Countably infinite

Countably infinite on the face of it appears to be a contradiction, which it is not.  In everyday language we may be using the term countable to signify a countable (and hence finite) number. But in mathematical terms countably infinite has a very specific and well defined meaning. In order to understand what countably infinite means,  we first need to understand certain concepts and a little about the history of counting.

Countably Infinite :history of Counting

Primitive man had no use for counting large numbers. While hunting or fighting other tribes he found it enough to indicate one, two or more. This sense of one, two or many is also found among the animals. You will almost never find a lonely animal picking a fight with a large group, while it will willingly fight a one to one battle with another animal. If you observe carefully, you will find that at times a lonely animal does fight two after an initial hesitation. This tendency shows that animals too do have a primitive sense of counting and numbers.

With the development of civilization various systems for counting developed, which have ultimately evolved as numbers the way we know them. The concept of infinity also came into existence with this evolution and the meaning of " infinite " was taken as something that cannot be counted.

Finite and Infinite

Concepts of finite and infinite became known and initially none imagined that there could be different degrees of  " infinite"  too. Researches on number theory, set theory and analysis have proved otherwise. Thus we know today that there are finite and infinite numbers but among the infinite numbers there too are various degrees .  Thus, in present day mathematics infinite cannot be just treated as a number that cannot be counted.

Definition of Countably Infinite

Countably infinite in mathematics means a set of elements which can be mapped one to one on to the set of natural numbers. In other words for which a one to one correspondence can be found between all its elements and natural numbers without skipping any natural number and without assigning either two elements to the same natural number, or assigning two natural numbers to the same element..

Not all infinite sets are countably infinite. It requires a very high degree of skills in mathematics and logic to understand how. But it has been established logically and mathematically that not only a higher degree of infinite exists beyond countably infinite, but for any degree of infinite, there exists one which is beyond it and hence larger.

Dividing Rational Numbers

To review, rational number is any number that can be written as a fraction of integers. We shall understand by using example, 3 and 4 are the integers so 3 divided by 4 would be considered as a rational number. Fractions as we know are rational numbers and so are whole numbers. Let us take 3 divided by 1 is 3 and is the same thing so dividing rational numbers using the multiplicative inverse.

Dividing Rational Numbers or How to Divide Rational Numbers – Rational numbers are divided using the multiplicative inverse. When we divide fractions, we find the multiplicative inverse of the divisor; often it is also known as reciprocal. Reciprocal is when we replace the numerator by denominator and denominator by numerator then the resultant fraction. For example: - For a fraction 7/8, its reciprocal will be 8/7.

To Divide Rational Numbers, we find the multiplicative inverse or reciprocal of divisor. For example, if we have to divide 2/5 by ¾ then ¾ is considered as dividend and 2/5 as the divisor. So in this case reciprocal of divisor will be 5/2. Now instead of dividing we will directly multiply 5/2 by 3/4. Therefore in simple words 3/4 is multiplied by the multiplicative inverse of 2/5. When we multiply fractions we multiply straight across the top, 3 times 5 is 15 and 4 times 2 is 8. Thus 15/8 is the solution.

We also understand that in Division of Rational Numbers, we have to turn the fraction upside down and then multiply the first fraction by the resulted reciprocal. If we have two rational numbers 2 /3 and 3/4 and we need to divide ¾ by 2/3 so ¾ will be called as divisor, its reciprocal will be 4/3, which is nothing but done upside down. Then next step will be multiplying the first fraction with the reciprocated fraction. That is 2/3 multiplied by 4/3. Here we shall multiply 2 by 4 and divided by 3 multiplied by 3 and get 8/9 as the solution. We also will understand how we simplify the fractions, 48/108 we simplify by 2 the whole fraction as it is divisible by 2 we get 24/54. Another simplification by 2 gives us 12/27. Here we understand that 12 and 27 will not be simplified by 2 anymore as it’s not divisible by 2, thus which could be another number, 3 is the other number which is divisible so we get 4/9. Thus we simplify the fractions.

Binary Numbers

Binary Numbers Tutorial – Binary numbers are used in computer programming and are used in all modern computer based devices. In Binary numeral system, there are two numeric values which are used to represent the numbers and those two values are 0 and 1. The numerical value represented in every case depends upon the value assigned to each symbol. For example: - 0 is denoted as 0, 1 is denoted as 1 and 2 is denoted as 10. Binary numeral system is also termed as base – 2 system because in this the base 2 is used to give a numeric value to a number.

Binary Representation of Numbers – Binary numbers are represented using two numerical values only that are 0 and 1. We know that in decimal system we use base 10 to represent numbers, the binary works on the same principle but the only difference is that it uses base as 2. For Understanding Binary Numbers, let me show you an example.
For example: - if we have a three digit number 432 then we know that number 4 holds the ones place which means 4 multiplied by 10^0, 3 holds the tens place which means 3 multiplied by 10^1 and 4 holds the hundreds place which means 4 multiplied by 10^2. In Binary system, we apply the same method but the only difference is we use base 2 instead of base 10. For this number we use 2^0, 2^1, 2^2 and so on.

