Variable and Expression

In the previous post we have discussed about Completing the Square and In today's session we are going to discuss about Variable and Expression. When we study about the algebra and finding the solution of the algebraic equations, we say that the child must have the knowledge of the variable and expression before we start the study of the equations. By the term variable, we mean the values which are unknown and the different values can be placed to these variables. Now we will say that the expression can be formed by joining the terms with the operators + and -.
So we say that x , y , z are the variables. On the other hand we have 3x, 5y, 6z as the terms which are used as the units for the formation of the algebraic expressions. So we say that 3x + 5y – 2z is the expression. We can perform the mathematical operations on the expressions.

In order to add the expressions, we need to remember that the like terms of the expressions can be added together. So if we have the two expressions 3x + 6y and 4x – 2y so to add the given expressions we will proceed as follows:
(3x + 6y) + (4x – 2y),
= ( 3x + 4x ) + ( 6y – 2y),
= 7x + 4y.
Thus we say that the coefficient of the like terms are added up. In the same way the operation of subtraction can also be performed. So if we take the two expression: 3x + 6y and 4x – 2y . If we need to find the difference of the two expressions we will write it as follows: (3x + 6y) – ( 4x – 2y).
= ( 3x – 4x ) + ( 6y + 2y),
= -x + 8y.
We can also perform the operation of multiplication and division on the algebraic expressions. Different rules are followed for solving such expressions.
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Completing the Square

In this blog we will see how to Completing the Square. To find the complete square of any term first we need to take a quadratic equation. For example: as we know that the equation which has highest power is 2 is said to be quadratic equation. The general form can be written as: ax² + bx + c = 0, then change the general form into this given form:

a ( x + d)² + e = 0.

If we want to find the value of variable ‘d’ then we use formula: the formula to find the vaule of ‘d’ given as: d = b / 2a and formula to find the vlaue of ‘e’ is given as: e = c – b² / 4a. To calculate the complete square we need to follow some basic steps so that we can easily understand the process.

Step 1: First of all we have to take a simple expression. Suppose we have simple expression as: x² + bx. Here in this expression value of x is twice.

Step 2: Then complete square in the given expression. So it can be written as: x² + bx + (b / 2)² = (x + b / 2)², it means if we add (b / 2)² then we get the square.

Let understand all the steps with the help of small example:

For example: p² + 6x + 7, in this expression the value of b is 6,

Solution: Given expression is p² + 6x + 7, to find its complete square we need to follow the above mention steps:

First of all we have to find the complete square or value of d. we can find the vlaue of d using the above given formula:

d = b / 2a = 6 / 2

So it can be written as:

p² + 6p + (6 / 2)² + 7 – (6 / 2)²

So it can be written as:

(p + 6 / 2)² + 7 – 9 on further solving we get:

(p + 3)² – 2. in this way we can find out the complete square.

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Irrational Numbers List

In the previous post we have discussed about Is 0 a rational or Irrational number and In today's session we are going to discuss aboutIn mathematics, any number written in the form of simple fraction is said to be irrational number or the number which is not rational is irrational number.

The ‘pi’ is the example of irrational number because its value after the decimal point is:

⊼ = 3.1415926535897……….. So it cannot be expressed in form of rational number.

Now we will see the irrational numbers list. The irrational number list is given below: A list of irrational numbers includes numbers like: (know more about Irrational Numbers List, here)

 √2, √3, √5, √7, √11, √13, √17, √19, √23, √29, √31, √37, √41, √43, √47, √53, √57, √59, √61, √63, √67, √69, √71, √73, √79, √83, √87, √89, √93, √97, √101, √103, √107, √109, √111, √113, √117, √119, √123, √129, √131, √137, √141, √143, √147, √153, √157, √159, √161, √163, √167, √171, √173, √173, √183, √187, √189, √193, √197, √201, √203, √207,√211………...and so on. The List of Irrational numbers include many other numbers. Let’s study some facts about irrational numbers:

·         Negative of an irrational number is also irrational number, meaning of this sentence is: if ‘y’ is irrational then ‘–y’ is also an irrational number.

·         If we add an irrational number and rational number then the sum we get is also irrational. Let ‘p’ is an irrational number and ‘q’ is a rational number then the sum (p + q) is also irrational.

·         In case of roots also the above step is applicable, let √7 is irrational and ‘e’ is rational then the sum √7 + e is also irrational. This is all about the irrational number list. Now we will Units of Momentum. Momentum can be defined as the product of mass and velocity of given object. The unit of momentum is kg.m/s. If you are prepression for IIT then please prefer online tutorial of iit sample papers. It is very helpful for iit exam point of view.   

