Learning Perfect Number

Any number which is a positive integer  is called a perfect number if the sum of the factors(or divisors) of that number is equal to that number itself. Obviously, the factors excludes that number itself. In other words, we can say that a perfect number is an integer whose sum of factors(or divisors) is double the number.


Example of a perfect number

6

Factors of 6 are 1,2,3,6

Sum of the factors of 6(excluding 6) = 1+2+3 = 6 (the number itself)

Hence 6 is a perfect number.

Similarly, 28( factors of 28 excluding itself are 1,2,4,7 and 14, 1+2+4+7+14=28) is a perfect number.

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General form of a perfect number


The general form of a perfect number is given as 2p-1 (2p - 1), where p is a prime number. It is important to note that not all numbers of that form are perfect number but all perfect numbers will be of that form.

for p =2, 2p-1 (2p - 1) = 21 (22  - 1) =2*3 = 6, a perfect number.

Discrete Mathematics Relation

Introduction to discrete mathematics relation:

The discrete mathematics deals with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.

Writing in set notation, if a is some fixed value:

` |{f(x)|x=a}|=1`

However, when we consider the relation, we relax this constriction, and so a relation may map one value to more than one other value. Having problem with Systems of Equations Solver keep reading my upcoming posts, i will try to help you.


Properties of discrete mathematics relation:-


In the following properties of discrete mathematics relation:-

Reflexive
Symmetric
Transitive
Antisymmetric
Trichotomy
Reflexive

Relation of the equality, = is reflexive. Examine that for all numbers a = a.So "=" is reflexive.

Symmetric

Relation is symmetric the values a and b:  a R b implies b R a.These type of relation is symmetric.

Transitive

Relation is transitive of all values of a, b, c: a R b and b R c implies a R c.These type of relation is transitive.

Antisymmetric

A relation is antisymmetric for all values a and b: a R b and b R a implies that a=b.These type of relation is antisymmetric

Trichotomy

A relation satisfy the all values a and b it holds true that: xRy or yRx.The  two relation numbers a and b, it is true that whether a ≥ b or b ≥ a (both if a = b).These type of relation is trichotomy. Please express your views of this topic How to Find Volume of a Cone by commenting on blog.


Example problems for discrete mathematics relation:-


Problem 1:-

`Let A = {1, 2, 3, 4, 5} and R : A harrA :-= {(a, b) : a |b}. What barR and R^-1?`

Solution:-

`{(1, 1) , (1, 2) , (1, 3) , (1, 4) , (1, 5) , (2, 2) , (2, 4) , (3, 3) , (4, 4) , (5, 5)}`

`barR = {(2, 1) , (2, 3) , (2, 5) , (3, 1) , (3, 2) , (3, 4) , (3, 5) , (4, 1) , (4, 2) , (4, 3) , (4, 5) , (5, 1) , (5, 2) , (5, 3) , (5, 4)}`

`R^-1 = {(1, 1) , (2, 1) , (3, 1) , (4, 1) , (5, 1) , (2, 2) , (4, 2) , (3, 3) , (4, 4) , (5, 5)}`

Problem 2:-

For each of the following relations of  pair which satisfies the relation ,and another pair which doesn't say whether each relation is reflexive relation,symmetric relation,transitive relation or anti-symmetric relation form discrete mathematics
(a) The relation on {1,2,3,4,5} defined by {(a,b)| a-b is even}
(b) The relation on {1,2,3,4,5} defined by {(a,b)| a+b is even}
(c) The relation on P, the set of all people,defined by {(a,b) | a and b have a common ancestor}

Solution:-

(a) The pair (4,2) satisfies the relation,(2,1) doesn't includes other relation.
(b) (3,3) satifies the relation,(3,2) doesn't includes other relation.
(c) (Bart,Lisa) satisfies the relation,(Homer,Marge) doesn't includes other relation.

How to Use Histograms

In statistics graphical representation of data by using bar graphs , histograms , frequency curves are important tools.

Introduction to how to use histograms:

Histograms is the method used in the analysis of graphical data. This is most widely applied in summarizing variables distributional information. Intervals or bins are the division of response variables, in equal size. The  occurrence number are calculated for each bin. Histograms is the composition of:  I like to share this Making Histograms with you all through my article.

Vertical axis = frequencies or relative frequencies.

Horizontal axis = response variable – refers to each intervals mid point.

The histogram is a visual representation of data , it shows the location of the measurement of data and how they are spread out.

It shows the highest frequency of the data by having a look at the highest rectangle.

The lengths of bases are same indicates that the class interval is same.

The data available in a table is not easy to interpret while the data on a histogram is easy. Comparing the data from two histograms is easier an less time consuming.

Tables of large sets of data make it complex to use while histograms are graphs , it is easier to organize and summarize using them.


