Degree of Vertex

The degree of a vertex v in this graph G, written deg (v), is equal to the number of edges in G which contain v, that is, which are incident on v. Since each edge is counted twice in counting the degrees of the vertices of G, The degree of vertex refers the the number of edge incident in a graph. The number of graph edges which touch in a graph is the degree of vertex.In a graph the minimum degree of vertex is denoted by G and denoted by σG and maximum vertex is denoted by ΔG
Theorem of Degree of Vertex

The Degree sum in the vetices of  graph G is equal to twice the number of edges in G.

degree of vertex

 In a graph the vertex of graph is odd  node if its vertex degree is odd.Here this graph has 10 vertices hence its degree is even

The number associated with each vertex is its degree;it is  defined to be the number of edges that enter or exit from it—thus, a loop contributes 10 to the degree of its vertex.The vertices of the simple graph shown in the diagram all have a degree of 10.In the above diagram here the maximum degree is 3 and minimum degree is 7
Degree Formulae

In a graph G={V, E} , the degree sum formulae is given by:

Σ Deg (v) =2 (E) , here E is the number of edge of graph.In a graph the vertices are in a ordered list that is defined to be the degree sequence.This formulae implies number of vertices which are even and odd degree.
Conclusion of Degree of Vertex

Degree of vertex is very important part of a graph  family because it represents them. Degree of vertex is also defined to be its Valency. and also known to be isolated vertex.

Sequence Comparison

In this article, we will discuss about the sequence comparison. It has two types of sequences. One is arithmetic sequence and next one is geometric sequence. Arithmetic sequence means that, the sequence of numbers such that the difference between two consecutive members of the sequence is a constant. Geometric sequence means that, the sequence of numbers such that the ratio between two consecutive members of the sequence is a constant. The sequence comparison formulas and example problems are given below.

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Formulas for Sequence Comparison:

Compare the formulas for arithmetic and geometric sequences.

Formula for arithmetic sequence:

nth term of the sequence : a_n = a_1 + (n - 1)d

Series of the sequence: S_n = (n(a_1 + a_n))/2

Formula for geometric sequence:

nth term of the sequence: a_n = a_1 * r^(n-1)

Series of the sequence: S_n = (a_1(1-r^n))/(1 - r)

Example Problems for Sequence Comparison:

We will compare the arithmetic and geometric sequence problems.

Example problem 1:

Find the 24th term of the given series 14, 16, 18, 20,....

Solution:

First term of the series, a1 = 14

Difference of two consecutive terms, d = 16 - 14 = 2

n = 24

The formula to find the nth term of an arithmetic series, a_n = a_1 + (n-1)d

So, the 24th term of the series 14, 16, 18, 20,.... = 14 + (24 - 1) 2

= 14 + 23 * 2

= 14 + 46

After simplify this, we get

= 60

So, the 24th term of the sequence 14, 16, 18, 20,... is 60

My forthcoming post is on prime numbers to 100 chart and Area of a Trapezoid will give you more understanding about Algebra.

Example problem 2:

Find out the 15th term of a geometric sequence if a1 = 15 and the common ratio (C.R) r = 2

Solution:

Use the formula a_n = a_1 * r^(n-1) that gives the nth term to find a_15 as follows

a_15 = a_1 * r^(15-1)

= 15 * (2)14

= 15 * 16384

After simplify this, we get

= 245760

The 15th term of a geometric sequence is 245760.

The above examples are helpful to study of sequence comparison.

Vertices Explanation

The basic unit as per which the graphs are made is the node. Techniclly speaking, vertices are indivisible and featureless , however, they could contain extra structure based on the purpose; an example for this is, a semantic mesh is that graph where the vertices depict categories of objects.Here you can see vertices explanation.
Brief Explanation of Vertices Explanation

Two vertices that make the edge would be the endpoints, the edge would be the event to those vertices. Vertex w is adjacent to vertex v in case the graph has an edge. A free set is the one where no two of these are next. However, a a vertex cover can be termed that set of vertices which the graph has endpoint of each edge.
Example of Vertices Explanation

Vertices are similar but not exact as vertices of polyhedra. Here the vertices are that of  the polyhedron, however, polyhedron vertices have an added structure that is not there in theory. In a going to graph, the further star of a node u are outgoing edges. Graph G vertices V plus the set of edges E, the forward star of u can be representaed as:

formula of vertices
Conclusion of Vertices Explanation

A graph can be identified as vertex-transitive if the symmetries it has map any vertex to another. It is vital to distinguish between labelled vertices and the ones that are not.  A former links to additional information which enables the same to be distinguished; one can call two graphs isomorphic only if the connection between their vertices pairs up vertices with equal labels. The latter is single and could be substituted for any other vertex based only on its adjacencies inside the graph and not on extra information.

