Standard Deviation and its Formula

The standard deviation is one of the important concepts in mathematics and can be very useful and also have business value. This can be used to compare two sets of dat. Data collection is one of the important tasks in any research. To carry out any research one has to collect dat. Once the data is collected they have to be analysed. There are many software tools that can be used to analyse dat. The statistical concepts are very useful in the analysis. One of the important concepts is the deviation and interpreting standard deviation correctly is very important. The interpretation part has to be done correctly and accurately, otherwise huge blunders might be possible in the research. The standard deviation sign is very useful and has to be used carefully and properly. An example of standard deviation can help explain the concept better. The examples always help in explaining the concepts better.

The standard deviation statistics can be very useful. There are many formulae that have to be studied in mathematics. Formulae can help one to solve the problems easily. If the formula is not known it has to be derived. The process of derivation can be cumbersome and also takes time. If the formula is known time can be saved. Time is a very important factor while solving problems. Most of the problems can be solved but the time to solve the problems is important. If more time is taken for solving a problem then the whole purpose of solving the problem is not achieved. The standard deviation calculation formula can be very helpful in finding the deviation. If the formula is not available it can be a tough task. To find it one must know the term variance. If one finds the square root of this term then the deviation can be easily found out. So, this is the basic formula to find the deviation. This is quite simple to use. There are many tough formulae in mathematics. This is one of the simplest and can be easily used.

The concept of mean is also very important to study. Only if the mean is found, this topic can be covered. This basically denotes the distance from the central value. First the central value has to be found out. The concept of median and mode has also to be studied. These two concepts are also very important and can also be used in various statistical analysis processes. There are many formulae in statistics. These have to be studied in order to solve the problems. The task of solving the problems can become simpler if these formulae are learnt thoroughly. The answers obtained must be verified also. If they are not verified then there can be problems in later stages. So, they have to be verified in the first place. This forms one of the important steps in arriving at the final answer. So, one has to be very careful while arriving at the final answer.

System linear equations

An equation of a straight line is called a linear equation. When we work with two or more linear equations it is called system of linear equation, the point of intersection of these lines is the solution. The possible number of solutions of a given system of equations depend on how the lines are, if they intersect at only one point there would be one solution, if the lines are parallel then there would be no solutions and if the line is the same then there is possibility of infinite number of solutions which is a rare case.

We can find the solutions of the given linear equations using one of the following methods, graphing method, substitution method, and elimination method. Even a graphic calculator can be used to find the solution.

An equation of the form a1x1+ a2x2 + a3x3+…….+ anxn=b where x1, x2, x3,……xn are the variables and a1, a2,a3,….,an and b are constants which are either real or complex numbers is called a linear equation. Here ai is the coefficient of the variable xi and b is a constant term in the equation.

Coming to a system of linear equations, they are linear equations which have the same variables. For instance, a linear system of m equations in n variables y1, y2, y3,……yn can be given as, a11y1 + a12y2+a13y3+………+a1nyn= b1;
a21y1 + a22y2+a23y3+………+a2nyn= b2 and so on, am1y1 + am2y2+am3y3+………+amnyn= bm. For any system of linear equations there are three possibilities of solutions, a unique solution, no solutions or infinitely many solutions. If the linear system has at least one solution it is said to be consistent and if it has no solution then it is said to be inconsistent. Example of linear systems of equations in two variables is, y=3x+2; y=5x-10

System of Linear Equation solver, while solving linear-equations with more than two variables graphing method of solving cannot be used so in such cases we use algebraic method of substitution or elimination.  In substitution method first one of the equations is written in the form something like ‘y=….’ Where y is one of the variables, next step would be to replace this ‘y’ value in the other equation and then solve the equations.

This method is repeated if necessary. In Elimination method a stepwise elimination of variables is done till there is only one variable left, the value of this variable is substituted in one of the linear equations to get the value of another variable. The method is repeated to get the final solution.

Proper Subset

As well known, a subset of A, say, set B, will have some or all the elements contained in the set A. But if the latter option is removed, then the subset is a proper set. Therefore the difference between a subset - proper subset is that the latter has at least one element less than the number of elements of the main set.
We can also define proper subset as strict subset, because of the analogy to the inequalities. The symbols ‘<’ and ‘>’ mean strict inequalities whereas the symbols ‘=’ and ‘=’ mean only just inequalities as they allow a situation of equality as well.  
On the other hand if B is the subset of A, then A is called super subset of B and if B is the strict subset of A, then A is called as proper superset of B. The subset proper set differences can be identified when the relations are expressed symbolically.
B ?  A (B is subset of A)  B ?  A (B is strict subset of A)B ?  A (B is subset of A)   B ?  A (B is strict subset of A)
Let us look into some proper subsets examples. The set {1, 2, 3} is a strict subset of the set {1, 2, 3, 4} because the former does not have the number 4 which is present in the latter. This is a simple example just to understand the basic difference. Let us see where we can see the difference more effectively. We know the domain of a logarithmic function y = ln (x) is (0,8),in interval form.
Suppose we introduce a set D as{0,8}. One can immediately recognize that x is a strict subset of D because, ‘x’ cannot assume the value of 0.
Similarly, we know the range of a sine function y = sin (x) is [-1, 1], in interval form. Suppose we introduce a set R as {-1, 1}. One can immediately recognize that x is a not a strict subset of R because, ‘y’ can assume all the value of between -1 and 1, both included.
Another practical example can be given from number system. We know all natural numbers are integers. But the converse is not true. That is, the set of integers contain natural numbers and in additionalso contain 0 and negative of whole numbers.
Therefore, set of natural numbers is a strict subset of set of integers. Same way, set of irrational numbers

