The Method to Solve Inequalities

The process of solving inequalities is similar to the process of solving equations. The linear equations can be easily solved with the help of the principle of transposing. In transposing only the terms that are unknown are kept on one side and the rest of the terms are brought to the other side. This will help in finding the unknown terms. The terms in an equation and inequalities consists of both constants and variables. The value of the constants does not change throughout the course of the equation or the inequalities but the value of the variables can take different values. The value of the variables can be found out if they are not given.

There are different methods to find the value of the unknown variable. In case of linear equations it is quite easy. But as the complexity of the equation increases it becomes more tedious to find the value of the unknown variable. Once the value of the unknown variable is found it should be checked whether the value obtained satisfies the given equation or the inequalities. If the equation or the inequalities is not satisfied the value of the unknown variable obtained is wrong. One must attempt the problem again and find the new value of the unknown variable.

The question how to solve inequalities is answered from the fact that they can be solved in the same manner the linear equations can be solved. This is because the basic difference between an equation and inequalities is nothing but the symbol used in them. The ‘equal to’ is replaced by the ‘greater than’ or ‘lesser than’ symbol. From this the method to solve inequalities can be learnt. The degree of equations plays a very important role in determining the solution of the equation. The same is true with inequalities as well.

The graphical method of finding the solution can be very helpful in this case. This is very good method of finding the solution. The solution obtained can be checked for its feasibility from the graph itself. From the graph first the feasible region and the rejection region are found out. The solution lies in the feasible region and not in the rejection region. So, once this is known the solution is obtained from the feasible and then a check is done for its feasibility. Every solution that is obtained must be checked for its feasibility.

Understanding square

The word ‘square’ in mathematics may refer to a square in geometry or the same in algebra. Both meanings are similar in some way and also different in some ways.

For the level of a 4th grader, the word usually refers to a plane shape of four sides as shown in the picture below. This can be called the square in geometry.

Shown above is one such. The properties of the same are as follows:
1. All sides are of equal length.
2. All angles are of equal measures.
3. It is also called a regular four sided polygon.

Some examples of a sq. can be stated as follows:
Any face of a cube is also same. A die used for playing board games is same shape. The base of Egypitian pyramids and the base of the Eiffel tower is also having the same shape.
Perimeter of a square: The perimeter refers to the sum of all the four sides of such figure. If the length of each of the sides of the same is ‘a’ units, then the sum of lengths of all the four sides would be = a + a + a + a = 4a. Thus perimeter of a sq. = 4a units.

Area of a sq. = If each of the sides of the same are ‘a’ units, then the area enclosed by the sq. polygon can be given by the formula:
A = a * a = a^2 sq. units.

Example: Find the area and the perimeter of the following:
Solution: Perimeter = 4a = 4*5 = 20 cm
Area = 5*5 = 5^2 = 25 sq cm.
Sq. value of a number:
In math, another concept for the same is regarding the square of a number. In general, such number refers to the area of a sq. figure such that the said number is the length of each of the side. Therefore if we say that we need the sq. of a number x, then the answer would be equal to the area of the same such that each of the sides of the sq. figure is x. So that would be x * x = x^2.

Finding Square of a number:

For smaller numbers, to find the sq. value is relatively easy. For example sq. value of 2 would be  = 2*2 = 4.
Sq. value of 7 is 7*7 = 49 and so on. For larger numbers, it can be found using various methods. Let us try to understand this better using an example.

Example: Find the sq. of the number 13.
Solution: We know that its nothing but 13 * 13
This could be rewritten as:
13^2 = 13*13
= (10+3) * (10+3)
= 10*10 + 10*3 + 10*3 + 3*3 (FOILing the terms)
= 100 + 30 + 30 + 9
= 100 + 60 + 9
= 169 <- answer="" p="">Alternatively, instead of FOILing the terms, we can also use the identity:
(a+b)^2 = a^2 + 2ab + b^2

What is Math Inequalitie

In algebra when we come across a mathematical sentence something like 3x+4y=0 it is called an equation which has an ‘=’ sign. It is not possible always to equate two values; sometimes the values are bigger or smaller relative to each other.
So there is another type of sentence which is used to show the relative size of two values which is called an inequality. The inequality math sentences use one of the following symbols, ‘>’ greater than, ‘<’ less than, ‘=’ less than or equal to and ‘=’ greater than or equal to. Inequality Examples are x+3<9 5-x="" x="-5;">9 etc.

