < -2x+2x+12

6x < 12

6x /6< 12/6

x < 2

Hope you like the above example of Inequalities.Please leave your comments, if you have any doubts.

Data Analysis and Interpretation

Data Analysis and Interpretation:
Let us understand how we can analyse the data and interpret it,in this blog let us understand the basic meaning of Data analysis and Interpretation.Next let us understand the meaning of Interpretation,the basic meaning of Interpretation is The power or explaining.
Analysis of data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, in different business, science, and social science domains.But the statistical data analysis is the data analysis plan to examine the research questions. Statistics, Data Analysis, and Probability introduce statistics as a problem-solving process. Different ways to organize, represent data, describe and analyze variation in data. In this introduction, we will briefly discuss those statistical concepts that provide the necessary foundations for more specialized expertise in any area of statistical data analysis.

Statistical Data Analysis:

To find what statistical data analyses are, first we go to define statistics. Statistics is a group of methods that are used to collect, analyze, present, and interpret data. Statistical data analysis gives hands to promote the use of statistical techniques to apply in order to make decisions in the problem solving in data analysis.

Steps for Statistical Data Applications:

In generally statistical data analysis involves four basic steps:

Step1. Defining the problem is step 1.

Step2. Collecting the data is the step 2.

Step3. Analyzing the data is step 3.

Step4. Reporting the results is last step.

Defining the Problem:

A correct definition of the problem is imperative in order to obtain exact correct data about it. It is very difficult to getting data without clear explanation of the problem.

Collecting data:

When the data are collecting and statistical methods have become easy.

Analyzing the Data:

Statistical data analyses are divides methods for analyzing data into two categories: 1. Eploratory methods, nd 2. Confirmatory methods.

Exploratory methods:

Exploratory methods are used for cover what the data seems to be simple arithmetic and draw pictures to summarize the data.

Confirmatory methods:

Confirmatory methods use ideas from probability theory for answer the specific questions.

Reporting the Results:

Through inferences, an estimate the characteristics of a population can be obtained from a sample. The results may report in the form a graph or a set (or) group of percentages.


Statistical Data Analysis Applications are Given Below:

* Statistical methods are used to help people for identify, study, and solve many complex problems.

* The statistical data analysis is used for so many applications; like that the statistical data analysis section will discuss the assumptions of regression, which include the removing of outliers from the data set, the examining of the linearity, and constant variance.

* The statistical data analysis section should know about how the regression will be interpreted. The statistical data analysis plan should state that the linear regression is being evaluated for the model fit,

* And then it is the part of the statistical data analysis is to discuss the R-square value of the linear regression.

* Then the next part of the statistical data analysis plan would be to examine the beta coefficient, the t-value,

* The last part of the statistical data analysis would discuss the significant beta in terms of the criterion variable happiness

* The statistical data analysis also approximates the sample size needed for the analysis.

Hope you like the above example of Data Analysis and Interpretation.Please leave your comments, if you have any doubts.

Correlation Theory and Regression Analysis

Correlation Theory and Regression Analysis:

The term correlation deals with the relationship between two or more variables. If a change in one variable effect a change in other variable, the variables are said to be correlated.

There are basically three types of correlation, namely,

* Positive correlation
* Negative correlation
* Zero correlation.

In this Blog, let us also learn about Regression.

Regression Definition:

A regression is defined as a statistical analysis assessing the association between two variables. It is used to find the relationship between two variables.


Regression Formula:

Regression Equation(y) = a + bx
Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
Intercept(a) = (ΣY - b(ΣX)) / N

where
x and y are the variables.
b = the slope of the regression line
a = the intercept point of the line and the y axis.
N = Number of values or elements
X = First Score
Y = Second Score
ΣXY = Sum of the product of first and Second Scores
ΣX = Sum of First Scores
ΣY = Sum of Second Scores
ΣX 2 = Sum of square First Scores.

Correlation study are frequently used in psychology research to explain about relationships between the variables. There is a necessary relationship between two variables that can be study about correlation, one variable causes a change in another variable does not proved by finding a correlation relationship. we can study that no equal causation in correlation.