Prominent Variables

Variables in math are very prominent. It is a part and parcel in math, especially in the topic of algebra. Let us try to figure out what is variable and in the process let us find the variables definition. It takes place of an unknown quantity.

Suppose we say that I am 10 years older than my brother, then at any point of time my age is (x + 10). Since at any point of time my brother’s age is also not known, I assumed that as x. Therefore it can be described as letter that represents the unknown quantity and any value can be assigned to that depending on the situation and need.

It is a general practice to use the small case letters of latter halves of English letters to denote the unknown quantities. Of course, none of the terms ‘small case’, ‘latter half’ or ‘English letters’ is a strict requirement. Though it is a general convention, there are exceptions in many cases. The unknown measures of angles of triangles are denoted in capital letters, to be consistent with the symbols of the respective vertices. Again in the very same case of geometry, the first halves of English alphabets are used for unknown quantities.

For example, the prominent statement of Pythagorean Theorem is,              a2 + b2 = c2. Also, there are many Greek letters are used for the unknowns, mostly in trigonometry. For example, the measure of angles are mostly expressed as ? (in case of units in radians) and as a (in case of units in degrees).

In general two or more of these are used in expressions or functions. As a simple example let us take the case of a linear function y = f(x), where f(x) = mx + b. This is a case of input and output function where variable(s) x denotes the input quantity and the ‘y denotes the corresponding output quantity.
Now by change of variables and solving for the same on the left we can find the behavior of the inverse of the function f(x). Let us now see in what way the variables and expressions are related. Expressions are mostly parts of functions. The degrees (the powers) of these decide the type of functions.

The function behaviors can be predicted from the degree and the sign of the leading coefficient of the function. For example, a quadratic function which of degree 2 and with a single unknown will always have rising characteristics on both ends for a positive leading coefficient.

The Method to Find the Triple Product

In mathematics there are basic operations like addition, subtraction, multiplication and division. Multiplication can also be known as the product of numbers. The numbers can be natural numbers, whole numbers, integers, and real numbers and so on. So, the product of two numbers is nothing but the multiplication of two numbers.

There are different types of quantities in mathematics. There are some which have magnitude and there are some which can have both a magnitude and direction as well. Different names have been allotted to these numbers. The former ones are known as scalars and the latter ones are known as the vectors.

The difference between the two is only about the direction they possess. The scalars do not have a direction attached to them. So, they can be handled more easily than the vectors or carrier. Since they have a direction attached to them a sign is used to represent the direction. This sign shows the difference in direction of two quantities of similar nature.

There is a concept in mathematics called the triple product. From the name itself it is understood three products or three quantities are involved in the operation. Now this can be carried out between scalar quantities or the carriers. Depending on the type of the quantities present in the operation the name of the product changes.

The triple vector product is nothing but an operation in which two cross-products are used. There are three of these are involved. The cross products between the first one and then the second and third are taken together. The vector triple product proof can be given mathematically. Even it can be represented geometrically. The triple vector product proof is easy to understand and requires the basic understanding of addition and subtraction.

A vector triple product example will explain the concept and make it clearer. The examples always make the concept easy to understand. Even difficult examples can be made easy with the help of examples. In geometry there exists a figure known as the parallelepiped.

When the scalar triple product is found out it helps in finding the volume of this geometrical figure. There is geometrical meaning attached to this type of product. It must be carefully understood. The scalar triple product can have different values. The values have different meanings. If it is found to be zero, then the volume of parallelepiped is found to be zero

Simultaneous equations

Simultaneous Equations: If two unknown values had to be solved at the same time then these type of equations are called Simultaneous Equations. Simultaneous forms of equations are two equations with two unknowns. They are called simultaneous because they must both be solved at the same time.

There are various methods to solve Simultaneous-equations.
Simultaneous Equation solver:
How to Solve Simultaneous Equations: These equations can be solved by  by elimination method,  Solving by substation method  and by squaring method.  These equations can also be solved by graphing.

Solving Simultaneous equations.

In the  elimination   method for solving  equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.
To add two equations, add the left hand expressions and right hand expressions separately. Similarly, to subtract two equations, subtract the left hand expressions from each other, and subtract the right hand expressions from each other.