The key words that are seen in the word problems and their meanings are, ‘At least’ means greater than or equal to; ‘Not more than’ means less than or equal to; ‘More than’ means greater than. Some more Examples of Inequalities are as given below:
The sum of x and 4 is greater than -5
Sum of x and 4 is ‘x+4’, the inequality used here is ‘>’, the resultant is -5
Finally the inequality would be, x+4 > -5
Subtracting 4 on both sides gives, x+4-4> -5 -4; x > -9 would be the final inequality. Here it means x can take all the values that are greater than -9. One point to remember here is -9 is not included as there is no equal to sign in the inequality.

The addition, multiplication and absolute principles of inequalities are as follows:

• If a>b then a+c>b+c
• If a>b and c is positive, then ac>bc and if c is negative, then ac
• If  X is any expression and c any positive integer such that |X|
• If X is any expression and c any positive integer such that |X|>c, this would be same as X>c, X>-c

Inequalities Word Problems
A taxi charges a flat rate of $1.85 in addition to $0.65 per mile. John has not more than $10 to spend on the ride. Write an inequality representing John’s situation and calculate the number of miles John can travel without exceeding his limit
Solution: Here the key words are ‘not more than’ so the inequality would be ‘=’
The variable here is number of miles= m
So, the inequality would be $0.65m + $1.85 = $10
Subtracting 1.85 on both sides, 0.65m+1.85-1.85 = 10-1.85
  0.65m = 8.15
Dividing on both sides with 0.65, 0.65m/0.65=8.15/0.65
m = 12.54
John can travel 12 miles without exceeding his limit

Progression definition

Progression definition – it means progress, growth, gain, move forward or instantaneous changes in forward direction. Electromagnetic waves travel forward. It means they are progressing. Plant is growing, it means it is progressing.
Patient health is improving or progressing. The student is progressing; it means that the student is passing his examinations from lower class to higher class. It is the progress in positive or negative direction. Speed may be increased or decreased. It may have acceleration or retardation both.

In mathematics there are three types of progressions.
Arithmetic progression(A.P) :- this take place according to a certain formula, for example
a, a+d, a+2d, a+3d, …………………a+(n-1)d
where a=initial or first term,
d= common difference between the two terms.
n= number of terms

Example1: 1,3,5,7………………..nth term
Sequence of terms is arranged in such a manner so that the difference of any two successive terms of the sequence remains constant

Geometric progression (G.P) :-The rule or formula for G.P is as under.
a,ar,ar²,ar³,…………………..ar^(n-1)
where a=initial or first term,
r= common ratio between the two terms.
n= number of terms
 Sequence of terms is arranged in such a manner so that the ratio of any two successive terms of the sequence remains constant
Harmonic Progression:- a H.P. is the reciprocals of an arithmetic progression. The form of HP is
1/a1,1/a2……………………
a1, a2, a3… is called an HP if  1/a1,1/a2..is an HP.
If a, b, c are in HP, then b is the harmonic mean between a and c.
In this case, b = 2ac/(a+c)
Examples are
12, 6, 4, 3,  , 2, … ,
10, 30, -30, -10, -6, -  , … ,

Progression In other fields:
Astrological:- It is used in Horoscope astrology for  forecast of future happenings
Horizontal:- the gradual movement from left to right or right to left of  a line .
Age:- the process of modification of a photograph of a person show  the  aging effect.
Tax: is a tax in which the amount of tax id directly proportional to the amount of income tax rate is increased as the income amount is increased.
Semantic:- It is the evolution of used word
Color:- the whole ranges of color for which the values changes smoothly through hues, and saturation, along with the illumination, or the combination of all these three factors.
Educational:- It is the progress of the  individual from lower classes to higher class.
Age:- the process of modification of a photograph of a person show  the  aging effect.
Tracking purpose:-It is used in video games to check the performance of the individuals.

Empty set

An empty set, as obviously indicates, is a set which is empty. In other words in an empty-set there are no elements. Sometimes it is also referred as null set. An empty set symbol can be any of the followings.      { } , (/),Ø. Of all these, the last symbol is very prominent.