Histograms:


Histogram are used for counting graphs  where number of pixels are calculated for each level between black and white. Black and white are on left and right side correspondingly. Depending on the number of bright pixels the height of graph varies. Lighter and darker images move the graph on right and left accordingly.I have recently faced lot of problem while learning Partial Fraction Example, But thank to online resources of math which helped me to learn myself easily on net.

Types of histograms


Histograms are classified into four types as follows:

1)      Histogram - absolute counts.

2)      Relative histogram - convert counts to proportions.

3)      Cumulative histogram.

4)      Cumulative relative histogram.


Uses of Histograms:


Calculate the range by finding the lowest and highest values from the given data.

Identify the number of bars that are to be used in histogram. The number of bars being used should not be too high or too low in order to provide the pattern in an effective way.

Width of each bar is calculated by dividing the range by the number of bars. Then, start with the low values, to determine the group of values to be contained or represented by each bar.

Compilation table is created, then filling the boundaries for each groups takes place.

Based on the data point counting the compilation table is filled for each bar. Total number of data points in each bar are also calculated.

Draw both the horizontal and vertical axes, and also perform labelling them

Draw in the bars to correspond with the total from the frequency table

The pattern of variation is identified and classified.

Functions Solving Online

Introduction to Functions solving online:

The mathematical idea of a function expresses dependence between two quantities, one of the produced is   independent variable, argument of the function, or it’s "input" and another one of the produced is dependent variable, value of the function, or "output". A function expression is associates a unique output with all input element from a fixed set, such as the real numbers.

The function of expression is f(x) =ax^2 + b x + c


Explain about Functions Solving online using quadratic functions


Quadratic function in online is helpful to know more about the simplest way to solving online functions which under  "ax^2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. While factoring could not always be successful, in online the Quadratic Formula can always find the solution by using the .

The general form is,

ax^2 + bx + c = 0

Where, x represents the variable, and a, b, and c, constants, with a ? 0. (If a = 0, the equation becomes a linear equation.) .

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Examples of Functions solving online using Quadratic functions


Using Quadratic functions solving some example problems are given below,

Ex 1 :   solving qudratic functions f(x) = x^2+6x+9 = 0

Sol :  To factorize the quadratic function, split the middle term (6x) into two terms so that the product of their coefficients is equal to the constant term (9).

Like 6x = (3x) and (3x)

So, 3x + 3x = 6x and

3 * 3 (coefficients of 3x and 3x) = 9 (constant term)

So, now the function becomes

x^2 + 3x + 3x + 6 = 0

Here ‘x’ in first term and 3 in last two terms commonly, by taking both of them commonly out, we get

x(x+3) + 3(x+3)  = 0

Now (x+3) in common

(x+3) (x+3)  = 0

Now x+3 = 0 or x+3 = 0

Ex 2 :  solving qudratic function f(x) =x^2+6x+8 = 0

Sol :    To factorize the quadratic function, we have to split the middle term (6x) into two terms so that the product of their coefficients is equal to the constant term (8).

Like 5x = (2x) and (4x)

So, 2x + 4x = 6x and

2 * 4 (coefficients of 2x and 4x) = 8 (constant term)

So, now the function becomes

x^2 + 2x + 4x + 8 = 0

Here ‘x’ in first term and 4 in last two terms commonly, by taking both of them commonly out, we get

x(x+2) + 4(x+2) = 0

Now we have (x+2) in common,

(x+2) (x+4)  = 0

Now x+2 = 0 or x+4 = 0

Mathematics Program

Introduction to solve algebra homework answers:

Algebra is one of the main branches of mathematics that deals with calculating unknown variables from the help of known values. Homework problem with answers helps us to understand the concept of algebra. It is the study of rules of operation and relations, algebraic expressions, conditions and polynomials. An algebraic expression represents a scale where all the arithmetic operations are carried out on both the sides of the scale. Algebra homework problem contains problems with complex numbers, matrices, vector etc. The solved homework problems with answers are given below.

solve algebra homework answers : Examples


The following are the solved example problems for algebra homework.

Ex 1:

Solve the algebraic expression

6(c -3) + 5d - 2(c -d -2) + 1

Sol:

Given algebraic expression is

6(c -3) + 5d - 2(c -d -2) + 1

Multiplying the integer with above terms

= 6c - 12 + 5d -2c + 2d + 4 + 1

Grouping the above terms

= 6c + 7d – 7

Ex 2:

Solve the algebraic equation.

x 2 - 3x = 0

Sol:

Given equation is
x 2 - 3x = 0

Take X factor as common
x (x - 3) = 0

So the product x (x - 3) to be equal to zero, then we get

x = 0 or x - 3 = 0

Solve the above simple equations to obtain the solutions.
x = 0
or
x = 3

X= 0 or 3 is the solution.