Take for example two equations.  x +y=10 and x-y = 2.
See  if by adding or subtracting one variable can be  eliminated. When we add these two equations  +y and –y gets cancelled.
we get 2x =12.

Hence  x = 6  Now plug in the value of x in any one equation say x + y=10
We get 6+y=10, so y = 10-6 which equals 4.  So the solution is  x =6 and  y=4

To solve by substitution , we take one of the equations and solve for x in terms of y  or solve for y in terms of x  Then this value is substituted in the unknown equation and solved for the next variable.
For example  x = 8-y and  2x – y  =  7.
Substitute 8-y  for x in the  2x – y = 7  and solve for y.
2(8-y) - y=7 . now  distribute it.
16 – 2y -y = 7  which equals 16 – 3y  = 7.
We have -3y = -9  Hence  y = 3 .

To find  x, put  value of  y  in the equation 2x – y =7.  We get  2x -3 = 7   so 2x = 10 and x = 5

To solve by graphing bring the two equations to the   y = mx + b form  where m is the slope and b is the y intercept.  Using this first mark the  y  intercept and the slope points.  Plot the points and draw  a straight line connecting  the dots .  Note the point where the two  lines intersects  one another.  From this point of intersection of the lines we can be found the values of  x and y.

Calculating Percentage Formula Using a Formula

There can always be a loss or profit in business. This concept is well explained in mathematics. A profit is obtained only when a product is sold at a greater price than its original cost. A loss is obtained in a business when the product is sold at a lower price than the cost of the product. This is the basic difference between the terms profit and loss. So, the selling price and the cost price of a product are very important in deciding the profit or loss obtained by selling a product. Various techniques are used to earn profit or avoid loss. Both these processes are same in nature. One can earn profit only by avoiding loss.

Now when a loss is obtained in a particular transaction, percent loss can be calculated from it. This helps in better understanding of the whole situation. The percentage loss can be calculated only after the loss in a transaction is found out. So, first the loss is found out. This is found by subtracting the price at which it is sold to the price it is produced. The price at which it is sold is called the selling price and the price at which it is produced is called the cost price. On using this formula in the case of profit a positive value is obtained and in the case of a loss a negative value is obtained. The negative value indicates that the value obtained is loss.

The percent loss formula can be derived once the pct is found out. To calculate percent loss formula, it is necessary to calculate the loss first in a transaction. Once the loss is calculated it is easy to calculate percentage loss formula with its help. The pct can be calculated only when the price at which it is produced is known. The cost at which a product is produced can be taken as the cost price. First the loss is calculated and then it is divided by the cost price.

For the calculation of the pct the answer obtained is always multiplied by 100. Only then the answer obtained is converted into pct. Similar is the case with the calculation of the gain pct. There can be either a gain or a loss in a transaction. Maximum emphasis is given to obtain profit. Profit is nothing but the gain. Both can be used interchangeably.

Rational expressions

Rational expressions are those involving ration of two polynomials or division of two polynomials. Rational expression: P(x)/P(y). Here P(x) and P(y) are polynomials. Here Q(x) cannot be zero.

Dividing Rational Expressions include division of two rational-expressions each of the form given above such that denominators of both the expressions are not zero.

Let us see How to Divide Rational Expressions now:

It would be easy for you to first understand division of rational numbers first before going to division of rational-expressions. To divide a rational number p/q by r/s we do multiplication of p/q and s/r. Actually, we need to reciprocate one of the numbers (second number) and then multiply it with the other number. Multiplication of rational numbers is done by simply multiplying the numerator by numerator and denominator by denominator.

Example: divide 4/3 by 9/5.
Solution: reciprocate the number 9/5 to get 5/9 and multiply it with 4/3.
4/(3 )÷9/5
 =  4/3.5/9
 =  ((4).(5))/((3).(9) )
 =  20/27

We divide rational form of expressions in the similar way as we divide rational numbers. Just reciprocate the second number and multiply it with the first number. Also check if the numerators  and denominators have any factors. If they have then write them in that form and try to cancel out the common factors of numerator and denominator.

If first rational-expression is f(x) = P(x)/Q(x) and second is g(x) = M(x)/ N(x), then f(x)/ g(x):
 = (P(x))/(Q(x))  (N(x))/(M(x)). here we have reciprocated g(x) and multiplied it with f(x).
 = (P(x)  .  N(x))/(Q(x)  .  M(x))

Let us see some examples to Divide Rational Expressions so that we are able to perform divisions of rational-expressions easily without any problem:

Example 1) divide  (y + 5)/3y by  (y+5)/(9y^2 ).
Solution 1) As shown in above explanation, just reverse the rational expression (y+5)/(9y^2 ) and multiply it with (y + 5)/3y :
=  (y + 5)/3y. (9y^2)/(y + 5)
= now here we see that y + 5 term is common in denominator and numerator so cancel it out. Also 3y is common on 3y and 9y2, so cancel one y also to get:
= 1/1. (3 y)/(y + 5)
= (3 y)/(y + 5)