The Purpose of Correlation Study:

Correlation study used for the purpose of representing the relationships between variables. Correlation has three possible way of study: a negative correlation, a positive correlation and no correlation and no correlation. The measurement of correlation co-efficient of correlation strength and the range can be measured from –1 to +1.

*

Positive Correlations: Both variables are increase or decrease at the same time. A correlation coefficient nearer to +1 represents a strong positive correlation.
*

Negative Correlations: Indicates the increase in the amount of one variable, the other decreases (and vice versa). A correlation coefficient nearer to -1 represents a strong negative correlation.
*

No Correlation: It Indicates no relationships between two variables. The correlation coefficient of 0 represents no correlation.

Types of Correlation Study:

1. Naturalistic Observation:

Naturalistic observation explains about recording and observing the number of variables interested in the natural environment without manipulation or interference by the experimenter.

2. The Survey Method:

The most common methods for surveying and questionnaries used in psychological research. In this survey method, the participants can be selected by random sampling can complete a survey, test, or question that is related to variables of interest. Ensuring generalizability of survey results is one of the vital part for random sampling.

3. Archival Research:

Archival research can be performed by the method of analyzing study and this is conducted by researchers of other area or they looking for historical patient records.
Purpose

The correlation is a different way to measure how that is associated with two related variables. The research looks at things that already exist and determines if and in what way those things are related to each other. By doing correlations the purpose is to allow us to make a what we know about one variable based on prediction for another variable.

Hope you like the above example of Correlation Theory and Regression Analysis.Please leave your comments, if you have any doubts.

Curve Fitting

Curve fitting equation:

Introduction to curve fitting equation:

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data.
The study of ways of constructing functions whose graphs are curves that "best" approximates a given collection of points.We consider approximations by linear functions, polynomials of degree 2 or 3 and exponential functions using the method of least squares. After working through these materials, the student should be able

* to calculate using a graphing calculator or computer software the least squares line, parabola or third degree equation which best fits data.


Curve Fitting Equation:

Built-in curves are able to be use as an assist for data revelation, near assume value of a purpose wherever no data are obtainable; also to go over the associations with two or else further variables. Extrapolation illustrates to the make use of a fixed curve away from the collection of the practical data, with is the topic to a larger degree of indecision given that it might reproduce the way use to make the curve as greatly when it reproduce the experiential data.



Equation for Curve Fitting:

Permit establish through a 1st degree polynomial equation:

y=ax+b

This be a line through slope a. We identify to a line determination attach whichever two points. Thus, a 1st degree polynomial equation is an accurate fit during whichever two points.

Condition we raise the order of the equation toward a next degree polynomial, we obtain:

y=ax2+bx+c

Determination accurately fit an easy curve toward three points.

But we enlarge the order of the equation toward a 3rd degree polynomial, we obtain:

y=ax3+bx2+cx+d

These resolve accurately well four points.

A further common statement would exist toward declare it resolve accurately well four constraints. Every constraint be able to be a point, angle, otherwise curvature. Angle with curving constraints is mainly a lot additional toward the split ends of a curve; also during such belongings be call end conditions.

The curve fit for the polynomial equation is

Y (x) = P(x, n) = a0 + a1*x + a2*x2 ... + ... an*x n

The curve fit for the nonlinear approximation is

F(y) = a0*f0(x) + a1*f1(x) + ... + an*fn (x)

Hope you like the above example of Curve Fitting.Please leave your comments, if you have any doubts.

Standard Probability:

Standard Probability:

Introduction:

A standard probability normal table is also known as “Unit Normal Table" is a numerical table for the values of Φ, the cumulative distribution task of the sharing. They are used to locate the probability that a statistic is practical below, above, or connecting values on the standard normal distribution, and by addition, any normal distribution.Let us now learn about the uses of standard Probability spaces,standard probability spaces are used routinely in ergodic theory, which cannot be said on probability theory. Some probabilists hold the following opinion: only standard probability spaces are pertinent to probability theory, thus, it is a pity that the standardness is not included into the definition of probability space. Others disagree, however:

Arguments against standardness:

* the definition of standardness is technically demanding;
* the same about the theorems based on that definition;
* it is possible (and natural) to build all the probability theory without the standardness;
* events and random variables are essential, while probability spaces are auxiliary and should not be taken too seriously.

