Absolute Values

In algebra and in vectors we come across numbers in negative directions although the magnitudes of such quantities have no signs or supposed to be positive.  Let us take a closer study.



What is Absolute Value in Math
Let us consider the numbers 5 and -5. Algebraically, they are not same. But when you plot these numbers on a number line, they are no doubt on the opposite sides of the origin 0. But they are at the same distance from the origin. That is -5 is also 5 units away from the origin on a number line. Hence considering only the fact of ‘how far’, we can say that the absolute value 0f -5 is 5.
Thus we can define that an absolute value of a number is only its physical value and hence it is always referred as positive. Symbolically the absolute value of a variable ‘x’ is denoted as lxl and it is always equal to + x.

Finding Absolute Value
As explained earlier an absolute value of a number cannot be negative. Thus, the easiest method of finding absolute value of a number is just consider the number only, ignoring the sign before it. That is for finding absolute value of -10, just consider only the number 10. That is l-10l = 10.

Limit of Absolute Value
Since the absolute value of a variable is always positive, the upper limit of the absolute value is infinity. However, because an absolute value can never be negative, it cannot cross below 0. Therefore, the lower limit of absolute value is 0. Thus the limit of absolute value of a variable can be expressed as [0, ∞).

Absolute Value Practice
The concept of absolute value is important. For example, A is situated 10 miles from his office and B is situated 5 miles from the same office but exactly in the opposite direction. Now to calculate the distance between A and B, we cannot algebraically say 10 + (-5) = 5 miles or 5 + (-10) = -5 miles. Here you need to apply the absolute value practice and say the distance is l10l + l5l = 10 + 5 = 15 miles.

Properties of Absolute Values
The important properties of absolute values are,
lxl = x for x ≥ 0, but = -x for x < 0.
If lf(x)l = a then there can be two cases. That is f(x) = a and f(x) = -a.
Suppose lf(x)l + a = b and if a > b, then there is no real solution to the equation.

How to Draw Line Plot?

What is a Line Plot in Math?
The definition of a line plot states that it is a graphical representation of frequency of occurrence of each and every data on a data line. Line plot is graphically represented using some marks such as x or dot.

How to do a Line Plot?
A line is drawn with the data values marked in it as we mark values in the x axis of the graph. The number of occurrences of the data values is marked above the number line individually as the variable x to make a Line Plot.  So the number of “x” marked vertically above each data value shows the number of occurrences of the data value in the given data set. Thus this gives the details of how frequently a data value occurs in the data set.

Example of a Line Plot
Let us consider the marks scored by Students of grade VII in their exam.  Let us assume that there are 10 students in the class and the maximum mark they can score is 100. The marks scored by the ten students are 90, 99, 76, 89, 55, 86, 98, 68, 40, and 35 respectively.

Note that the school follows the grade system as shown below:
• The marks 90-100 will lie in grade A
• The marks 80-89 will lie in grade B
• The marks 70-79 will lie in grade C
• The marks 60-69 will lie in grade D
• The marks 50-59 will lie in grade E
• The marks below 50 are considered as ‘fail’ and they will lie in grade F.

Now let us map the students’ marks in the above grade range:
• The marks lying in grade A are 90, 99, and 98
• The marks in grade B are 89, and 86
• The mark in grade C is 76
• The mark in grade D is 68
• The mark in grade E is 55
• The marks in grade F are 40, 35

From this analysis, we can see that three students have scored grade A, two of them have scored grade B, one student each have scored the grade C,D and E and two students have failed and obtained grade F.  Now, let us mark the line plot with the values A, B, C, D, E, and F:

x
x x x
x x x x x x

<--------------------------------------------------------->
| | | | | |
A B C D E F
Grade
The line plot drawn above gives the graphical representation of the frequency of the occurrence of data in every grade.

Examples of Complementary Angles

What are Complementary Angles?
Two angles are said to be complementary angles, if the sum of the two angles measure 90 degrees.  Thus we can say that two complementary angles together from a right angle measuring 90 degrees. But these two angles need not necessarily be adjacent angles i.e. next to each other.

Example of Complementary Angles
One of the best examples of complementary angles can be seen in right angled triangle.  In a right angled triangle, one angle is the right angle. It is a fact that the sum of all the three angles in a right angle is equal to 180 degrees. Thus, it is clear that the sum of the other two non-right angles will be equal to 90 degrees.  This means that the two non-right angles in a right angled triangle are complementary angles. In other words, we can say that these two angles complement each other.