To convert a Binary number to a decimal number, we follow the same pattern. For example: - if we have 100111 then it can be expanded as 1 X 2^5 + 0 X 2^4 + 0 X 2^3 + 1 X 2^2 + 1 X 2^1 + 1 X 2^0 which can be simplified as 32 + 0 + 0 + 4 + 2 + 1 = 39. Hence 39 can be written as 100111.

Multiplying Binary Numbers – Binary numbers are multiplied the same way as we multiply numbers in the decimal system. For example: - 1 X 1 = 1, 1 X 0 = 0 and 0 X 1 = 0. Therefore if we have to multiply 101 and 11 then it will be 1111

Dividing Binary Numbers – Binary numbers are divided the same way as we divide the numbers in decimal system. For example: - 11011 divided by 101 gives 101 as a quotient and 10 as a remainder.

All about Box and Whisker Plot

Definition of Box and Whisker Plot
Statistics tells us that any group of data given for statistical analysis is clustered around a middle or central value.  We can define Box and Whisker Plot as a box drawn graphically by marking the given data points on the graph in such a way that the box so obtained shows the middle part of the data values. The box and whisker plot is represented in box and whisker graph. Definition of Box and Whisker plot also states that it is the representation of data distribution along a graphical number line using quartiles and median. The Box and Whisker plot can be drawn either horizontally or vertically as needed.

How to Create a Box and Whisker Plot?
Once the given data is arranged in ascending order, we can create a Box and Whisker plot by finding the median of the data (Q2) which divides the given data into two halves. The median of the two halves, Q1 and Q3 is found out and it divides the data into quarters or quartiles. Now a box is drawn from Q1 to Q3. The whiskers are drawn at the first and last data value and then they are attached to the box using a line. As the Box and whisker plot visually represents median, the 25th quartile, the 75th quartile, the smallest data value and the highest data value of the given data values, it is also called as five-number summary. These values represent the middle or the centre of the data distribution, the spread of the data and the overall distribution range.

Example of Box and Whisker Plot 
We can see a wide range of applications of box and whisker plot in real life.  Some of the real life examples of the box and whisker plot are comparison of the marks, weight or height of the students in a class, comparison of the rate at which the different commodities are sold, comparison of the weight and the operating time of the phones, comparison of the height climbed by skyscrapers, comparison of cow’s daily milk yield etc.  The box and whisker plot can also be applied on data showing the rainy days, weight of the dogs, weekly earning etc.

Box and Whisker Plot Online
Box and whisker plot can be drawn using free online tools available in the internet. Box and Whisker Plot Online software helps us to draw the box and whisker plot easily for the given data. It is very useful for students to learn for the data required. The data can be user specified or the students can even use the inbuilt data available.

Line Equation

A line equation is nothing but an algebraic equation where every term in that equation will either be a product of a constant and a single variable or simply a constant. These equations will have one or more than one variables. They play an important role in applied mathematics. Equation of a line may simply be said as the linear equation which will not include exponents. In modeling many phenomenons, these equations are used in reducing non linear equations to linear equations from the assumption that the interested quantities will vary from background state only to a small extent.

The most common form used in order to find an equation of a line is given by, Y= mx + b.

The above given equation is used to describe the straight line on a particular coordinate plane. There are many equations representing the equation of a straight line, but this is the common form. This representation is known as Slope and Intercept form. In above said equation, ‘x’ and ‘y’ represents the coordinates of any points lying on the line, ‘m’ is known as the slope of the line(steepness) and ‘b’ is known as the Intercept which is a point formed when the line is crossing the y axis.

Standard Form Line Equation
There are many ways in order to express the equation of a line which are having their own pros and cons. There is a standard form in order to represent the equation of the line, which is given as Ax + By = C, where both A and B are not equal to zero. The equation finally is represented in this form giving out the standard form to represent. This standard form is essential when one needs to graph out the line or in order to find the line y intercept or x intercept. Also, some of the other forms for finding equation of a line are given as follows:

• Point-slope form: y – y1 = m(x-x1), where x1 and y1 are any points present on the line and ‘m’ is the slope, which is nothing but the proportionality constant. 
• Two point form: When there are two points lying on the line, then the formula to find the equation of the line is (x2–x1) (y-y1) = (y2-y1) (x-x1). Here x1, y1 and x2, y2 are two points present on the line, where x1 and x2 are not equal. Here the value y2-y1 divided by x2-x1 is nothing but the slope formula, which when simplified will give the point slope form. 
• Intercept form: This equation is a modification of standard form, where A is 1 divided by ‘a’ and B is 1 divided by ‘b’. This equation states that the sum of x divided by ‘a’ and y divided by ‘b’ will be equal to one.