Arguments in favour of standardness:

* conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure;
* the same for measure-preserving transformations between probability spaces, group actions on a probability space, etc.;
* ergodic theory uses standard probability spaces routinely and successfully;
* being unable to eliminate these (auxiliary) probability spaces, we should make them as useful as possible.


Normal Distributions:

Normal distributions are symmetrical, bell-figure distributions that are useful in describing actual-earth information. The standard probability normal distribution, represent by the text Z, is the normal distribution has a denote of 0 and a standard deviation of 1. Since probability table cannot be printed for every normal distribution, (there are infinite), it is general practice to change a normal to a standard normal, and use a Z table to find probabilities.

Reading the Table:

Tables use at least 3 different conventions, depending on the understanding of the meaning of an entry such as 1.58:

Cumulative: This is nearly all common, and gives Prob (Z ≤ 1.58)

Complementary cumulative: The complement (1–x) of more than: Prob (Z ≥ 1.58)

Cumulative from zero: The cumulative probability, initial from 0: Prob (0 ≤ Z ≤ 1.58)

These can easily be check by inspect a number like 2.99:

* If this is just about 1 , next it display cumulative probabilities;
* If this is just about 0 , next it display complementary probabilities;
* If this is just about 0.5 , next it display cumulative from 0 probabilities.



Printed tables typically give cumulative probabilities, the possibility that a statistic takes a number less than or the same to a number, from at smallest amount.


Hope you like the above example of Standard Probability.Please leave your comments, if you have any doubts.

Simplex

Simplex:

Introduction:

Let us understand the meaning of Simplex Method.
The Simplex method is a method which goes from one BFS or extreme point of the feasible region of an Linear programming problem expressed in tableau form to another BFS, in such a way as to continually increase or decrease the value of the objective function until optimality is reached.The simplex method moves from one extreme point to one of its neighboring extreme point.

In geometry,a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope with n + 1 vertices, of which the simplex is the convex hull. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron. A single point may be considered a 0-simplex, and a line segment may be viewed as a 1-simplex.

Definition of a Simplex is given below:

A simplex may be defined as the smallest convex set which contains the given vertices.A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.

Simplex method Problem 1:

Consider

x1 + x2 - x3 + x4 = 5

2x1 – 3x2 + x3 + x5 = 3

-x1 + 2x2 - x3 + x6 = 1

Solution:

The initial tableau is given by

Tableau 1:

x1 x2 x3 x4 x5 x6 b B1

x4 1 1 -1 1 0 0 5 1 0 0

x5 2 -3 1 0 1 0 3 0 1 0

x6 -1 2 -1 0 0 1 1 0 0 1

The current basic solution is [ 0, 0 , 0, 5, 3, 1]T which is feasible. Suppose we choose a1,1 as our pivot element. Then after one pivot operation, we have

Tableau 2:

x1 x2 x3 x4 x5 x6 b B1

x1 1 1 -1 1 0 0 5 1 0 0

x5 0 -5 3 -2 1 0 -7 2 1 0

x6 0 3 -2 1 0 1 6 -1 0 1

We note that the current basic solution is [ 5, 0, 0, 0, -7, 6]T which is infeasible. Using the new (2, 2) entry as pivot, we have

Tableau 3:

x1 x2 x3 x4 x5 x6 b B1

x1 1 0 -2/5 3/5 1/5 0 18/5 1 0 0

x2 0 1 -3/5 2/5 -1/5 0 7/5 2 -3 0

x6 0 0 -1/5 -1/5 -3/5 1 9/5 -1 2 1

The current basic solution is [ 18/5 , 7/5, 0, 0, 0, 9/5]T and is feasible. Finally, let us eliminate the last slack variable x6 by replacing it by x3.

Tableau 4:

x1 x2 x3 x4 x5 x6 b B1

x1 1 0 0 1 -1 -2 0 1 1 -1

x2 0 1 0 1 -2 -3 -4 2 -3 1

x3 0 0 1 1 -2 -5 -9 -1 2 -1

The current basic solution is [ 0, -4, -9, 0, 0, 0]T which is infeasible and degenerate.

