Introduction to Algebraic Fractions

Let us study about Algebraic Fractions,
Introduction to Algebraic Fractions :

Algebraic fractions are fractions using a variable in the numerator or denominator, such as 3/ x. Because division by 0 is impossible, variables in the denominator have certain restrictions. The denominator can never equal 0. Therefore, in the fractions
Be aware of these types of restrictions.
I hope the above explanation was useful.

The Binomial

The Binomial

A discrete variable that can result in only one of two outcomes is called binomial. For example, a coin flip is a binomial variable; but drawing a card from a standard deck of 52 is not. Whether a drug is either successful or unsuccessful in producing results is a binomial variable, as is whether a machine produces perfect or imperfect widgets.

Binomial experiments

Binomial experiments require the following elements:

* The experiment consists of a number of identical events ( n).
* Each event has only one of two mutually exclusive outcomes. (These outcomes are called successes and failures.)
* The probability of a success outcome is equal to some percentage, which is identified as a proportion, π.
* This proportion, π, remains constant throughout all events and is defined as the ratio of number of successes to number of trials.
* The events are independent.
* Given all of the above, the binomial formula can be applied ( x = number of favorable outcomes; n = number of events):
I hope the above explanation was useful.

Gaussian Elimination

Gaussian Elimination :

The purpose of this article is to describe how the solutions to a linear system are actually found. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Once this final variable is determined, its value is substituted back into the other equations in order to evaluate the remaining unknowns. This method, characterized by step-by-step elimination of the variables, is called Gaussian elimination.

Example 1: Solve this system:


Multiplying the first equation by −3 and adding the result to the second equation eliminates the variable x:


This final equation, −5 y = −5, immediately implies y = 1. Back-substitution of y = 1 into the original first equation, x + y = 3, yields x = 2. (Back-substitution of y = 1 into the original second equation, 3 x − 2 y = 4, would also yeild x = 2.) The solution of this system is therefore ( x, y) = (2, 1), as noted in Example 1.

Gaussian elimination is usually carried out using matrices. This method reduces the effort in finding the solutions by eliminating the need to explicitly write the variables at each step. The previous example will be redone using matrices.
Hope the above explanation was useful, now let me explain about matrices.

Regular Polygons

Let us learn about Regular Polygons,
There is really no limit to the number of sides a polygon may have. The only practical limit is that unless you draw them on a very large sheet of paper, after about 20 sides or so, the polygon begins to look very much like a circle.

Parts of a regular polygon
In a regular polygon, there is one point in its interior that is equidistant from its vertices. This point is called the center of the regular polygon. In Figure 1, O is the center of the regular polygon.

Figure 1 Center, radius, and apothem of a regular polygon.

The radius of a regular polygon is a segment that goes from the center to any vertex of the regular polygon.

The apothem of a regular polygon is any segment that goes from the center and is perpendicular to one of the polygon's sides. In Figure 1 , OC is a radius and OX is an apothem.

Finding the Perimeter

Because a regular polygon is equilateral, to find its perimeter you need to know only the length of one of its sides and multiply that by the number of sides. Using n-gon to represent a polygon with n sides, and s as the length of each side, produces the following formula.

Finding the Area

If p represents the perimeter of the regular polygon and a represents the length of its apothem, the following formula can eventually be shown to represent its area.

Example 1: Find the perimeter and area of the regular pentagon in figure 2 , with apothem approximately 5.5 in.

Figure 2

Finding the perimeter and area of a regular pentagon.

Hope the above explanation helped you.

Ratio and Proportion

Let us learn about ratio and proportion,
Ratio is a concept that you have probably encountered in other math classes. It is a comparison of sizes.

Ratio :

The ratio of two numbers a and b is the fraction , usually expressed in reduced form. An alternative form involves a colon. The colon form is most frequently used when comparing three or more numbers to each other. See Table 1 .

TABLE 1 Ratio Formats Ratio Written Form 3 to 4 3/4 or 3 : 4 a to b, b ≠ 0 a/b or a : b 1 to 3 to 5 1 : 3 : 5

Example 1: A classroom has 25 boys and 15 girls. What is the ratio of boys to girls?




The ratio of boys to girls is 5 to 3, or 5/3, or 5 : 3.
Example 2: The ratio of two supplementary angles is 2 to 3. Find the measure of each angle.

• The angles have measures of 72° and 108°.
  
• 2 x to 3 x reduces to 2 to 3.
  
• 2 x + 3 x = 180° (The sum of supplementary angles is 180°.)
  
• 5 x = 180°
  
• x = 36°
  
• Then, 2 x = 2(36°) and 3 x = 3(36°).
  
• So, 2 x = 72° and 3 x = 108°

The angles have measures of 72° and 108°.

Hope the above explanation helped.

Classifying Triangles by Sides or Angles

Let us study classifying triangles by sides or angles,
Triangles can be classified either according to their sides or according to their angles. All of each may be of different or the same sizes; any two sides or angles may be of the same size; there may be one distinctive angle.


The types of triangles classified by their sides are the following:



1. Equilateral triangle: A triangle with all three sides equal in measure. the slash marks indicate equal measure.

2. Isosceles triangle: A triangle in which at least two sides have equal measure

3. Scalene triangle: A triangle with all three sides of different measures


The types of triangles classified by their angles include the following:





1. Right triangle: A triangle that has a right angle in its interior

2. Obtuse triangle: A triangle having an obtuse angle (greater than 90° but less than 180°) in its interior.

3. Acute triangle: A triangle having all acute angles (less than 90°) in its interior

4. Equiangular triangle: A triangle having all angles of equal measure.

Hope the above explanation helped you, now let me explain polygons.