Compare and Order Integers

A number line is good place to start when describing integers. As integers are all the counting numbers so starting from 0, the numbers to the right side of zero are called positive integers, and from 0 to the left side, all the numbers are called the negative integers.

Comparing Integers - We can Compare Integers with their Opposites which means the numbers on the opposite sides with equal distance from zero on the number line. For example, on a number line, 4 and -4 are the opposites. The number zero is the opposite of itself, and it is considered neither positive nor negative. Let us graph the integers and their opposites. Let’s graph 4 and its opposite on the number line. Firstly graph 4 by placing a dot at positive 4 on the number line. We know the opposite of positive 4 is negative 4. So place a dot at a negative 4 on the number line to complete the problem. Next example lets graph -2 and its opposite on the number line. First, graph negative 1 by placing a dot on the negative on the number line. Next step we know the opposite of negative 1 is positive is 1. So we place a dot on number line to complete our graph.

Ordering Integers – Ordering integers means Ordering Integers from least to Greatest and for this we write integers in the ascending order. The numbers on left of number line is smaller than the number on the right. Comparing and Ordering Integers can be done by plotting them on the number line. For example: - If we have numbers -3, 0, 5,-5,-1 and 4 and we need Compare and Order Integers, we first plot them on the number line and see the integers to the left on the number line are smaller to those on the right. So reading the dots on the number line from left to right gives us the order from least to the greatest. Here we have to do is read the dots from left to right and we have our order. The smallest number is here on the extreme left will be -5 and then it’s -3, then -1, then 0, then 4 and then 5 which is on extreme right. So we can write the order as -5 < -3 < -1 < 0 < 4 < 5.

Numerator and Denominator

Numerator and Denominator Definition - In mathematics, when we study fractions, the numerator and denominator are very important words. In a fraction, the top part is known as the numerator and the bottom part is termed as the denominator.  To understand the concept better, let us draw a circle that is divided into four equal parts, we call them fourths where one of the parts has been shaded. Then we say that one fourth of the circle has been shaded and it is written as ¼, here 4 is the denominator and 1 is the numerator. In Fractions Numerator and Denominator are represented like this Numerator/Denominator.

How do we Define Numerator and Denominator- The denominator describes the number of parts, and the numerator describes the number of parts that are shaded.

How to Remember Numerator and Denominator - Where do these names come from and how do they make sense? Denominator means the name. It is the name for the fraction. Well there is a history to the word denominator, which helps to understand the connections. When we go to the church we might say it has a certain denominations, Luther is one denomination, catholic is another denomination. There is another use of word denomination, when people have bills, a dollar bill or five dollar bill, those are different denominations, or the name that they have.

In school election, we might have noticed, that we nominate somebody to be the president if require. It is a similar word nominate. Nominate means to name somebody to be the president. Or in Spanish, the word nombre that means the name, so nombre, nominate, denomination they all relate to the word denominator. They come from the same land root of denomino.

So denominator is just a name. Whatdoes numerator mean? The number of parts that we talking about. So we have drawn four equal parts, how many shaded parts is the number that represents a numerator. Let us understand it by using an example that is identifying the numerator and denominator in the fraction 3/4.

The numerator is the number on the top in fraction. Here we have ¾ where 3 is on the top so 3 is the numerator and the denominator is the number in the bottom of the fractions. It is 4 here. This fraction represents 3 out of 4 pieces of a pie. The denominator shows total number of pieces of pie and 3 is represented as the part of those 4 pieces.

Definition of Mean

It is also called as arithmetic average, or simply average. I is defined as the arithmetic average of the set of data value and is calculated by the sum of all the values from the data and divide by size of the data. Especially the term arithmetic average is used in mathematics and statics, since it help us to differentiate it from other methods such as harmonic and geometric average. It is most frequently used in mathematics and statistics fields such as economics, sociology and history.

It is generally sample average or population average. If the set of observation is a statistical sample, then that average is called as sample average. If the set of observation is a statistical population, then that mean is called as population average.  The sample average is denoted by the symbol ( ) which also called as x bar and the population average is denoted by the symbol of Greek letter (µ). Both the sample and population average is calculated by adding all the observations and divided by the number of observations.