Hope you like the above example of Simplex.Please leave your comments, if you have any doubts.

Symmetry

Symmetry:

Introduction:


In mathematics, Symmetry usually conveys two primary meanings. The first meaning is aesthetically pleasing proportionality and an imprecise sense of harmonious. The second meaning is "patterned self-similarity" that can be proved according to the rules of a formal system.Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.

The "precise" notions of symmetry have various measures and operational definitions. For example, symmetry may be observed:

* with respect to the passage of time;
* as a spatial relationship;
* through geometric transformations such as scaling, reflection, and rotation;
* through other kinds of functional transformations; and
* as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.

The best way to understand about symmetry is by learning about the Types of Symmetry:

Properties of Reflection symmetry:


Reflection symmetry is symmetry which includes mirror symmetry, mirror-image symmetry, or bilateral symmetry. In 1D, the point of symmetry is available. I

In 2D, an axis of symmetry is available. In 3D, the plane of symmetry is available. Mirror symmetric is nothing but an object or figure which is indistinguishable from its transformed image.

The symmetry of a two-dimensional shape is a line, if any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. When any of the shape was to be folded in half over the axis then the two halves would be identical.

Symmetry of isosceles is the triangles and the kites and the isosceles trapezoids are the symmetry of quadrilaterals.


Properties of Rotational symmetry:


Rotational symmetry is also symmetry in which some or all rotations in m-dimensional Euclidean space.

Rotational symmetry is direct isometrics.

In the rotational symmetry we can take that point as origin. The rotational symmetry forms the special orthogonal group in which the group of m×m orthogonal matrices with determinant 1.


We will discuss about the various types of Symmetry in Detail in the next blogs to come,here is a list of the various types of Symmetry in Geometry:

Symmetry in geometry

* Reflection symmetry
* Rotational symmetry
* Translational symmetry
* Glide reflection symmetry
* Rotoreflection symmetry
* Helical symmetry
* Non-isometric symmetries
* Scale symmetry and fractals

Hope you like the above example of Symmetry.Please leave your comments, if you have any doubts.

Linear Programming

Linear Programming:


Let us learn about what we mean by the term Linear Programming,and let us also explore the standard forms of Linear Programming.
Linear Programming is the most popular and widely accepted technique of mathematical programming. The aim of linear programming is to utilize the scarce resources such as man power, man, energy, material and so on. A linear programming problem is one that seeks to maximize or minimize an objective function subjects to constraints. The problem is referred to as linear programming if both the objective function and constraints are linear.


Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear equations.

Structure of Linear Programming Model:


1. Identification of decision variable
2. Define the decision variable
3. Define the objective function
4. Illustrate the constraints to which the objective function should be optimized.
5. Add the non-negative constraints from the consideration


Let us now learn about the standard form of Linear Programming:

Standard form is the usual and most intuitive form of describing a linear programming problem. It consists of the following three parts:

* A linear function to be maximized

e.g., Maximize: c1x1 + c2x2

* Problem constraints of the following form

e.g.,

a1,1x1 + a1,2x2 ≤ b1
a2,1x1 + a2,2x2 ≤ b2
a3,1x1 + a3,2x2 ≤ b3

* Non-negative variables

e.g.,

x1 ≥ 0
x2 ≥ 0.

* Non-negative right hand side constants

bi ≥ 0

The problem is usually expressed in matrix form, and then becomes:

Maximize: cTx
Subject to: Ax ≤ b, x ≥ 0.

Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problems involving negative variables can always be rewritten into an equivalent problem in standard form.

Hope you like the above example of Linear Programming.Please leave your comments, if you have any doubts.

algebra of complex numbers

Let us study algebra of complex numbers,
A complex number is an ordered pair of real numbers with addition defined
by
(a, b) + (c, d) = (a + c, b + d)

and multiplication defined by
(a, b) × (c, d) = (ac − bd, ad + bc),

where a, b, c, and d are any real numbers.
We will let i denote the complex number (0, 1). Then, by our definition of multiplication,
i2 = (0, 1) × (0, 1) = (0 − 1, 0 + 0) = (−1, 0).



