Correlations

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

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

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

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

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

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

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

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

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

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

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

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

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

Vectors

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

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

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

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

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

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

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

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

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

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

Algebra Word Problems

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

Get easy steps to solve Algebra Problems (click here)


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

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

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

ALGEBRA WORD PROBLEMS

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

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

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

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

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


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

Rational Inequalities

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

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

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

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

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

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

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

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

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

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

Set Operations

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

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

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

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