Formula for Mean
Formula for Sample average
= (∑¦x)/n
Where,
- Sample average
Sx - Sum of all the sample observation
n - Number of sample observation
Formula for Population average formula
µ =  (∑¦x)/N
Where,
µ - Population average
SX - Sum of all population observation
N - Number of population observation


Define Mean
It is defined as it the average of numbers from given set of data and is calculated by just addition of all numbers from set of data and divides them by the size of the data. Simply it is a just average of the given numbers. Average is represented by the symbol .
  = (∑¦x)/n
Where,
- Mean
S - Symbol of summation
x - Values present in given set of data
n - Number of data or size of data or how many numbers present in set of data

Definition of Mean
It is an arithmetic average of set of a data values and is not to be a middle value or any one value from the set of data values. It is a central value of a set of numbers or data. Calculation of average is very simple. Averages is calculated by adding all the numbers and divide them by how many numbers there are, in other words, it is the sum of the numbers divided by the count of numbers.
For example to find the average for given set of data such as 6, 11, 7, 4. To calculate average first add all the numbers,
6 + 11 + 7 + 4 = 28
Then dived the resulting number by how many numbers are their in data set, here there are 4 numbers are their, so
28/4 = 7
The average value is 7.

Sample Mean Formula
Where Sn = x1, x2, x3,………xn random sample of size n, then the formula for sample average is,
= (∑¦x)/n
Where,
- Sample average
S - Symbol of summation
x - Values present in the sample set of data
n - Number of sample data

How to find Directional Derivatives?

In calculus, the directional derivative is a differentiable function which multivariate along a vector V with a point P. It represents nothing but the rate of change of a function which moves in the direction of V, through the point P. The directional derivatives are the generalization of the partial derivatives; where in one of the coordinate axis is always parallel in direction.

Mathematical Definition of Directional Derivatives
The directional derivative is defined as the rate of change of the scalar function f(x) = f(x1, x2, x3... xn) along the unit vector u = (u1, u2, u3….. un), in which the unit vector is defined by a limit in terms of h. Therefore the mathematical definition for the directional derivative can be written as,

 Du f(x) = lim h->0+ f(x+hu) – f(x) / h.

Derivation of Directional derivative

It is simple method to find the directional derivative formula which will be used to simplify the problems. Let us define a function that is made up of a single variable, such as g(z) which equals f(sum of x0 + az and y0 + bz). Here x0, y0, a, b are some numbers which are fixed. Now, z is the letter which does not represent a fixed number.  Then, by the actual definition, for a single variable function we know that, g’(z) equals the limit which tends to h->0 of g (sum of z and h) – g(z) whole divided by h.  Now applying the value of the derivative z as zero, we will get, g’(0) which equals limit which tends to h->0 of subtraction of g (h) and g(0) whole divided by h. Substituting the equation of g(z) mentioned earlier we will get g’(0) as lim h->0 of f(x0 + az, y0 + bz) – f(x0,y0) whole divided by h. Then this equation will be obviously equal to the Du f(x0, y0). Therefore we will finally come with a relationship of g’(0) which equals Du f(x0, y0). Now we can rewrite g(z) as g(z) = f(x, y) where x will be equal to x0 + az and y will be equal to y0 + bz. Then, applying the chain rule we will get the equation as,  g’(z) = dg/dz = Sum of (multiplication of df/dx and dx/dz ) and (multiplication of df/dy and dy/dz), which will be equal to sum of fx (x,y) a and  fy (x,y).After taking z=0, we will get x as x0 and y as y0 and applying these, we will get, g’(0) as sum of            fx (x0,y0) a and fy (x0,y0) b. Now equating this equation with the g’(z) equation of very first we derived, we will arrive with a formula as shown below:

Du f(x, y) = fx (x, y) a + fy (x, y) b.