Probability vs Odds

Probability:

Probability is defined as the possibility of occurring the events in random. Probability is given by the ratio favorable outcome to the possible outcome.

Probability = `n/N`

Where

n = Number Of favorable outcome

N = Number of possible outcome

Odds

Odds is a part of probability, which is defined as the ratio of probability of events favorable outcome to the probability of unfavorable outcome.

Odds = `P/(1-P)`

Where

P= Probability

Odds is also given by

Number Of favorable outcome
Odds = --------------------------------------
Number of unfavorable outcome

Let us see the problems for probability and odds calculate in detailed clear explanation.
Example Problems for Probability Vs Odds

Problem for probability vs odds:

When dice are thrown random calculate the probability and odds value if the event occurred is 1, 4.

Solution:

Total number of events in a dice, N = 6

Number of events occurring 1, 4 in a dice, n = 2

Therefore number of favorable outcome = 2

Number of unfavorable outcome = 6-2 = 4

Probability = `n/N`

= `2/6`

Probability = `1/3`

Number Of favorable outcome
Odds= ----------------------------------------
Number of unfavorable outcome

=` 2/4`

Odds=`1/2`

Problem for probability vs odds:

When dice are thrown random, calculate the probability and odds value if the event occurred is not prime number.

Solution:

Total number of events in a dice, N = 6

Number of events occurring even numbers in a dice is not prime number = 2

Therefore number of favorable outcome = 2

Number of unfavorable outcome = 6-2 = 4

Probability = `n/N`

=`2/6`

Probability = `1/3`

Number Of favorable outcome
Odds=  ----------------------------------------
Number of unfavorable outcome

= `2/4`

Odds = `1/2`

Problem for probability vs odds:

Calculate the odds if the probability of event occurring is 1/15

Solution:

Given, probability, P = `1/15`

The relation between even and odds is given in the formula,

Odds = `P/(1-P)`

= `(1/15)/(1-(1/15))`

= `( 1/15)/(14/15)`

= `(1/5)*(15/14)`

Odds = `1/14`

PPM Calculations

is an abbreviation for “parts per million” calculations, which plays a pivotal role in exchange and recording of information relating to impurities in a substance. Impurities at a grosser level can be easily dealt with using percentages, which is a well known concept and is nothing but an acronym for “per hundred”. It may be noted that PPM calculations are usually with reference to mass unless expressed otherwise, while percentage expressions could be with reference to mass, volume or count that needs to be explicitly stated.
Utility of Ppm Calculations:

In many scientific experiments a very high degree of purity is required for reactants or substances used. Expressing purities in terms of percentage is no doubt possible in such cases but such expressions become lengthy decimals. This difficulty must have led scientists and mathematicians to devise calculations in terms of PPM, i.e. parts per million. In case of still lower levels of impurities concepts of PPB (parts per billion) or PPT (parts per trillion) are used.

It needs no explanation that it is much easier to say and understand 1 PPM than saying and understanding 0.0001%. PPM calculations are particularly useful in case of environmental pollution since a large number of toxic pollutants become life threatening at barely detectible levels.
Ppm Calculation Formula:

The formula for parts per million is analogous to percent by mass, but is more convenient when there is only a small amount of other substance present in a mixture or solution. PPM is defined as the mass of the component in mixture or solution divided by the total mass of the solution multiplied by 106 (one million) as shown below.

Mass of impurity

Parts per million = ----------------------------------------   X 106

Mass of mixture or solution

PPM value can be easily converted to percent mass by multiplying it by 10,000. Similarly a percent mass value can be converted to parts per million through a division by 10,000.

Pi (π) Fraction Approximation

In this article we shall discuss about pi fraction approximation. A number Π (sometimes note down as pies) is a numerical constant whose value is the ratio of every circle's circumference to its diameter in Euclidean space; this is the like worth as the ratio of a circle's area to the four-sided figure of its radius. It is just about equivalent to 3.141593 in the common decimal notation.