Example 2) Divide (x^2+ 2x-15)/(x^2-4x-45)  by  (x^2+ x-12)/(x^2- 5x-36).
Solution 2) we can write (x^2+ 2x-15)/(x^2-4x-45)  ÷  (x^2+ x-12)/(x^2- 5x-36) as:
 (x^2+ 2x-15)/(x^2-4x-45).  (x^2- 5x-36)/(x^2+ x-12)
= ((x +5)(x - 3))/((x - 9)  (x + 5)).((x - 9)(x + 4))/((x + 4)  (x - 3))
See that all the terms are cancelling out in numerator with denominator. So we get
 (x^2+ 2x-15)/(x^2-4x-45)  ÷  (x^2+ x-12)/(x^2- 5x-36) = 1

Importance of Math Tutoring during Exam

The first and foremost benefit of online Math tutoring is that a student can take unlimited sessions at any time from the comfort of home. Online tutors not only teach students simple steps to solve difficult Math problems, but also give useful exam tips.


It has been observed that many students get worried by the thought of Math exam as it requires more concentration with logical thinking. Students actually need to use their brain while solving tough and tricky Math sums. By using correct methods, a student can easily get an accurate answer to any Math problem. Learning problem solving techniques from an online tutor can help a student to score better in exam. Having stressed and anxiety during examination is a common problem among students. But proper guidance can reduce anxiousness of students up to a certain extent. Online tutoring websites are the best option to get exam help and assignment and homework help. In an online learning session, well qualified tutors will work on your problems and also give you last minute tips before exam.

Online Math tutoring assists student to solve problems in several ways by using correct techniques. In a virtual learning session, students get to know each step for their math problem through a whiteboard. A whiteboard comes with features like math symbols, attachment option,  chat option, etc. that makes a learning session more adaptable for students. Take unlimited sessions in a secure web environment and improve  your problem solving skills. Buy a math tutoring package, just by log-in to a tutoring website and create a loin-Id and password to experience a smart and fun way of learning.

Scoring good marks in Math exam is always a matter of concern for parents as well as tutors. But performing well in exam is not that easy for the students who feel incapable doing tricky Math problems. An online tutor will not only give you enough time to understand logic behind a math problem, but also teach you some useful steps to solve a sum in fast way. During examination it is apparent that students become stressed and feel blank. Hence, it is crucial to have a proper tutoring session where students can work on their queries and clear their doubts from an experienced tutor.

An online Math tutoring program is an ideal solution for students who suffer from exam fear. With the help of a qualified tutor, students can ask as many questions as they need and want. Apart from this, regular practice of Math problems under the guidance of an online tutor can improve a student logical reasoning and make him or her confident during examination. In a web environment,  a session can be scheduled from any location and at student's convenient time so as to give them comfortable learning.

Vector Subtraction

Vector Subtraction is a special case of vctor addition. 
Let there are two Vectors A and B then subtraction of Vector B from A is A – B which can also be written as A + (-B).
So, A – B = A + (-B)
Here (–B) is a Vector that is parallel to Vector B but is in reverse or opposite direction of Vector B.
According to parallelogram law of addition, if there are two Vectors such that they form the two consecutive sides of a parallelogram then the sum of the Vectors is represented by the diagonal of the parallelogram passing through the point at which the tails of two Vectors meet. The geometrical representation of A-B is given in following diagram.

The above method is known as head to tail method.
You can find Subtraction of Vectors by using perpendicular components method. In this method you need to split each Vector into perpendicular components. We normally split the Vectors in x and y directions.
Suppose a Vector of magnitude 5 makes an angle of 30o with horizontal then its vertical component will be 5sin(30)j and horizontal component will be 5cos(30)i. i and j are unit Vectors along the x and y directions respectively. If a component is in downward or left direction assign a negative sign to it. Similarly Split all the given Vectors in such a way. Now add all the horizontal components together and add all the vertical components separately. Subtract a component if its magnitude is negative. Now calculate the resultant magnitude of the Vectors using Pythagoras theorem R2 = A2 + B2 . Here A is sum of horizontal components, B is sum of vertical components and R is resultant.
The angle that the resultant Vector makes with the horizontal is given as: Ө = tan-1|B|/|A|  . Here|A| and |B| are magnitudes of the Vectors.
Subtracting Vectors is similar to addition of Vector. The difference is just of the negative sign.
Let us see some Vector Subtraction Examples now:
Example 1) Subtract Vector B = 9i+7j+2k from A =3i + 5j + k.
Solution) To subtract B from A we write them as A-B = (3i + 5j + k) – (9i - 7j + 2k)
Now add the Vectors in the direction i, j, k separately. As the given Vectors are three dimensional we are considering z direction also. i, j, k refers to unit Vectors in x, y and z direction.
(3-9)i = -6i
(5- (-7))j= 12j
And (1-2)k = -k
A-B=(-6i+12j-k)
|A-B| = √(〖(-6)〗^2+〖12〗^2+〖(-1)〗^2 ) =√(36+144+1) = √181

Venn diagram

An illustrative representation of relationship between and among the given sets that have something in common are called Venn Diagrams. These diagrams are represented using circles, the most commonly used ones are the two circles and three circle Venn-diagrams. It is used mainly to depict the intersections of sets which mean to show the objects or things common to all the sets. The Picture of Venn diagram with two circles or three circles consists of a rectangle which has in it two circles or three circles respectively. Intersection of sets is shown as overlapping circles, if no common objects then the circles are shown separately.