This equation will provide a simple way of calculation than the previous limit based definition.
Directional derivatives are used in solving the matrix symmetrical valued functions; especially the second directional derivative and it plays a vital role in solving non linear type problems.

Algebra Help

Algebra is the first branch of math which students come across, that is a complete change from arithmetic and basic geometry that students are used to studying. Algebra introduces the concept of the unknown variable and teaches students how the unknown can be derived from known parameters. Making use of both numbers and alphabets, algebra is a whole new ball game and needs complete attention right from the beginning .

How do keep up with the subject from the day you start learning it? To begin with, algebra is not as difficult as many students claim it is. Generally, students who require a little more time and explanation to understand algebraic concepts, seldom get it and as a result, come to the conclusion that it is hopelessly difficult. Needless to say, this leaves them with precious little motivation to continue learning the subject.

One of the first steps to learning algebra is to ensure that you understand each concept or equation when it is taught. Too many students tend to make light of their lack of clarity, telling themselves they'll take the time to clear it later on. Instead, take a proactive approach and ask a lot of questions in class. You don't have to worry about sounding dumb since it's a new subject and everyone is just as clueless as you are. Moreover when students ask questions, it helps teachers gauge the level of understanding among the students and proceed accordingly.

Memorize any equations or formulas immediately. You will be using them a lot so it's good to familiarize yourself with them and learn when and how to use these formulas. The best way to commit them to memory is to write them down. Practice sessions are very important so try to make time for them on a daily basis. Include different types of questions and don't just stick with the ones your teacher provides you with. As you get more comfortable with algebra and learn the formulas, vary the difficulty level to challenge yourself and learn new methods. It will also ensure that you are prepared for any question during the exams.

If you find that things are going above your head, then get help with algebra straightaway. Irrespective of whom you go to for help, they will find it easier to start from the beginning rather than having to finish the whole lot two weeks prior to the exam. For minor doubts you can approach your teacher or a classmate to point you in the right direction. For more extensive help, use a tutoring service which provides good quality, experienced tutors who will explain the theory, clear any doubts, and provide practice material. Algebra help online can be found on a number of websites which feature detailed tutorials as well as one on one tutoring, as per students' convenience.

Surface area of an ellipse cross section

Area of ellipse:
Like a circle or any other closed figure, an ellipse is also a closed figure. Therefore it is possible to find its area. Area of an ellipse can be found using the values of a and b. But what are these a and b? That can be understood using the general equation of an ellipse.

General equation of an ellipse:
[(x-h)/a]^2 + [(y-k)/b]^2, where a is the semi major axis and b is the semi minor axis. (h,k) is the centre of the ellipse.

Area of ellipse formula:
From the above general equation we know that the semi major axis is a and the semi minor axis is b. The formula for the area of an ellipse can now be written as follows:
Area = A =  pi * ab. The semi major axis can also be termed as the distance between the centre and the vertex, which is ‘a’ here. The semi minor axis can also be terms as the distance between the centre and the co-vertex, which is ‘b’ here.  See the picture below:

Implicit equation of an ellipse:

Another method for finding the area of an ellipse is using its implicit equation. The general form of the implicit equation of an ellipse is:

Px^2 + Qxy + Ry^2 = 1

To calculate area of an ellipse if the equation of the ellipse is given in this form, we use the following formula:

Area = A = 2 pi/√(4PR – Q^2)

Let us now try to understand how to calculate the area of an ellipse using an example.

Example 1: 
Find the area of an ellipse with centre at (2,3), length of major axis being 6 cm and length of minor axis being 4 cm.

Solution:
 In this problem, major axis = 6, therefore semi major axis = a = 6/2 = 3 cm
Minor axis  = 4, therefore semi minor axis = b = 4/2 = 2 cm
Using the formula for area of ellipse,
A =pi* ab =  pi * 3 * 2 = 6 pi cm^2

Statistics Mode

Mode definition (math):
In statistics, a branch of mathematics, mode of a set of observations (also called data) is the particular observation that occurs most number of times. In other words, the observation with the maximum frequency is the mode of the data.