It may be noted that it is not same as the Greek letter F. There is an interesting point in referring about empty-set. A set when empty is in a way common to all the sets that we think of. Therefore a set with no elements is better referred as ‘the empty set’, the word ‘the’ is more apt than the word ‘an’.

The null set, apart from the fact of having no elements has other properties too. The empty set can be a subset of any set. This may be a bit difficult to understand but a Venn diagram will be helpful to understand clearly. The union of the null set with any set A is the set A itself. It is obvious because we add only a 0 to the number of elements of A.

The intersection of the null set with any set A is again the null set. It is very clear that the intersection set cannot have any elements and it has to be empty. There can be only one subset of the null set which is the null set itself.

Now let us discuss about power set of the empty set. Before that let us see what a power set is. A power set of a set A is the set of all possible subsets of set A, including the null set. For example, if A is {a, b, c}, then all the possible subsets are { }, {a},{b},{c},{a, b},{b, c},{c, a},{a, b, c}. Therefore, the power set AP(A) is defined as {{ }, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}. Thus we find there are 8 subsets for a set A having three elements.

That is, 23 subsets for 3 elements and in set language the power set P(A) in this case can be expressed as P(A) = 23.  Extending this concept in general for a set A having ‘n’ number of elements, P(Ø) = 2n. Therefore, in case of a null set the power set of the null set is P(A) = 20 = 1. That is the null set itself. In other words, P(Ø) = { }.

Learning About the Field of Vectors

The quantities in mathematics can have different features. The features can be magnitude and sometimes direction as well. All the quantities do not have the same feature. This difference in features can be the difference between the quantities. The quantities that have only magnitude are known to be scalar quantities and the quantities that have both the magnitude and a direction can be called vectors or carriers.

This is the basic difference between the two types of quantities. This difference must be understood to understand the difference between the two types of quantities. This difference can be very helpful in understanding the difference between the quantities. The vector space over a field can be understood only when the concept of carriers is clear and understood. It is part of a mathematical in which the carriers which can be multiplied with a scalar.

The operation of addition is also possible within this space. The scalars can be any numbers. Sometimes they can also be real numbers or can also be complex numbers. The complex numbers have an imaginary part attached to them. The flux of vector field is nothing but a flow property of carriers. This property is concerned with a unit area.

The flow is per unit area. This concept is very important in the field of physics also. Both mathematics and physics are related to each other. So, this concept appears in both physics and mathematics. The knowledge of physics will help in understanding of the concept. The mathematical plot vector field will explain the concept.

The pictorial representation of the concepts will help in the understanding of the concepts. So, the pictorial representations are always helpful.

The flow of a vector field is related to the flux of the field. These both concepts are interrelated and the understanding of one concept will help in the understanding of the other concept. The plot can be shown in the Cartesian plane using the ‘x’ and ‘y’ axes.

The vector field graph shows the orientation of the various carriers in the Cartesian plane. The carriers are shown aligned in the different directions. So, the orientation of the carriers can be easily understood with the help of this graph.

This graph shows the different directions associated with the carriers. The carriers are shown oriented in different directions and this can be understood from the graph. So, the graph can be very useful.

Introduction to Linear Algebra

Linear Algebra is one of the branches of Mathematics.  These helps with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are found in modern math thus, it is widely used in both abstract algebra and functional analysis.
It is also found in solving problems in analytic geometry.
 Its application is found extensively  the natural sciences and the social sciences.  This forms an intro to Linear Algebra.

 The mathematical approach of solving for variables in equations is Algebra.  
It does the manipulation of these variables using mathematical structures.
It is like taking algebra to another level by manipulating these variables using mathematical structures. This also helps to write the entire system of equations in the form of matrices. When expressed in the form of matrices this to study in depth of the individual parts of matrix which is known as vertex. A vertex is simply a representation of a coordinate.

Linear algebra is a branch of mathematics.  The creation of analytic geometry led to  the system of linear equations  acquire a new significance.
Linear-algebra is important for both pure mathematics and applied mathematics.    
Linear form of algebra when combined with calculus facilitates the solution of linear systems of differential equations.

Analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences   use the techniques of Linear-algebra.
Linear mathematical models are sometimes used to approximate Non-linear mathematical models because linear algebra is such a well-developed theory.
The real world applications need Linear-Algebra.  Algebra is all about linear system of equations and their solutions Linear-Algebra usually consists of the linear set of equations as well as their transformations on it.  Linear Equations  consists of  topics  like  Linear Equations , Matrices, Determinants, Complex numbers, Second degree equations, Eigen values ,Linear Transformations.