Understanding Reduced Fraction is always challenging for me but thanks to all math help websites to help me out.

I Need Help with Pi (π) is an Irrational Number:

The need help with pi (π) is an Irrational number. So, Irrational number is given below that,

Designed for a lot of centuries previous to the real evidence, mathematicians have consideration that pi was an irrational number. The initial effort at confirmation was through Johann Heinrich Lambert in 1761. From side to side a complex technique he recognized that if x is rational, `tan(x)` have to be irrational. It goes after that if `tan(x)` is rational, x have to be irrational. Because `tan (pi/2) = 1` , `pi/2` should be irrational; thus, pi data should be irrational.

A lot of people saying Lambert's evidence as too simplified an answer for such a complex and long-lived problem. In 1794, though, A. M. Legendre establishes one more evidence which reverse Lambert up. This original evidence as well go as far as to establish that π2 (pi2) were also irrational.
I Need Help with Notation for Pi (π) Fraction Approximation:

The need help with notation for pi (π) fraction approximation is given below that,

My forthcoming post is on Simplifying Expressions Calculator and Finding Unit Rate will give you more understanding about Algebra.

We cannot note down an easy division that equals Pi. The pi fraction approximation denoted as `22 / 7.`

At the present, infinite series value of pi (π) fraction approximation is given below that π (Pi) = `22 / 7 ` = 3.14159265358979323.

The accepted rough calculation for numbers like pi (π) data = 3.141592653589793238462643383 is closed but not precise.

Example for numbers like pi (π) data irrational number π (Pi) = 3.14159265358979323846264…,

The pi (π) fraction approximation value is `22/7.`

General Vector Spaces

In general vector spaces, the idea of associating a column matrix with a vector leads to the concept of vector spaces. Let E^n denote the set of all n xx 1 column matrices, or vectors with n elements. The nonempty set E^n is a vector space because it satisfies the following axioms for vectors u, v, w in E^n and scalars a, b.

Basic Rule for General Vector Spaces:

Closure

u + v in E^n, au in E^n

Addition

Associative law

(u + v) + w = u + (v + w)

Zero vector

u + 0 = u

Existence of negatives

u + (-u) = 0

Commutative law

u + v = v + u

Multiplication

Distributive law

a(u + v) = au + av

(a + b)u = au + bu

Associative law

(ab)u = a(bu)

Unity law

1u = u
More about Vector Space for General Vector Spaces:

In general vector spaces, a set of vectors {u^(1), u^(2), . . . , u^(n)} is linearly dependent if and only if the sum

sum_(i=1)^n c_iu(i) = 0

is satisfied for a set of scalars {c_1, . . . ,c_m} that are not all zero. The set of vectors {u^(1), u^(2), . . . ,u^(n)} is linearly independent if above equation is satisfied only when c_i = 0 for all values of i.

A basis for the set E^n is a set of n linearly independent vectors that belong to E^n. If we write the basis as {u^(1), u(2), . . . , u(n)}, then any vector u in E^n can be written as a linear combination, or superposition, of basis vectors such that

u = sum_(i=1)^nc_iu^((i))

where {c_i} is a unique set of numbers for general vector spaces.

The inner product of two n-dimensional vectors u, v is defined as

(u, v) = sum_(i=1)^n u_i * v_i = (u*)^T = u xx v

where * denotes complex conjugation. Two vectors u^(i), u^(j) are orthogonal if their inner product is zero; that is, (u^(i), u^(j))=0. Vectors u^(i), u^(j) are orthonormal if their inner product satisfies

(u^(i), u^(J)) = delta_(ij)

Where the Kronecker delta is defined by

if i!= j then delta_ij = 0.

if i= j then delta_ij = 1.

The Euclidean length of a vector u with elements {u_i: i = 1, . . . , n} is the inner product of vector u with itself; thus

||u|| = (u, u)^(1/2) = [sum_(i=1)^(n) (| u_i |)^(2)]^(1/2)

The Euclidean lengths of two vectors u, v satisfy the Schwartz inequality

|u xxv| <= ||u|| ||v|| for general vector spaces.

Range of a Function Calculator

The range of a real function of a real variable is the set of all real values taken by funtion f(x) at the points in its domain. In order to find the range of a real function f(x) using calculator we will use the following algorithm.