The total number of elements in the set is shown at the right corner of the rectangle using a symbol ‘µ’ or ‘E’. The object or elements which do not belong to any of the sets is shown outside the circle within the rectangle. A glance at the Venn diagram help to understand the complete picture of the relationship among the sets given and also help to find the unknowns values using  the diagram.

Example of Venn Diagram is as follows: set X={1,2,3,5,7,9} and set Y={7,9,11,13}, the intersection of these sets given by X∩Y={7,9}.
The Venn-diagram showing the relationship between these two sets is as given below, the pink region shows the intersection of the sets, X∩Y.It also shows the elements only in the set X {1,3,5} in green color and the elements only in the set Y {11,13} in yellow color.

Venn diagram compare and contrast the relationship between the elements of the sets, they are basically used to visualize the relationship between two or three sets. They are also used to compare and contrast the characteristics of the elements of the sets.  An outline about the sets can be easily created using the Venn-diagrams. Venn Diagram with three circles is a type of diagram which involves three sets.  Let us learn more about this using a simple example: A survey was done on a group of children as how many like to play baseball, football and ice hockey shown by the Venn-diagram below,

The above Venn-diagram gives the following information about the three sets, F, B and I.

• Total number of children (E) would be sum of all the numbers on the three circles, 5+14+8+2+16+5+20=70, so E=70.
• n(F)=29, n(B)=47, n(I)=31
• only n(F)=5, only n(B)=20, only n(I)=16
• n(F∩B∩I)=8, n(F∩B)=22, n(B∩I)=13, n(F∩I)=10
• only n(F∩B)=14, only n(B∩I)=5, only n(F∩I)=2

Greatest common divisor

The numbers when multiplied together to get another number are called the factors of the number. For example consider the number fourteen, 2 when multiplied with 7 gives 14 and hence 2 and 7 are the factors of the number 14 which includes 1 and also 14. The factors of 6 are 1, 2, 3 and 6.
The factors of 9 are 1,3 and 9. The common factors of 6 and 9 are 1 and 3. So, we can define common factors as the factors which are common to all the numbers considered.

 Let us now learn about Greatest common factor which is also called Highest Common Factor and at times Greatest Common Divisor. It can be defined as the greatest integer which divides the number evenly that is without any remainder. In short it is called GCD.

For example, GCD of 12 and 18 is 6 as 6 is the greatest integer which divides 12 and 18 evenly. Let us now find the Greatest Common Divisor of the numbers 36 and 72. The first step to find the GCD would be to write all the factors of each of the numbers, factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36; factors of 72 are 1, 2, 3, 4, 6, 9, 12, 18, 36 and 72.

The next step is to identify the greatest integer which is common to both the lists, 36 is the largest integer common to both the lists and hence GCD of 36 and 72 is 36.

There are different methods to find the GCD of two or more numbers the simplest being the prime factorization method where prime factors of each numbers are listed and then the greatest prime factor common is determined.

The other method is Euclid’s algorithm which involves a process of repeated division given by the Greatest Common Divisor formula, gcd(a,b)=gcd(b, a mod b) where [a mod b] is got by { a – b[a/b]}, mod here means modulus. There are variations in the formula according to the values of a and b.

Let us now learn about properties of GCD which can be given as:

• For any integers ‘a’ and ‘b’ there exist integers ‘p’ and ‘q’ such that [p.a+q.b]=gcd(a,b)
• If ‘a’ and ‘b’ are relatively prime integers then there exist an integer ‘p’ such that pa=1(mod b)
• If ‘a’, ‘b’ and ‘c’ are relatively prime integers and if a divides bc then a divides c

Estimation and Rounding

Estimation and rounding is a important concept in basic mathematics. Number system is the concept that defines everything about numbers and estimation and rounding is a concept covered within number system.

Let’s have a look at both the concepts in this post.
The process of rounding a number estimating its nearest value is called estimation and rounding. For example: After she started breathing exercises during pregnancy , her medical issues lessen to about 59.8%. Here, we can also say the same thing in another way: After she started breathing exercises during pregnancy , her medical issues lessen to about 60%. This is rounding a number estimating to its nearest value. This is an approximate value of the given number.