Is 0 a rational or Irrational number

Before comparing that Is 0 a rational or Irrational number, it is necessary to learn about the rational and irrational. Rational number can be defined as a number which is written in the fraction form or written in p/q (in ratio). For example: 1.2 the number is rational because it is also written in the fraction form. So we can write it as 6/5. An irrational number can be defined as a real number which cannot be written in the fraction form or cannot be written in p/q form. For example: pi the value of pi is 3.14 it is not in the ratios. So pi is included in the categories of irrational number. In word we can say that the numbers are not rational are all irrational numbers.
Now we will see Is 0 a rational or Irrational number? According to the definition of rational number and irrational number we can easily say that ‘0’ is rational number because it can be written in the form of p/q or in form of fraction. If we write 0 in fraction form then we can write it as:
⇨ 0 / 1 = 0
We can easily compare any number using the definition of rational number and irrational number.  Suppose we have given some number 9.5, 5, 1.75, √2, 0.111, √3, √99, now find which number is rational number and which one is irrational number. (know more about Is 0 a rational or Irrational number, here)
According to the definition of rational number and irrational number we can easily compare the given number.
9.5 can be written as 19/2, so it is rational number. 1.75 can be written as 7/4 so it is also rational number. √2, √3 and √99 cannot be written in the fraction form so these numbers are irrational number. And 0.111 can be written as 1/9 so it is also a rational number. So this is all about the rational and irrational number.
Now we will see the Units of Density, unit of density is given as kg/m³. Before entering in the examination of 10^th class please focus on 10th maths question paper. It is very helpful for exam point of view and In the next session we will discuss about Irrational Numbers List. 

How to Find Percentage

In the previous post we have discussed about Proportions and In today's session we are going to discuss about Percentage, It defines how a number is present in form of fraction. It is define as a ratio that is based on the whole number. We can describe percentage as a value occurred on per 100. As the name describe per cent in which cent means hundred. In this blog we are going to know about how to find percentage. When we define any number in terms of percentage as 25% means 25 / 100 or 25 per 100. It also express the method of changing any percent value in the whole number.

We can easily express a number in form of percentage , first generate a fraction that have numerator as the value for which we want to calculate percentage and denominator define as the whole value and then multiply this fraction with 100 .It gives the percentage as if there is part x and its whole value is y then percentage is expressed as x / y * 100.

We can describe whole process of generating percentage in form of some steps as:

(1): Find the whole value

(2): define the portion for which percentage will be calculated.

(3): Generate the fraction

(4): calculate the percentage by multiplying the fraction value with the 100.

All the steps are described by an example as if the value is 25 and whole value is 50 than percentage is calculated as put the whole value and partition value in form of fraction as 25 / 50 and then multiplied it with 100 as 25 / 50 * 100 = 50 % that means 25 is 50 % of 50 that means 25 is just half of 50.

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Proportions

In algebra proportion is define as the special form used for comparison of two ratios . When we talk about the ratios it define as relationship between two or more things. But when we talk about the proportion it is the method of setting two ratios equal. We can say it in other words as when two ratios are equal to each other then their proportion are also equal. It is explained as 1 / 2 is equal to the 2 / 4 or 14 / 28.

Proportions is used when there is one part is missing in the given ratio as if a ratio is equal to other ratio means there proportion are equal then we can easily find the missing value. (want to Learn more about Proportions, click here),

As if there are two ratios a / b that is equal to x / y then these are stated as

a / b = x / y means these ratios have same proportions.

Sometimes values are missing from ratios but having the same proportion as x / 10 and 1 / 2 have these ratios are stated as x / 10 = 1 / 2, so for finding the value of x we use the proportion as

x = 10 * 1 / 2

x = 10 / 2 = 5.

We can also depict the proportion as the comparison that shows the relative relation between two or more things in terms of quantity , quality and any other sort of measurement.

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Consecutive Odd Integers

In the previous post we have discussed about How to solve Consecutive Integers and In today's session we are going to discuss about Consecutive Odd Integers. Let us first talk about the integers. The numbers which can be expressed in the form of the whole numbers and their additive inverse are called integers. All the integers can be expressed on a number line. A number line contains the series of all positive and negative numbers marked at the equal interval, where we could observe the number zero at the center, positive numbers at the right of the number line and negative numbers at the left of the number line.  Now we will learn about the Consecutive Odd Integers. By the consecutive numbers, we mean the numbers appearing one after another. Thus 1, 2, 3 . . .  are consecutive integers.  Now if we talk about the consecutive odd integers, we mean the series of odd integers which occur one after another but are odd.  So we say that the series of odd consecutive integers are 1, 3, 5, 7, 9 , . . . . . . etc.  In case of writing the series of odd integers in the form  of general series we will assume any number x. We know that 2 * x will always be the odd number. Now on another hand we say that if we add 1 to the odd number, it becomes the odd number. For this we say that the number 2x + 1 is any odd number. Now to write the series of the odd consecutive integers, we write 2x + 1, 2x + 3, 2x + 5, 2x +7 . . . . etc. (know more about Integer, here

 In order to learn about the Variance Calculator, we will visit online math tutor and learn more about the related topics. Central Board of Secondary Education Sample Papers is also available in all the subjects and they can guide us to understand the   pattern of the question papers in the previous years. It will guide the students to prepare for the fore coming examinations.