Put y=f(x)
Solve the equation y =f(x) for x in terms of y. Let x =phi(y)
Find the values of y for such that the values of x, obtained from x= phi(y) , are real and are in the domain of function f
The set of all such values of y obtained in the above step  is the range of function.


We use the above steps to find the range of a function using calculator.


I like to share this Fractions Calculator with you all through my article.

Examples on Range of Function Calculator:

Ex -1 Find the range of the function f(x) given by 1/(sqrt(x-5))

Solution We have f(x) = 1/(sqrt(x-5))

Let y = f(x)

y=1/(sqrt(x-5))

rArr y^2=1/(x-5)

rArr x-5 = 1/y^2

rArr x = 1/y^2 +5

It is clear that x assumes all real values except y=0

The range of the function is (0,oo )

Ex -2 Find the range of the function f(x) = 3/(2-x^2)

Solution: Given f(x) = 3/(2-x^2)

Let y = f(x) = 3/(2-x^2)

rArr y = 3/(2-x^2)

rArr 2-x^2 = 3/y

rArr 2-3/y = x^2

rArr (2y-3)/y = x^2

Taking square root on both sides

rArr x = sqrt((2y-3)/y)

x will take all values except -sqrt(2) and sqrt(2) , if (2y-3)/y>=0

rArr y in (-oo,0)U[3/2,oo)

This is the range of the function.


I am planning to write more post on What's a Polygon and help with math online free. Keep checking my blog.

Practice Problems on Range of Function Calculator:

Pro 1 Find the range of the function f(x) = (x^2-9)/(x-3)

Pro 2 Find the range of the function f(x) = (4-x)/(x-4)

Ans

1) Range of the function is R-{6}

2) Range of the function is {-1}

Definition of Median Age

In this article shall we discuss about definition of median age. The middle number is also indicating as a median. Now, we have to place the numbers in ascending order. And then locate the middle number. If the set age data contain two middle numbers’ we have to  find the average of both the middle numbers. This also denoted as a median (middle number).

Example Problems Based on Definition of Median Age:

The Example problems based on definition of median age is given below that,

Example 1:

How do you determine median of a following age’s set is 13, 9, 11, 10, 15 and 8?

Solution:

Step 1:

The given data set is 13, 9, 11, 10, 15 and 8.

Step 2:

Now, we have to find the median.

Step 3:

So, arrange ascending order of given data set.

Therefore, the ascending order is 8, 9, 10, 11, 13 and 15.

Step 4:

Here, 10 and 11 as a middle number.

So, now we have to find out average of two numbers.

Step 5:

Therefore, the average of two numbers is `(10 + 11)/2 ` = `21/2` = 10.5

The median value is 10.5

Example 2:

How do you determine median of a following age’s set is 7, 9, 10, 12, 11, 14 and 8?

Solution:

Step 1:

The given data set is 7, 9, 10, 12, 11, 14 and 8.

Step 2:

Now, we have to find the median.

Step 3:

So, arrange ascending order of given data set.

Therefore, the ascending order is 7, 8, 9, 10, 11, 12 and 14.

Step 4:

Here, 10 is a middle number.

Therefore, the median value is 10.


Practice Problems Based on Definition of Median Age:

The practice problems based on definition of median age is given below that,

Problem 1:

How do you determine median of a following age’s set is 8, 10, 11, 7, 12 and 13?

Answer:   The median is 10.5

Problem 2:

How do you determine median of a following age’s set is 14, 21, 20, 27, 26 and 29?

Answer:   The median is 23.5

Problem 3:

How do you determine median of a following age’s set is 17, 8, 16, 14, 13, 15 and 11?

Answer:  The median is 14.

Fraction Converter Chart

A certain part of the whole is called as fractions. The fractions can be denoted as a/b, Where a, b are integers.By using fraction converter we can convert the fraction as decimals, equal fraction, mixed fraction and percentage. In this article we are going to see fraction to decimal chart and some problems on fraction converter chart.
Fraction Converter Chart :

The following figures shows the fraction converter chart .

The fraction to decimal chart

Fig(i) The fraction converter  chart

The fraction to decimal chart

Fig(ii) The fraction converter  chart

The fraction to decimal chart

Fig(iii) The fraction converter  chart

Let us see some problems on fraction converter chart.