There are certain rules to be followed while incorporating estimation and rounding, two are mentioned as below:
Rule 1: If the number that needs to be rounded is followed by a number which is greater or equal to 5, it can be rounded to its nearest greater number.

• Example 1: The hospital bag checklist of her NGO includes about 68.8% of things for baby and child care.
The hospital bag checklist of her NGO includes about 70% of things for baby and child care.

• Example 2: These are my most comfortable shoes ; my feet stays almost 95% relaxed wearing the same.
These are my most comfortable shoes; my feet stays almost 96% relaxed wearing the same.
Rule 2: If the number that needs to be rounded is followed by a number which is less than 5, it can be rounded to its nearest smaller number.

• Example 1: She bought apples for 61 rupees per kg for her daughter.
She bought apples for 60 rupees per kg for her daughter.

• Example 2: I have gained almost 6.3 kg of weight after getting married.
I have gained almost 6 kg of weight after getting married.
These are the basics about estimation and rounding.

Finding Determinant Value of 3x3 Matrices

Matrices, like any other expressions,are also subjected to basic algebraic operations. Multiplication is one of those. In addition to normal restrictions on matrix operations, multiplication of two matrices has additional restrictions and the method is a bit strange.

Suppose A and B are two matrices, their product is defined only if the number of columns of A and the number of rows of B must be same. Else, the multiplication is not defined. The algorithm for multiplying 3x3 matrices is described below.

The first element of first row and first column of the product C of matrix A and matrix B is the sum of the products of the elements of the first row with the corresponding elements of the first column of B.

For the second element in the same row, the products are now first row of Avs second column of B. Like this the entire elements of C are found.

With the restrictions and strange algorithm of the process, it can clearly be concluded that multiplication of matrices is not commutative.

A matrix just conveys the details of the data and it does not give any solution to any question on the relations between the items. On the other hand if the same arrangement of data indicates an operation for a solution, then that particular arrangement is called the determinant of the same matrix.

A matrix is denoted by containing the items of a data within a set of elongated square brackets. In a determinant the same are enclosed by two thin vertical bars (similar to absolute value symbol). The following is an example.

The determinant of the matrix indicates the solution by the operation (a1 *b2- a2*b1).
The determinant of a matrix with 3 rows and 3 columns is called a determinant 3x3 matrix or just as determinant 3x3.
To find determinant of 3x3 matrix, first rewrite the elements in the same order but with the symbol of a determinant. To find the value of determinant of a 3x3, the following is the procedure.
Consider a11, the first element in the first row and column. By deleting the row and column that includes a11, you are left with a 2 x 2 determinant which is called as the minor of the element a11and denoted as, M11.
Same way, for the second and third elements in the first row a12 and a13 , the minors are M12 and M13 respectively. Now the formula for the value of the determinant D is,
D = (a11*M11  -a12*M12 + a13*M13)

Set theory

We deal with many types objects in mathematics and when we combine them and out them together, we call them Sets Math and study of that collection of objects is the theory of sets.

The objects of the sets are usually called elements.

Set Theory Notation– We know that the collection of elements is termed as sets. We usually write a st in curly brackets.
For example: - A st of first five even numbers can be written as {2, 4, 6, 8, and 10} or the saet of first seven natural numbers is written as {1, 2, 3, 4, 5, 6, 7}.

The intersection of two sets is symbolized by inverted U and the union of two sets is written as with symbol U. Union of sets A and B means the saet of all objects that is member of both the sets and this can be written as A U B.

The intersection of two sets that is A intersection B is all the elements that are part of both A and B. If A is contained or part of B than we can call A as subset of B or if B has A contained in it then we can call B as superset of A.

We use different types of axioms in the theory of sets. Axioms can be logical or non-logical. Logical axioms tells us the universal truth which is like a + b = b + a.

An axiom act as a starter to derive the proofs similarly in Set Theory Proofs, we make use of Axiomatic Set Theoryto derive various proofs. 

There are many types of axioms that are used in the axiomatic theory which are axiom of existence, axiom of equality, axiom of pair, axiom of separation, axiom of union, axiom of power, axiom of infinity, axiom of image, axiom of foundation and axiom of choice.

Set Theory Problems are simplified using the axioms of the theory.

We have different types of special sets that we use in mathematics and that are used in the theory of sets.
For example the saet of all integers is denoted by Z where Z = {…-3, -2, -1, 0, 1, 2, 3…} and st of all prime numbers is denoted by P where P = {2, 3, 5, 7, 11, 13, 17, 19, 23…}

LCM (least common multiple)

LCM is an abbreviation for Least Common Multiple’. It is also called as ‘lowest common multiple’ and less popularly as ‘smallest common multiple’. For a given set of two or more integers, the least common multiple is the least integer that can be divided by all the integers in the given set.