Linear algebra help to mathematics: It is a useful branch of mathematics.   Below listed is some of the applications of Linear form of Algebra.  Constructing curves, Least square approximation, traffic flow, Electrical circuits, Determinants, Graph theory, Cryptography.
It   is used to draw graphs. Most equations of linear algebra will   be a straight line. For example draw a graph for a vehicle travelling at a constant speed at various time intervals.
The graph helps in determining the    unknown variable that is the distance by plotting in on the graph.  This can also be used for   a multitude of different functions that is a ready tool for  lots of different real life functions.

Learning to Solve Algebraic Equations

There are various disciplines in the field of mathematics. It is a very interesting subject to learn and can be fun. To simplify algebraic equations, the concept of equations has to be clear. In equations there are both constants and variables present.

There is difference between both these. The value of constants almost remains same throughout but the value of variable changes. Equations can be solved with the help of various methods. The method of elimination can also be used.

It is very simple to use and can be easily learnt. One of the variables is eliminated to find the value of the other variables in this method. The equation algebra can be an interesting subject to learn if one has interest to solve equations. Sometimes equations in algebra can be very tricky to solve and one has to be careful while solving them.

The concept equations algebra is nothing but the equations that are present in this discipline of mathematics. There can be various methods to solve these equations. Many methods are available online to solve them. These methods can be easily learnt and applied while solving them. The algebra equations online will help in explaining the concept better.

Once the equations are solved, various values of the variables present in the equations are obtained. These values must satisfy the whole equation; otherwise they cannot be taken as the solution of the equation. These solutions must be feasible and must be acceptable.

An equation can have more than one solution. Once the solutions are obtained, it can be checked whether the solutions are right or not. Sometimes even complex numbers can be obtained as part of the solutions. In a complex number there is an imaginary part attached to it. There is also a real part also to the complex number. Both the real and imaginary part forms the complex number.

The art of solving equations can be very interesting. The equations can be of different types. There can be linear and quadratic equations. The difference between the two is only in their degree. The linear equation is very simple to solve compared to the quadratic equation.

The linear equation can be easily solved with the help of basic arithmetic operations and the principle of transposing. Once these concepts are known the linear equations can be easily solved. Once solved the solutions could be easily checked.

Boolean Algebra Tutorial

This Boolean Algebra Tutorial gives us a holistic view of what Boolean functions and calculations are and will surely aid in solving many mathematical problems related to Boolean algebra
Basic operations in Boolean algebra are as follows:
Conjunction: also denoted by an inverted V this operation basically is similar to the intersection operation. In this we have a specific formula. Actually this formula is derived by basic algebra intersection operation:
A intersection B = A + B – A V B - - - - - - -(1)
Here V is the union operation.
In Boolean Algebra we only deal with true or false here. So anything that is not true will be a false. We can also denote it by zero and one. Here one will denote a true and zero will denote a false.
Now let us state the complimentary operation. This is also called as the negation operation. It is a singular operand operation. That is in this we will need just one operand to calculate the result. Let us take an example. Say A = True. Now negation A will be false. This operation can be stated as :
Let A = True;
~ A
Output A. Now the value of A will be false.
Union operation has already been discussed. In Boolean we have certain Boolean Algebra Rules to be followed for the same. Let us take a quick look at them.
True union True = True
(True union False) or (false union true)= True
False union False = false

Similarly for intersection we have
True intersection true = true
True intersection false = false
False intersection false = false
As we can see both the operation are just opposite to each other. These are Boolean form of algebra rules. Let us state them in a more simplified way.
A + 1 = 1
A + 0 = A
A . 1 = A
A . 0 = 0
So it is all about the multiplication or the dot product and the addition with zeros and ones. A here can have any value that is one or zero. Boolean Algebra Examples can be:
Let a = 1
B = 0 so a . b = 1 . 0 = 0
Similarly let b = 0, ~ b = 1. This is an example of complimentary operation.
Other rules for the Boolean forms are:
~(~A ) = A
A + ~A = 1
A. ~A=0
~(A+B) = (~A) .(~B)
~(A.B) = (~A) + (~B)

Help your Child to Score Better in Exam with Online Tutoring

Online tutoring helps students to attain pinnacle of success in the academic field. Get a one-on-one learning session from a preferred tutor any time just by using a computer and internet connection. This learning mode is a great way to overcome student's exam anxiety.