Finding mode:
The following steps are to be followed when calculating the mode of a set of observations of ungrouped data.
Step 1: Arrange the data in ascending order
Step 2: Look for the entry that occurs the most number of times.
Step 3: The number found in the step 2 above is your mode.

Mode Math examples:
To properly understand what is a mode in math or statistics, let  us try the following examples:

Example 1: The height of 10 students in centimeters is as follows, find the mode of the data.
145, 142, 143, 146, 144, 146, 143, 141, 140, 143
Solution:
Step 1: Arrange the data in ascending order. So now the numbers would look like this:
140, 141, 142, 143, 143, 143, 144, 145, 146, 146
Step 2: Look for the entry that occurs the most number of times. In the above data, the number 143 occurs thrice, whereas all the other numbers occur either once or twice.
Step 3: The number found in step 2 is the mode. Therefore for this problem, mode = 143 since it occurs most number of times.

Example 2: The lengths of screws manufactured by a firm in inches are as follows:
1.55, 1.50, 1.52, 1.53, 1.50, 1.52, 1.51, 1.54, 1.56
Find the mode of the data.
Solution:
Step 1: Arranging the data in ascending order we have:
1.50, 1.50, 1.51, 1.52, 1.52, 1.53, 1.54, 1.55, 1.56
Step 2: From the above arrangement we see that two numbers 1.50 and 1.52, both occur twice.
Step 3: Therefore this data has two modes: 1.50 and 1.52. Such a data is called bimodal.

Example 3: 
The ages of students in college algebra class is as follows:
25, 23, 24, 22, 23, 25, 22, 21, 27, 28, 20, 29
Find the modes.
Solution:
Step 1: Arrange the data in ascending order
20, 21, 22, 22, 23, 23, 24, 25, 25, 27, 28, 29
Step 2: There are three numbers, each occurring twice. They are 22, 23 and 25.
Step 3: These three numbers are therefore the modes. Such a data is called trimodal.
If there are more than three modes in a data set, it is called multimodal data.

Identifying conic sections

See the picture above. Consider a double cone as shown in the picture. Also consider a plane. Let this plane intersect the double cone. The way in which the plane intersects the double cone, would give rise to conic sections. As we can see in the picture the axis of the cone is vertical.

1. Conic sections circles:  If the intersecting plane is perfectly horizontal, that means that if the intersecting plane is perpendicular to the vertical axis, then the conic section thus produced would be a circle.

2. Conic sections ellipse: If the intersecting plane is inclined to the vertical at an angle that is between α to 90 degrees, where α is the angle between the vertical axis and the slant length of the cone, then the conic section thus produced would be an ellipse.

3. Conic sections parabola: If the intersecting plane is inclined to the vertical axis at an angle that is between 0 to α (α is same as described above), then the conic section produced would be a parabola.

4. Conic sections hyperbola: If the intersecting plane is parallel to the vertical axis, then the conic section thus produced would be a hyperbola.

Identifying conic sections:
Conic sections can be identified by two methods. They are, graphical and algebraic.
Graphical method of identifying conic sections:
The graphs of various conic sections are as shown below:

1. Graph of a circle:

The graph of a circle would look like above, it would have a centre and the radius on all sides would be equal.

2.Graph of an ellipse:

The graph of an ellipse can look like any of the two above figures. One is a horizontal ellipse and other is a vertical ellipse. An ellipse would have a major and a minor axis, two vertices and two co-vertices.

3.Graph of a parabola:

A parabolic curve would look like above. It would essentially have a vertex, a focus, a directrix and an axis of symmetry.

A hyperbola can be horizontal or vertical. Algebraic method for identifying conic sections:
In a conic sections practice problem, the equation of the curve should be similar to any of the following general equations:

Equation of the curve	Type of conic section
(x-h)^2 + (y-k)^2 = r^2	Circle
((x-h)/a)^2 + ((y-k)/b)^2 = 1	Ellipse
y-k = 4a(x-h)^2	Parabola
((x-h)/a)^2 - ((y-k)/b)^2 = 1	Hyperbola