Some of the other examples of complementary angles are:
55 degrees, 35 degrees
40 degrees, 50 degrees
67 degrees, 23 degrees

How to Solve Complementary Angles?
Now let us see how to find complementary angles.  As we know the definition, finding complementary angles is very easy by applying the definition. If the value of one angle is given, then another angle complementary to that angle is found out by subtracting the given value from 90 degrees.

Example 1:
Consider two angles which are complementary to each other.  If one of the angles is 48 degrees, find the other angle.

Solution: If x is the unknown angle, then 48 added to x will give 90 degrees.  Therefore, x is given by subtracting 48 from 90. i.e., X = 90-48 = 42 degrees.
Therefore, the other angle is 42 degrees.

Some Complementary Angles Problems will be in the form of slightly confusing word problems.

Example 2:
If one of the complementary angles is six more than twice the other angle, find the angles.

Solution:  If the variable Y is considered as one angle measure, then, as per the given statement, the other angle is given by 2Y+6. We know that the sum of the two complementary angles is 90 degrees.  So,
Y+2y+6 = 90
3y = 90-6
Therefore the value of Y will be obtained if 84 are divided by 3. Thus, the value of Y will be 28. If y = 28, then the second angle will be (2*28) + 6 which results in the value 62. Thus, the measures of the angle are 28 and 62.

Free Math Help

Plenty of students find it necessary to find good math help in order to understand the subject and get good grades. Math does not come easily to all and while the subject itself is fairly simple, different students understand it in different ways. Basic math skills are necessary since people use them all the time. Many occupations require an understanding of math or at least specific areas of math.

To learn math effectively, study well right from the beginning. Understanding concepts as and when they are covered, helps students retain them longer. Since each lesson in math builds on what was taught earlier, this ensures that students understand what they are taught as they move further into each topic or chapter.

Going through the lesson before class enables students to get the most out of each class. Even if you don't understand what you are reading, hearing the same thing again will make it much clearer in your mind. It will also enable you to ask better questions and jot down good points. Make it  a habit to go back and study what you have covered in class the very same evening, when it's still swimming around in your memory.

There are a lot of math resources at students' disposal, especially on math help websites. Make use of free math help to learn concepts and get ahead in the subject. Math resources online consist of free worksheets, games and quizzes. There are any number of tutorials covering all the topics in school and college level math, written in simple terms and illustrating each concept with plenty of examples.

Online calculators are very useful tools which can give you the answer for just about any type of question. If you're stuck with a problem, try using any of the free online calculators to get the answer immediately. It is also a great way to cross check your homework and make sure you have everything right. Students can also enlist the help of live tutors who will solve the problem and send you the steps to the solution as well. Practice is key to getting good grades in math and online worksheets provide students with plenty of options to choose from. All the sites provide the answers to their worksheets, with some providing the complete solution too.

Practice of statistics

Statistics in the plural are statistical facts systematically together with some definite object in view, in any field of enquiry, whatsoever of observation, measurement or experiment; for example, statistics of the population of a country, males and females, refugees, births and deaths, heights and weights, income and expenditure, food production, etc. The statistics deals with every aspects of this, consists not only the set, analysis and interpretation of data. The statistics is a technique used to obtain, analyze, summaries, compare and present the numerical data. In this article, example problems and practice problems for learning practice statistics exam is given.

Example Problems for Practice of Statistics:-

Example Problems for learning practice Statistics exam are given below:

1) In statistics the daily maximum temperature recorded in degree C. At New york during the first week of July, 2005 was as under, 39, 37, 38, 28, 30, 35, and 36. Find the mean temperature recorded.

Sol:-

Mean temperature = sum of observations / no of observations

= `(39 + 37 + 38 + 28 + 30 + 35 + 36) / 7`

= `243 / 7`

= 34.7 degree C.

2) find the arithmetic mean of the numbers 3, 0, -1, 7, 11 in statistics.

Sol:-

Mean = sum of observations / no of observations

= `(3 + 0 + (-1) + 7 + 11) / 5`

= `20 / 5`

= 4.

3) In statistics what is the median weekly salary of worker in a firm whose salaries are Rs. 84, Rs. 60, Rs. 50, Rs. 40, Rs. 45, Rs. 42, Rs. 38, Rs. 65, Rs. 71?

Sol:-

1) first arrange the salaries in order: Rs. 84, Rs. 71, Rs. 65, Rs. 60, 50, 45, 42, 40, 38.

2) Next, count the number of salaries. It is 9.

The fifth salary (Rs. 50) has the four salaried which are less than it and four salaries above it. Therefore, Rs. 50 is the middle or median salary.