Online tutoring is the ideal learning resource as it helps students to score good marks in exam. It has been observed that parents are more concerned about their child's education and usually they impose lot of pressure on them to do well in exam. The high expectations of parent sometimes create an exam anxiety in student, which leads to poor performance. To attain academic goal, a student should opt for extra learning resource. Online learning is a wonderful option to get a thorough knowledge on a specific topic. In a virtual classroom, students get personalized attention from a tutor, which is extremely comfortable and secured way of learning.

It is apparent that while studying in classroom, many students come across doubts but they don't feel like resolving it from the tutor due to hesitation and embarrassment. In a secure web environment, students have the privilege to clear any type of doubt any time from their online tutors. Besides this, students also get assignment and homework assistance in a step-by-step manner. Experience a streamlined and interactive mode of learning that will surely improve your learning skills, makes you understand a particular topic and also guide you during examination time.

Exam is the yardstick that measures the academic performance of every student therefore it is important to get a good learning guidance. Online tutoring not only caters to the educational requirement of students but also make them efficient while dealing with difficult questions. These one-one-one learning sessions are channelized through a personal computer. A student can get an instant connection with a tutor by log-in to a tutoring website. Apart from this, every topic is well explained by the tutor with the help of a whiteboard. Moreover, to make the session more active, a student can also ask question to the tutor through chat.

Classroom sessions are undoubtedly important but acquiring knowledge from additional learning resource is extremely beneficial for students of all grades. Attending classes on regular basis help students to understand a concept in a better manner. But learning a subject in a web environment will absolutely make you confident during exams. No matter how much time students spent in their studies, learning a subject with strong concentration can actually help you out to understand and remember each and every concept. With online learning, a student can obtain comprehensive knowledge on a particular topic and can perform better to attain academic objective.

Vector Components

As we know Vector is a quantity which has some magnitude and direction. Any Vector can be split into two Vector components such that the two Vector  component are perpendicular to each other.
For example as shown in the figure given below a Vector in the north west direction can be split into two components which are in north and west direction. Similarly a Vector in north east direction can be divided into two components which are in north and east direction. Note that the magnitude of these two components will not be same as that of the initial Vector.

Vector Components and the Vector which is divided into components together form a right angled triangle such that the two components are base and altitude of the triangle and original Vector if hypotenuse of the triangle. Given below a Vector ‘a’ which is divided into two components ax and ay such that ax, ay, a form the base height and hypotenuse of the triangle. ay component shown in figure can be shifted in right direction parallel to form the height of the triangle.  
Components of a Vector can be found out by using properties of trigonometry which are based on a right angled triangle.

According to trigonometry, in a right triangle if hypotenuse makes an angle Ө with the base then,
sinӨ = height or perpendicular/hypotenuse
cosӨ = base/hypotenuse.
For previous illustration sinӨ = ay/a
cosӨ = ax/a
So, Vector Component ax and ay will be:
ax = a(cosӨ)   ……….(component of Vector ‘a’ in x direction)
ay = a(sinӨ)  ………….(component of Vector ‘a’ in y direction)
Vector a can be written as: a = axi + ayj = a (cosӨ)i + a (sinӨ) j
If a Vector is given as a = a1i + a1j, then the Vector components can be read directly. Here a1 is component in x (horizontal) direction and a2 is Vector in y (vector) direction.
Let us take an example of Components of Vectors :
Example) Find the component of a Vector with magnitude 5 and makes an angle of positive 60o with horizontal.
Solution) let the Vector be P.
components of Vector P in x direction will be: Px= PcosӨ
= 5cos(60o)
= 5(1/2)
= 5/2 = 2.5
components of Vector P in y direction: Py= P(sin60o)= 5(√3/2)
So Vector P = 2.5i +5(√3/2)j
Note) If the Vector makes 60o with the vertical then Py = Pcos(60o) and Px = Psin(60o) or Py = Psin(30o) and Px=Pcos(30o) (This is because in this case the Vector will make 30o with horizontal)

Subsets

A collection of things which have something in common as per the rule is called a set. For instance set of colors would be {red, green, yellow, white, blue} this set represents the collection of five different colors, these are called the elements of the set.