The fundamental method of finding the smallest common multiple for a set of integers is first list out the multiples of each integer. Pick up the common integers that appear in all such lists. Now figure out the lowest among them. It is the least common multiple. For example, let us try to find the least common multiple of 4 and 6.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 …
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42 …
The integers that are seen common in both the lists are: 12, 24, 36 …
The smallest integer among these commons is 12.
Therefore, the smallest common multiple of 4 and 6 is 12.

There is another method to find the lowest common multiple by using the following formula.
LCM of x, y = (x*y)/(GCF of x and y)
Let us try this method to find the least common multiple of 12 and 16.
The GCF of 12 and 16 is 4.
Therefore, the least common multiple of 12 and 16 = (12*16)/(4) = 3*16 = 48.

In cases where you need to find the lowest common multiples of many numbers, especially large, you can find that by prime factorization. The method here is to do the prime factorization of each number. The lowest common multiple is the product of the highest power of each prime number, out of all factors. Let us find the least common multiple of 9 and 12 by this method.
The prime factorization of 9 is 3*3 = 32.
The prime factorization of 12 is 3*4 =3*2*2 = 3* 22.
The product of the highest powers of prime numbers is 32* 22= 9*4 = 36.
Hence, the smallest common multiple of 9 and 12 is 36.

The knowledge of lowest common multiple for a given set of integers are greatly helpful in finding lowest common denominator for a given set of fractions. The lowest common denominator is the lowest common multiple of the denominators of the given fractions. Therefore, LCM problems are mostly found in such applications. For example the lowest common divisor of the fractions (1/3) and (1/5) is 15, which is the lowest common multiple of 3 and 5.

Word Problems on Multiplication

Word problem is one of the most important concepts in basic arithmetic. A word problem is nothing but a text representation of a mathematical operation such as addition, subtraction, multiplication and division. Word problems make it easier for one to solve a mathematical problem. Let’s have a closer look at word problems on multiplication.

Word problem on multiplication is a textual representation of multiplication operation. For example: There are 18 babies in this day care home. Every baby requires two metal sipper bottles each. How many metal sipper bottles one needs to buy? This is the text representation of multiplication problem: (18*2=?). The answer is 9.

Solving Word Problems on Multiplication
Solving word problems on multiplication includes certain steps. Firstly, the numbers to be multiplied by and with needs to be identified. Secondly, the word problem needs to be converted to a mathematical problem and then finally multiply the number. For example: Grade v has 30 students and each student needs 12 Crayola crayons. How many Crayola crayons I need to buy in total? Identifying the numbers, we get to be multiplied with i.e. 30 and the number to be multiplied by i.e. 12. Converting the word problem to a mathematical operation and then multiplying, (30*12=360). Therefore, the answer is 360 Crayola crayons.

Examples of Word Problems on Multiplication

1. Maria has three children. She wants to buy three pairs of socks for each from online shop baby store. How many pairs of socks she needs to buy from online shop baby store?
Answer: 3 children, 3 pairs of socks each
3*3 = 9 pairs of socks.
Therefore, Maria needs to buy 9 pairs of socks.
2. There are 50 people in the show. How many apples do one needs to get so that each person get at least 5 apples?
Answer: 50 people, 5 apples each
50*5 = 250 apples.
Therefore, the answer is 250 apples.
3. The man has two sisters. Each needs five notebooks. How many notebooks do the man needs to buy in total?
Answer: 2 sisters, 5 notebooks each
2*5 = 10 notebooks.
Therefore, the answer is 10 notebooks.

Adjective and its types

Adjective is one of the eight parts of speech. Adjective is a part of speech that says something about the noun or describes a noun. For example: Disney store India collection has many fun toys for kids. Here, ‘many’ is the adjective that is talking something about the noun. There are different types of adjectives, namely, adjectives of quality, adjectives of quantity, adjectives of number, demonstrative adjectives and interrogative adjectives. Let’s have a closer look at each of these types along with examples in this post.

Adjectives of Quality

These types of adjectives answer to the question “of what kind”. Common examples of adjectives of quality are: beautiful, ugly, big, small and more. For example: Arun bought some very beautiful kids’ toys and accessories from online Disney store India collection.

Adjectives of Quantity

These types of adjectives answer to the question “how much”. Common examples of adjectives of quantity are: some, little, any, enough and more. For example: DK books India collection has some good collection of story books and science books for kids.

Adjectives of Number

These types of adjectives answer to the question “how many”. Common examples of adjectives of number are: one, two, three, four and more. For example: I bought three encyclopedia books from DK books India collection yesterday.

Demonstrative Adjectives

These types of adjectives answer to the question “which type”. Common examples of demonstrative adjectives are this, that, these, those and more. For example: These pen packs from ELC India store is of real good quality.

Interrogative Adjectives

These types of adjectives are used to ask questions about the noun. Common examples of interrogative adjectives are: what, which, whose and more. For example: What did you buy from ELC India collection?