Ex 3:

Solve the algebraic expression

4(a -1) + 2b - 5(a -b -4) + 5

Sol:

Given algebraic expression is

4(a -1) + 2b - 5(a -b -4) + 5

Multiplying the integer terms

= 4a - 4 + 2b -5a + 5b + 20 + 5

Grouping the above terms

= -a + 7b + 21

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Solve algebra homework answers : Practice problems


Find h (4) and g(4) and h(4) / g(4) and the functions g and h is given as

h (x) = 3x - 8 and g (x) = x 2 - 12

Solution:

Calculate h(4)

h(4) = 3(4) - 8 = 4

Calculate g (4)

g (4) = 4 2 - 12
= 16 -12 = 4

h (4) / g (4) =4/4 =1

Solve the algebraic expression

4(c -1) + 2d - 5(c -d -4) + 5

Solution:

Given algebraic expression is

4(c -1) + 2d - 5(c -d -4) + 5

Multiplying the integer terms

= 4c - 4 + 2d -5c + 5d + 20 + 5

Grouping the above terms

= -c + 7d + 21

Solve the equation     5(-2x - 2) - (-2x - 4) = -4(4x + 4) + 15

Sol:

Given the equation

5(-2x - 2) - (-2x - 4) = -4(4x + 4) + 15

Multiplying the integer with above terms.
-10x - 10 + 2x + 4 = -16x - 16 +15

Grouping the above terms.

-8x - 6 = -16x - 1

Add -8x - 6 to both sides, the above equation becomes

-8x = 7

X= -7/8

The practice problems are given below for homework.

1) Solve the algebraic equation     6(-8y - 3) - (-5y - 5) = -8(2y + 4) + 9

Answer: y = 8/27

2) Solve the algebraic equation.   x 2 - 10x = 0

Answer:   x= 0 or 10

Perfect Squares

Introduction:

Square of number is multiply the same number twice.A × A =A2, here square of  A  is written as A2 Here A is called the base and 2 is called the index or the power.Now observe the following examples:

0^2 = 0 ×0 = 0

1^2 = 1 × 1 = 1

2^2 = 2 ×2 = 4

These examples are square the same number.The square of 0,1,2,3,4 are 0,1,4,9,16 respectively. These square numbers are known as perfect squares. I like to share this Transformations Geometry with you all through my article.


Explain perfect square with examples:


Examples:

1). Is 625 a perfect square?

Yes, because 625 can be expressed as the product of two same numbers as 25 × 25.

2). Is 10 a perfect square?

No, 10 is not a perfect square since 10 cannot be written as the product of two same numbers.

3). Is 144 a perfect square?

Yes, because 144 can be expressed as the product of two same numbers as 12 × 12.

4). Is 70 a perfect square?

No, 70 is not a perfect square since 70 cannot be written as the product of two same numbers. Understanding Graphing Calculators is always challenging for me but thanks to all math help websites to help me out.


Perfect square Examples:


(1). Find the Perfect square of 20

Solution:

20 ^2 = 20 * 20

= 400

(2). Find the Perfect square of 111

Solution:

111 2 = 111 * 111

= 12321

(3). Find the Perfect square of 13

Solution:

13^ 2 = 13 * 13

= 169

(4). Find the Perfect square of 81

Solution:

81^ 2 = 81*81

= 6561

(5). Find the Perfect square of 100

Solution:

100 ^2 = 100 * 100

= 10000

Perfect square Exercises:

(1). Find the Perfect square of 32

(2). Find the Perfect square of 15

(3). Find the Perfect square of 09

(4). Find the Perfect square of 723

(5). Find the Perfect square of 40

Answers:

(1). 1024

(2). 225

(3). 81

(4). 522729

(5). 1600

Perfect Number Examples

Definition:

A perfect number is the positive integer in that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently of a perfect number is that a number is half the sum of all of its positive divisors (including itself), or σ(n) = 2n.


Even perfect numbers

In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2p − 1 with p a prime are prime

In the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number:

for p = 2: 21(22 − 1) = 6

for p = 3: 22(23 − 1) = 28

for p = 5: 24(25 − 1) = 496

for p = 7: 26(27 − 1) = 8128.

Noticing that 2p − 1 is a prime number in each instance, Euclid proved that the formula 2p−1(2p − 1) gives an even perfect number In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes. I have recently faced lot of problem while learning Definition of Rational Numbers, But thank to online resources of math which helped me to learn myself easily on net.


Odd perfect numbers

It is unknown whether there are any odd perfect numbers. Various of results have been obtained, but none of that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has been presented by a heuristic argument in which suggests that no odd perfect numbers exist.[4] Also, it can been conjectured of that there are no odd Ore's harmonic numbers, except for 1. If true in this would to imply that there are no odd perfect numbers.