Prime Numbers

Prime Numbers:

Introduction to meaning of Prime Numbers:
We are often faced with the questions like-What is the meaning of Prime numbers??In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The first twenty-five prime numbers are: Prime Numbers:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.Condition: - If x is the prime number then the next factors of the number x is 1 and X. Let us now see the meaning of prime numbers.A prime number is one that has only two factors namely 1 and itself and a composite number has factors besides 1 and itself.



In this blog we can get a clear understanding of Prime Numbers.A natural number greater than 1 that has no divisor between 1 and itself is said to be prime, hence called a prime number or simply a prime. Every natural number greater than 1 has at least the two distinct divisors 1 and itself; a prime has no others.

The number 2 is a prime, there being no candidate divisors between 1 and itself; from it, all even numbers thereafter are non-prime, i.e. 50% of all subsequent numbers. The numbers 3, 5, and 7 are all prime, meaning that, of the first six such subsequent numbers, precisely half are prime, half non-prime. However, of any subsequent six consecutive numbers, at least one of the odd values must be divisible by 3; including the three even numbers this means that at least 66% must be non-prime.So the trend goes; as we look further afield, with an accumulating collection of primes to be divisors, the density of primes declines progressively. But, no matter how far up the numbers we travel, we never exhaust the primes, nor is there any known point above which all further primes are spaced by more than the minimal value of 2 .

Example Problems - Meaning of Prime Numbers:

The easiest way to understand the prime numbers is by solving problems related to Prime numbers.

Problem 1:

Find out the number 29 is prime number or not?

Solution:

Here the number 29 is divisible by one and itself only. It has no more factors other than this. So 29 is considered as a prime number.

Problem 2:

Find out the number 53 is prime number or not?

Solution:

The getting number 53 is not divisible by two. 53 has only two factors. Those factors are one and itself only. So we can say the given number is prime number.

Hope you like the above example of Prime Numbers.Please leave your comments, if you have any doubts.

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.