The elements are separated using commas (,) and every element is unique in the set. The notation uses parenthesis which are curly or in other words the flower brackets {} and usually a set is denoted using a capital letter.  Let ‘E’ be a set denoting all the even numbers less than ten; so set E={2,4,6,8}. While learning about sets we come across finite set, infinite set, universal set, empty set, sub-set and power set.

In this article we shall learn more about a Sub-set. Consider a set A={1,2,3,4,5,6} and set B={2,4,6}. When we compare both the sets it is clear that all the elements of set B are present in the set A and hence we can call set B as the sub-set of set A.

Definition of a Subset can be given as, a set B is a sub-set of the set A only if every element of set B is in the set A. The subset sign is ‘⊆’.

For instance, set A={orange, pineapple, apple, grapes, kiwi, mango} and set B={apple, kiwi, orange, mango}. Here each element of set B is in the set A and hence Set B is a sub-set of Set A and is denoted as B⊆A. Suppose Set P={2,3,4,5,8} and set Q={1,3,7,9}; each of elements of set Q are not in the set P and hence we cannot call set Q a sub-set of set P.
So, subset meaning is the set which is a part of the whole set.

So, a subset in math can be better explained using the following example, a set P is a sub-set of set Q if and only if all the objects or elements of set P is in set Q.

If set P={x, y, z} and set Q={x, y, z, p, q, r}, every element of P is in Q and hence P⊆ Q. Some of the subsets examples are, list of all the sub-sets of the set A={a, b, c} can be given as { },{a},{b},{c}, {a,b},{a,c},{b,c}, {a,b,c}.

The number of sub-sets of a given set can be given by the formula 2^[n(S)] where [n(S)] is the number of elements in the set. If there are three elements, the number of sub-sets is 2^3 which is 8 sub-sets in all.

Interval notation

A set in mathematics can be defined as collection of elements. There are different ways in which a set of numbers with end points can be described, roster form, set builder form, interval notation, graphing on a number line and using venn diagrams. Interval notation is a method of representing a set of numbers which describe the span of numbers that lay along an axis namely the x-axis.

An interval can be defined as a subset of the numbers. There are two symbols used in interval notation, an open bracket or parenthesis denotes an open interval in which the number is not included and a square bracket or parenthesis denotes a closed interval in which the number is included.

For instance, (-3,3) is an open interval notation in which both the numbers -3 and 3 are not included in the span of numbers and a closed interval notation is given by [-3,3] where both the numbers -3 and 3 are included in the span of numbers.

For more interval notation practice let us consider the following examples, [-4,1)= {-4, -3,-2,    -1 ,0,1} , this is a half open interval. (-2,1] is a half open interval which is equivalent to set of numbers
 {-1,0,1}.
Let us now learn the steps in solving inequalities in interval notation given the inequality x>-4, here the inequality symbol is greater than but -4 is not included, so it is an open interval and also it is never ending which means it extends till infinity.
So, the interval notation would be (-4,-infinity). Let us now graph and write in interval notation the inequality x<=-2.

In the above graph the closed red dot over -2 shows a closed half interval notation which means -2 is also included in the interval and as the inequality is less than -2 it is towards the left on the number line which shows the other end point is never ending and hence is minus infinity.
Together the half interval notation of the given inequality would be (-infinity, -2]. If there is an open dot over the number on a number line it denotes the number is not included and hence the symbol ‘(‘ or ‘)’. So, inequalities in interval notation are very simple when used with the appropriate parenthesis according to the given inequality.
For instance, 0<=x<6 as="" be="" can="" shown="" x="">-1 would be (-1,+infinity). Absolute value interval notation of |x|< 8 is equivalent to any real number between -8 and 8, the double inequality would be -8