Geometric representation of a complex number

If we identify the real number a with the complex number (a, 0), then we have
ai = (a, 0) × (0, 1) = (0 − 0, a + 0) = (0, a).
Then for any two real numbers, we have
(a, b) = (a, 0) + (0, b) = a + bi.

That is, a + bi is another way to write the complex number (a, b). In particular, with this
convention, becomes
i2 = −1,
that is,
i = p−1.
Moreover, we may write as
(a + bi) + (c + di) = (a + c) + (b + d)i
and (7.1.2) as
(a + bi) × (c + di) = (ac − bd) + (ad + bc)i.
In fact, we may view the latter as a consequence of the ordinary algebraic expansion of
the product
(a + bi)(c + di)
combined with the equality i2 = −1. That is,
(a + bi)(c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i.
It also follows from this formulation that if r is a real number, which we identify with
r + 0i, and z = a + bi is a complex number, then
rz = r(a + bi) = (r + 0i)(a + bi) = ra + rbi.
Hope the above explanation helped you.

Exterior Angle of a Triangle

Let us learn what is Exterior Angle of a Triangle,
An exterior angle of a triangle is formed when one side of a triangle is extended. The nonstraight angle (the one that is not just the extension of the side) outside the triangle, but adjacent to an interior angle, is an exterior angle of the triangle (Figure 1 ).





Figure 1 Exterior angle of a triangle.

In Figure 1 , ∠ BCD is an exterior angle of Δ ABC.

Because m ∠1 + m ∠2 + m ∠3 = 180°, and m ∠3 + m ∠4 = 180°, you can prove that m ∠4 = m ∠1 + m ∠2. This is stated as a theorem.

Theorem 26: An exterior angle of a triangle is equal to the sum of the two remote (nonadjacent) interior angles.

Example 1: In Figure 1 , if m ∠1 = 30° and m ∠2 = 100°, find m ∠4.

Because ∠4 is an exterior angle of the triangle,

Hope the above explanation helped you.

Subtracting integers using numberline

Let us learn how to subtract integers in the number line: The first number in the expression tells you the initial position, the second number tells the number of ‘jumps’ you need to make in the number line and, the minus sign tells the direction of the jump which is to the left of the first number. For example to subtract 3 from 2, (in symbol, 2 – 3), you will end at -1 after jumping 3 units to the left of 2.

The problem arises when you will take away a negative number, e.g., 2- (-3). For the process to work, the negative sign is to be interpreted as “do the opposite” and this means jump to the right instead of to the left, by 3 units. This process is also symbolized by 2 + 3. This makes 2 – (-3) and 2 + 3 equivalent representations of the same number and are therefore equivalent processes.
1. the interpretation of the operation sign (to the left for minus, to the right for plus);

2. the meaning of the numbers (the number your are subtracting as jumps, the number from which you are starting the jumps from as initial position);

3. the meaning of the negative sign as do the opposite of subtraction which is addition, and;

4. finding an expression that also represents the process in the case of taking away a negative integer

Hope the above explanation helped you.

Surface Area of a Combination of Solids

Let us learn what is surface area of a combination of solids,

Let us consider the container seen in Fig. 1.1. How do we find the surface area of
such a solid? Now, whenever we come across a new problem, we first try to see, if
we can break it down into smaller problems, we have earlier solved. We can see that
this solid is made up of a cylinder with two hemispheres stuck at either end. It would
look like what we have in Fig. 1.2, after we put the pieces all together.
If we consider the surface of the newly formed object, we would be able to see
only the curved surfaces of the two hemispheres and the curved surface of the cylinder.
So, the total surface area of the new solid is the sum of the curved surface
areas of each of the individual parts. This gives,
TSA of new solid = CSA of one hemisphere + CSA of cylinder
+ CSA of other hemisphere
where TSA, CSA stand for ‘Total Surface Area’ and ‘Curved Surface Area’
respectively.
Now me give you some examples on Surface Area of a Combination of Solids.