Coordinate Planes

Coordinate Plane:

In this Blog before we get into the details of coordinate planes,let us first understand the definition of a coordinate plane.The basic meaning of a coordinate plane is Coordinate plane is a plane formed by the intersection of a horizontal number line.

Definition of Coordinate Plane:

Coordinate plane is a plane formed by the intersection of a horizontal number line with a vertical number line. They intersect at their zero points. This point of intersection is called the origin and written as (0, 0).

On a coordinate plane, the horizontal number line is called the x-axis and the vertical number line is called the y-axis.

This is a coordinate plane. It has two axes and four quadrants. The two number lines form the axes. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The center of the coordinate plane is called the origin. It has the coordinates of (0,0). Locations of points on the plane can be plotted when one coordinate from each of the axes are used. This set of x and y values are called ordered pairs.




















With the help of another example we can get a clear understanding about the coordinate Planes.


In the coordinate plane there are four quadrants.

Quadrant 1.

In quadrant 1 both X axis and Y axis values are positive. [( +x, +y )]

Quadrant 2

In quadrant 2 X axis values are negative and Y axis values are positive. [( - x, +y )]

Quadrant 3

In quadrant 3 X axis values are positive and Y axis values are negative. [( + x, - y )]

Quadrant 4

In quadrant 4 both X axis and Y axis values are negative. [( - x, - y )]
Hope you like the above example of Coordinate Planes.Please leave your comments, if you have any doubts.

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.