Even and Odd numbers

Even numbers and odd numbers are two of the most important concepts in basic mathematics taught in primary school learning. Even and odd numbers are two basic classifications of numbers. Let’s have a look at both the concept along with relevant examples in this post.
Even Numbers
Even numbers are those numbers that can be evenly divided into two parts. In simple terms, any number that is divided by 2 is an even number such as 2, 4, 6, 8, and 10 and so on. For example:
• After comparing the 2 Christmas gift ideas given by Maria and You, I found yours better. (Here, 2 Christmas gift ideas is an even number).
• I have listed 8 common hobbies for children for my next article. (Here, 8 common hobbies for children is an even number).
• Are you turning 42 this year? (Here, 42 is an even number).
• I have bought 4 kgs of apples. (Here, 4 kgs is an even number).
• She bought the shoes for Rs.350. (Here, Rs.350 is an even number).
Odd Numbers
Odd numbers are those numbers that cannot be evenly divided into two parts. In simple terms, any number that is no divided by 2 is an odd number such as 3, 5, 7, 9, and 11 and so on. For example:
• He bought 3 gifts for his baby last week. (Here, 3 gifts is an odd number).
• The flight ticket’s range to my native is currently Rs.6789 (Here, 6789 is an odd number).
• I have bought 3 pairs of Yonex badminton shoes for my kid and his friends. (Here, 3 pairs of Yonex badminton shoes

is an odd number).• She is turning 7 years old this year. (Here, 7 years old is an odd number).
• Maria’s weight was 57 kg recently. (Here, 57 kg is an odd number)These are the basics about even and odd numbers.

Matrix: Cofactors

Consider a matrix that is a square matrx. That means that the matrx is n X n matrx. The number of rows and the number of columns in such a matrx are equal. Each term of the matrx can then have a cofactor. The position of each of the element of a matrx is described by (i,j). Where ‘i’ is row number (ith row) and j is the column number (jth column). For example consider the matrx below:
[a b c]
[d e f]
[g h l]

Here the position of the element a is (1,1) since it is the first element in the first row and first in the first column. Similarly the position of the element f would be (2,3) since it is the third element in the second row. So row number of f is 2 and column number is 3.

Cofactor of matrix:
To be able to find the cofactors of a matrix, we first need to find the minors of each of the elements. This can be done as follows:
If we take any entry, say ‘a’ in the above matrx and remove the row and the column containing a and keep the other entries in the same order, we get the determinant
|e f|
|h l|

This determinant is called a minor of ‘a’. Thus, removing the column and the row containing a given element of a matrx and keeping the surviving entries as they are, yields a determinant called the minor of the given element.

If we now multiply the minor of the entry in the ith row and jth column by (-1)^(i+j), we get the co-factor of that element. Therefore in the above matrx, the minor of h is
|a c|
|d f|
And multiplying this by (-1)^(3+2) = (-1)^5 = -1, we get the cofactor of the element h. This cofactor, therefore is
(-1) * |a c| = -(af – cd)
           |d f|

Note: The co-factor of an element is precisely the factor by which that element is multiplied in the expansion of the determinant of that matrx.

The co factors of a,b,c,… etc are denoted by Ac, Bc, Cc,… and so on.

If every element in a matrx is replaced by its cofactor, the resulting matrx is called the cofactors matrix. Therefore for our matrx above, the cofactors matrx would look like this:
[Ac Bc Cc]
[Dc Ec Fc]
[Gc Hc Lc]

Where, Ac = +(el-fh), Bc = -(dl-fg), Cc = (dh-eg), Dc = -(bl-ch) and so on.

Solving by elimination method

The elimination method is one of prominent methods in solution to the system of linear equations with two variables. The elimination method for solving equations may be apparent in certain cases where, the coefficients of a particular variable or variables are same.

In such a case you automatically use this technique. These cases of equations are called solving elimination equations. For, example, if x + y = 5 and x – y = 1, you eliminate ‘y’ by just adding both the equations to get the solution as x = 3. Similarly you eliminate ‘x’ by subtracting the second equation from first and figure out y = 2. But all systems are not as simple as this. Hence let us take a broader look on solving by elimination method.

Suppose, you have a system of equations in which all the coefficients of the variables are different. Select a variable for elimination. Your skill and experience will tell you which will be the ideal variable to eliminate. Take the LCM of the coefficients of that variable and multiply each equation with the ‘missing’ factor of the LCM in each case.

Now the given set of equations is transformed in such a way that the new set has the same coefficient for the selected variable in each equation. Now do the subtraction or addition operations on the transformed equations, so that the selected variable is eliminated and you get the solution of the remaining variable. Subsequently, plug in that in any of the given equations to figure out the solution of the other variable.