More Examples:

They live in a beautiful house. (Beautiful is an adjective of quality.)
She had enough suffering from him. (Enough is an adjective of quantity.)
I got three educational toys from ELC India collection. (Three is an adjective of number.)
Those apples we bought last Monday are very nice. (Those are demonstrative adjective.)
Which is your purchased toy from online? (Which is the dear one?

These are the basics about adjectives and its types.

Set Theory and Logic

Set Theory and Logic is branch of mathematics which concerns with the study of sets. Sets are the collection of objects or data which are known as elements of the set(s).
 A is said to be a subset of B if each element of A is an element of B. we denote it as: A ⊂ B.
 There are different binary operations applied on two sets as follows:

Union of sets A and B (A∪B) gives all the elements which are either in A or B. eg: if A={4,5,6,7} and B={12,3,4,7} then A∪B is {3,4,5,6,7,12}.

Intersection of sets A and B gives the elements which are common in both sets. Eg:if A={4,5,6,7} and B={12,3,4,7} then A∩B={4,7}.

Difference of sets A and B give the elements of A that are not in B.  Eg:if A={4,5,6,7} and B={12,3,4,7} then A-B={5,6}.

Symmetric difference (A∆B)of sets A and B gives the elements which are member of either of the sets but not of both. Eg:if A={4,5,6,7} and B={12,3,4,7} then {3,5,6,12} is symm. diff.

Set Theory Venn Diagram is a diagram which projects all possible relationships among some sets. Sets  in a diagram are represented by a closed curves, normally circle in a plane. Above operations can also be represented through these diagrams as shown:

(A-B)

Set Theory Formulas 
A∪A=A
A∩A=A
A∪B=B∪A
A∩B= B∩A
(A∪B)∪C=A∪(B∪C)
(A∩B)∩C=A∩(B∩C)
(A∪B)C= AC∩BC
(A∩B)C=AC∪BC
E(A∪B)=E(A)+E(B)–E(A∩B)
E(A∪B∪C)=E(A)+E(B)+E(C)–E(A∩B)–E(A∩C)–E(B∩C)+E(A∩B∩C)

Infinite Set Theory deals with infinite sets. Infinite sets are those which are not finite or the number of elements is infinite. These sets are countable or uncountable. Eg: set of positive integers is countable set while that of real numbers is uncountable.  

Probability Set Theory explains probability of outcome of an experiment. The possible outcomes of an experiment are represented using sets. For example when a coin is tossed its possible outcomes are: head and tail. So outcome is given as: S ={head, tail} or {H, T}. Similarly, if two coins are tossed then possible outcomes are given as: {HH, HT, TH, HH}.
The formulas of sets theory are also applicable here. If A and B are two events then P(A∪B)=P(A)+P(B)-P(A∩B). Here P is probability of an event to occur.

Set Theory Proofs Examples to understand this topic clearly.
Example) Let P, Q and R are three sets. If P ∈ Q and Q ⊂ R, then P ⊂ R is true or false? If not true, give example.
Solution)
False. Let P={3}, Q={{3}, 2} and R={{3}, 2, 4}. Here P∈Q as P={3}
and Q⊂R. But P⊄R as 3∈P and 3∉R.

components of algebra

In this article components of algebra, we will discuss algebra and the components of algebra. Algebra is a part of mathematics, which mainly deals with the expressions, variables and arithmetic operators. Algebra leads to understand the mathematical concepts from the childhood. Generally any part of the mathematical word problems is converted to algebraic equations prior since which is the easiest method to find the solution by solving the algebraic equation. Let us discuss some example for components of algebra.

Components of Algebra:

Consider the following algebraic equations given below,

`x+5(x+8)=70`

The above line is fully named as algebraic equation or algebraic expression

In the above algebraic equation,

x is variable

‘+’ algebraic operator

Numbers are the constant

Consider the following algebraic equations given below,

`x+y=15`

Algebraic Equation may contain the single variable,two variable or it may contain multivariable .Here the above equation contains the two variable.

Consider the following algebraic equations given below,

`x+sin x +e^(x)=6`

Algebraic equation may contain some other functions such as trigonometric function exponential function or logarithmic function.

Example Problem for Components of Algebra:

Example problem 1- Components of algebra

An integer is six more than another integer, sum of the twice the smallest integer and four times the greatest integer is 54. What is the greatest integer?

Solution:

Consider Smallest integer x

Greatest integer x+6

2x+4(x+6) =70

2x+4x+24=70

6x+24=54

6x=30

x=5

Smallest integer x=5

Greatest integer x+6=11

Example problem 2- Components of algebra

The ages of Paarthi and Sabari differ by 12 years when comparing. If 3 years ago, the elder one be 5 times as old as the younger boy, find Paarthi and Sabari present ages.

Solution:

Consider the age of the Paarthi be x years

The age of the Sabari = (x + 12) years

:. 5 (x - 3) = (x + 12 - 3)

5x-15=x+12-3

4x=15+12-3

4x=27-3

4x=24

x=6

Paarthi present age is 6 years

Sabari present age is 18 years