4) Find the mode for the following data 52, 58, 58, 58, 65, 73, 73, 73?

Sol:-

Here 58 and 73 repeated three times.

So 58 and 73 are two modes.

Practice Problems of Statistics:-

Practice problems for learning practice statistics exam are as follows:

1) Find the arithmetic mean of the numbers 4, 2, 1, 0, 7, 10.

Answer: 4

2) Find the median salary of the following salaries of worker: Rs. 56, Rs. 89, Rs. 121, Rs. 38, Rs. 98, Rs. 70, Rs. 70, Rs. 72.

Answer: Rs. 71.

3) Find the arithmetic mean of the numbers -2, -1, 0 , 1, 4 , 10.

Answer: 2

4) Find the mode of the numbers 10, 11, 12, 12, 14, 15, 15.

Answer: 12 and 15

Exterior angles of polygons

Definition:
For any polygon, the angle formed between any one side of the polygon extended and the next consecutive side is called an exterior angle of that polygon.  See the figure below:

The sum of an internal angle and its corresponding exterior angle is always 180 degrees. In other words the interior angle and the exterior angle of any polygon are supplementary to each other.

A polygon has as many numbers of exterior angles as interior angles. The following figure shows the exterior angles of a pentagon.

Exterior angles theorem:

The exterior angles theorem can be stated as follows:
“The sum of exterior angles of a polygon is always 360 degrees. “
The sum of exterior angles does not depend on the number of sides of the polygon. The sum of interior angles would be different for different polygons. But the sum of exterior angles of a polygon is always 360 degrees.

Proof of the above theorem:

We know that the sum of the interior angles of a triangle = 180 degrees
The sum of interior angles of a quadrilateral = 360 degrees
The sum of the interior angles of a pentagon = 540 degrees
Therefore the sum of interior angles of a polygon of n sides = (n-2)*180 degrees
Therefore the measure of each interior angle = (n-2)*180/n degrees
Since each interior and its corresponding exterior angles are supplements of each other,
Therefore the measure of corresponding exterior angle = 180 - (n-2)*180/n
Therefore the sum of n such exterior angles
= n*[180 - (n-2)*180/n]
= n*[180n – 180n + 360]/n
= 180n – 180n + 360
= 360 degrees
Hence it is proved that irrespective of the number of sides of the polygon, the sum of exterior angles is always 360 degrees.

Exterior angles of a polygon formula:
The formula for the sum of interior angles of a polygon is
= (n-2)*180 degrees

Therefore each of the interior angle would be
= (n-2)*180/n degrees

Therefore the corresponding exterior angle for that polygon would be:
= 180 - (n-2)*180/n degrees

Thus the formula for finding the measure of each exterior angle of an n sided regular polygon is
180 - (n-2)*180/n degrees

Frequency table math

In statistics frequency table refers to the tabular representation of frequencies of a sample. The frequency distribution of a sample can be represented in three ways: (a) Textual representation, (b) tabular representation and (c) diagrammatic representation.

(a) Textual representation: This method comprises presenting data with the help of a paragraph or a number of paragraphs. The official report of an enquiry commission is usually made by textual presentation. The merit of this mode lies in its simplicity and even a layman can present data by this method. This method is however not preferred by statisticians simply because it is dull monotonous and comparison between different observations is not possible in this method.

(b) Tabular representation: We can define frequency table as a systematic presentation of data with the help of a statistical table having a number of rows and columns and complete with reference number, title, description of rows as well as columns and foot notes, if any. This method is any day better than the textual representation because it facilitates comparison between rows and columns. Complicated data can also be represented using a frequency table. To be able to make a diagram for the distribution we first need the data in tabular form. Only then can we convert it to a bar chart or a pie chart. All other parameters such as mean, median, mode, standard deviation, regression analysis etc are not possible unless we have the frequency distribution table.

Example of a frequency table:
The production of rice in a particular region for each year starting from the year 2001 to the year 2007 is given in the frequency table below:

Year	Quantity in MT (Metric tons)(frequency)
2001	25
2002	30
2003	32
2004	36
2005	35
2006	35
2007	37

In the above frequency table example we see that it is relative easy to understand the data. By merely looking at the table we can say that the production in the year 2004 was 36 MT. Also it is fairly simple to calculate the mean, median mode etc as well. Thus we see that representing frequency distribution in tabular has many merits.

(c) Diagrammatic representation of data: Another alternative and attractive representation of statistical data is provided by charts, diagrams and pictures. The various forms of diagrammatic representations are bar chart, histogram, pie chart, etc.