Algebra Problem

Algebra problems present a challenge to many students who continue to be mystified by the subject throughout high school. The key to understanding algebra is a combination of understanding the concepts, practice, memorizing the equations, and practice.
Since algebra is a new branch of math with concepts and methodology students are completely unfamiliar with, it’s quite important to start students off on a positive note. It’s highly likely that they have already formed a negative impression based on reports from older siblings and friends. The most effective thing that tutors can do is to present algebra as a simple subject which any student can learn with ease.

Of course, presenting the course material in a way that makes it comprehendible to every student is another story altogether. Some students take to algebra like a duck to water, after only a few classes while others tend to take longer. Many students go through high school math with a hazy overview of algebra concepts and many unanswered doubts.  Algebra problems in particular, take some time and guided instruction before students can start solving them on their own, which students may or may not receive in class.

With the number of students in each class, longer and more complicated curriculum to finish, and teachers pressed for time, individual attention for each student is largely an unmet criterion. On the bright side however, there are plenty of learning aids available in the market today which are designed keeping students’ learning issues and hurdles in mind. Many of the learning aids, products and services are focused on providing help with math, particularly algebra theory and problems.

Students who are keen on getting extra help with the subject should seriously consider using one of these services to learn better, understand concepts, solve algebra problems easily, and score better grades. Most of them are available on the internet so all you really need is a computer with an internet connection and you’re all set to access math help, anytime you need it. Students who need personalized instruction can make use of written and video tutorials or live tutoring services. If you’re looking at sharpening your problem solving skills, peruse hundreds of worksheets and practice questions which have different types of questions of varying difficulty. Algebra geniuses can make use of these services to keep challenging themselves, test their knowledge and learn advanced concepts, which may not be covered in class.

Introduction to Statistics Examples

The study of data is called Statistics.  Collections of observation of an individual or a number of individuals is called data.

Collection of data:  There are two types of data namely Primary data and Secondary data.

Primary Data:  The data which is collected by the investigator with a definite object for his own purpose is called Primary Data.

Secondary Data:  The data which is collected by someone other than the investigator is called Secondary Data.

Statistics Examples: Measures of Central Tendency

Measures of Central Tendency:

A numerical value which represents approximately the entire statistical data is called Measures of Central Tendency of the given data.

The different ways of measuring central tendency of a statistical data are

Mean,  Median  and  Mode.

Statistics Examples: Mean

Mean :

The mean of a set of data is the same as finding average.

Mean = `(Sum of all observations )/(Total Number of Observations)`

`Mean of ungrouped data:`

` Mean = ``sum_(i = 1)^n` `f_(i)` `x_(i)`
                ------------------------------
                   `sum_(i = 1)^n` `f_(i)`

Ex :

Find the mean of the following data:

x f
25 25
35 20
45 15
55 15
75 10

Solution:

Construct another tabe:

x f fx
25 25 625
35 20 700
45 15 675
55 15 825
75 10 750
85 3575

` Mean = ``sum_(i = 1)^n` `f_(i)` `x_(i)`
                 ---------------------------------
                    `sum_(i = 1)^n` `f_(i)`

Mean =  `sum`fx /  `sum`f
=3575 / 85
=42.06

Statistics Examples: Median

Median for Raw data:

Arrange the set of datas in ascending or descending order.  The middle most value is the Median.

Rule 1:  If n is odd, the median = `(n + 1)/(2)` th term

Rule 2 :  If n is even, there are two middle terms ie `(n)/(2)`  th term and  `(n)/(2)` + 1 th term.

In this case , the arithmetic mean of these two terms is the median.

Median =     `(n)/(2)`  th term  +  `(n)/(2)` + 1 th term
                        ----------------------------------------
                                                  2

Ex 1:
Find the median of 6, 7, 2, 5 and 10

Sol:
Arrange the given datas in ascending or descending order:
2, 5, 6, 7, 10
Here n= 5 ( odd number)
Median =  `(n + 1)/(2)` th term =  `(5 + 1)/(2)` th term
=   `(6)/(2)` th term
=  3 rd term
=   6

Ex 2:
Find the median of : 6, 11, 15, 7, 19, 8, 4, 10

Sol :
Arrange the given datas in ascending or descending order:
4, 6, 7, 8, 10, 11, 15, 19
Here n = 8 ( even)
Median =    `(n)/(2)`  th term  +  `(n)/(2)` + 1 th term
                     ----------------------------------------
                                              2

Median  =   `(8)/(2)`  th term  +  `(8)/(2)` + 1 th term
                     ----------------------------------------
                                             2

Median  =   `(4th term + 5th term)/(2)`
 = ``(8 + 10)/(2)`
= 18 / 2
= 9

Statistics Examples : Mode

Mode:  Mode is the repeated value of the given data

Ex: Find the mode for the given data:  34, 56, 21, 56, 71, 98, 22, 56

Sol: In the given data 56 is repeated thrice.  So the mode is 56.
Mode for tabulated data:

Number  7 8 9 10 11 12 13 14 15
Frequency 3 7 11 14 13 17 12 8 6

Sol:  Since the frequency of number 12 is maximum
Mode = 12