Let us elaborate with an example. Let the system be 17x + 2y = 49 and 19x + 3y = 54. It is prudent to decide to eliminate ‘y’. Eliminating ‘x’ not an incorrect step. But that will lead to cumbersome working. The LCM of 2 and 3 is 6.

Multiplying the first by 3 and the second by 2 (the missing factors of 6), the equations are transformed to 51x + 6y = 147 and 38x + 6y = 108.. A subtraction operation gives you the result as 13x = 39, means x = 3. Now plugging in x = 3 back, you can figure out y = -1.

The method of solving equations by elimination is the basis for the concept of use of determinants for solution of system of equations.

let us make a simple illustration. Suppose a1x + b1y = c1 and a2x + b2y = c2 . By elimination method you can establish, x = (c1b2 – c2b1 )/(a1b2 – a2b1 ), which is the concept behind Cramer’s rule of solution by using determinants.

Sequence and series

Sequence Series is a very important part of mathematics. When we write numbers in an order then we call it sequence of numbers.

The list of numbers follows a particular trend in that. The numbers in a seq are called terms of the seq. We cannot flip the terms in a seq.

Their order cannot be changed. We do the counting of terms from left to right.
For example: - 1, 3, 5, 7, 9 is a seq of five terms where first term is 1 and third term is 5 and last term is 7.

If we look at this seq we see that this seq follow a certain trend and that is every next term is two more than the previous term. Now what is a series?

A series is when we add the terms of a seq that constitutes to form a series. There are different types of seq and ser that exists and they are: -

1. Arithmetic Sequences and Series– In this type of seq and ser, the next term can be determined by adding the common difference to the previous term. The general arithmetic seq and ser is given as a, a+d, a+2d, a+3d, a+4d… where ‘a’ is the first term and d is the common difference. For example: - 1, 4, 7, 10, 13, 16. Here we see that first term is 1 and the common difference is 3

2. Geometric Sequences and Series– In this type of seq and ser, the next term can be determined by multiplying the common multiple by the previous term. The general geometric seq and ser is given as a, ar^2, ar^3, ar^4… where ‘a’ is the first term and r is the common multiple. For example: - 2, 6, 18, 54. Here we see that the first term is 2 and the common multiple is 3 which can be determined by dividing any term by its previous term.

3. Infinite Sequences and Series– In this type of seq and ser, the last term is unknown to us. We do not have fixed number of terms in that as it is an infinite ser. For example: - 2, 6, 10, 14…..

Series and Sequences Formulas are used to determine the first term, last term, sum of terms, number of terms or the common difference or ratio.

Equivalent Decimals

Decimals are nothing but a whole numbers but broken into pieces, so example we have one whole block, and we cut it in 2 pieces, and we call it as Tenths. How to do equivalent decimals, so how do we write it ? We write it as 1 .2 tenth (Here decimal represents AND). But if we break that down even further into more ten pieces and take out and add 3 then we call these as Hundredths.  So this way we have once, tenths and hundredths. So this way we will write it as 1.13 (one and thirteenth, remember here the last piece that we saw. Here we will name the last 1 and 3 as thirteenth, the last digit even if it is zero we call it that). Now late us take one more piece of cube and break it down into ten equal parts that will be called as thousands. So we have ones, tens, hundredths and thousands. Let us say we have this number up in broken to two pieces .we will call it as 1.232 that is one and two thirty two thousands, so we read the decimal like this way.

Now moving up to equivalent decimals examples, what if we have only one block so we will call it as 0.1( here we will read it as zero and one tenth.) but let us say we want to turn it to hundredths. The whole one piece when broken into ten equal parts has the same value for 0.1 as well. However they are hundredth. so all we need to say hundredth we need to add a zero. Because when we read decimals we need to say the last digit. Thus it will be 0.10(we call it as zero and one tenth one hundredth.) so one tenth is equal to tenth hundredths.

How to find equivalent decimals? We can by taking the denominator and dividing it with the numerator of the same fraction. Now for example we have ½ as a fraction so, 2 have to get divided by 1 so we will add a zero and bring up a decimal point. This way we have 10 which will be divisible by 10. As 2 times 5 is equals to 10.  And 10 subtracted by 10 and remainder comes to zero. So we have got the solution as 0.5 is the equivalent decimal.
What is a Equivalent decimal are nothing but the decimals which have the equal proportions.