I like to share this Add Fraction with you all through my article.

Problems on Fraction Converter Chart :

Problem 1:

What is `12 / 3` as a mixed fraction ?

Solution:

Given, Fraction 9/3

We need to find the mixed fractions equals to `12/3`

To find mixed fraction equals to `12/3` , divided 9 by 3.

_____

3 )   12   (  4           ------------- > Quotient

12

0                   -------------  > Remainder

The mixed fraction equals to `12/3`   = Quotient (Remainder / Divisor) = 4 `0 /3`

Verification:

We can verify the answer as shown in below,

Quotient `(Remainder) / (Divisor)`   = `(( quotient * Divisor ) + Remainder ) /(Divisor)`

`4 0 / 3` = ( ( 4 *3) + 0 ) / 3

= `12 / 3`

Problem 2:

Convert the following fraction into decimal  7634 / 2600

Solution:

Given, 7634/ 2600

We need to find the decimal expansion

_____

2600) 7634   ( 2

5200

2434

Here 2434 can't be divide by 2600.

So multiply 2434 by 10 and add one decimal place with the quotient,

_____

2600) 7634   ( 2.9

5200

24340

23400

940

My forthcoming post is on Proportion Formula and Geometric Mean Example will give you more understanding about Algebra.

Here 940 can't be divided by 2600.So multiply it by 10.We already add one decimal place. So here no need to add anything with the quotient.

_____

2600) 7634   ( 2.93

5200

24340

23400

9400

7800

1600

The approximate answer for the given problem is 2.93

Answer: 7634 / 2600 = 2.93

Problem 3:

Find the 3 equivalent fractions of `2/15` .

Solution:

Given, The fraction `2/15` .

We need to find the equivalent fraction to `2/15`

Multiply `2/15` by 2 on both numerator and denominator,

`2/15` = `( 2 * 2 ) / (15 * 2)`

= `4/ 30`

Equivalent fraction to `2/15 ` = `4 / 30` .

Multiply  2/15 by 5 on both numerator and denominator,

`2/15 ` = `( 2*5) / ( 15*5 )`

= ` 10/ 75`

Equivalent fraction to `2/15` = `4 / 30` = `10/75`

Multiply ` 2/15` by 3 on both numerator and denominator,

`2/15` = `( 2* 3 ) / (15 * 3)`

= `6/ 45`

Answer: 3 equivalent fraction of `2/ 15` is `4/ 30` , `10/ 75` , `6 /45`

Is this topic algebra questions and answers hard for you? Watch out for my coming posts.

Practice Problems on Fraction Converter Chart:

Problems :

1.Convert the following fraction into decimal   27 / 30

2.Convert the following fraction into decimal  156/243

3.Convert the following fraction into decimal  45 / 78

4.Convert the following fraction into decimal  28 / 93

Answer key:

1. 0.9

2. 0.6419

3. 0.5769

4. 0.301

Molecule Element Compound

Matter is chemically classified as Elements, compounds and mixtures.

Dalton had proposed that atoms of the same or different elements combine with one another to form compound atoms. These compound atoms later came to be known as molecules.  Avogadro made a clear distinction between atoms and molecules. According to him, atom is the smallest particle of an element which can take part in a chemical reaction. It may or may not be capable of independent existence.

Molecule Element Compound:molecule

Molecule is the smallest particle of matter(element or compound) which can exist independently. It may be made up of the atoms of the same or different elements For example, N2 contains only Nitrogen and it is one molecule of Nitrogen. CO2 is one molecule of carbon dioxide. It contains one atom of carbon and two atoms of oxygen.

Atoms combine to give molecules, and molecules combine to give elements. Elements combine to give compounds.
Molecule Element Compound : Elements

Elements are substances which cannot be reduced to simpler materials either by physical or chemical changes. Examples are iron, copper, gold, silver etc. At present 105 elements are known. About 90 of the elements can be extracted from natural sources but the rest can be prepared artificially by means of nuclear processes.

Molecule Element Compound : Compounds

Compounds are substances which have been formed by the combination of two or more different elements in a definite ratio.Millions of compounds are known because of the different combinations of the 105 elements.  A few examples are water, ammonia, methane, and benzene. Out of these, water and ammonia are examples of inorganic compounds, and methane and benzene are examples of organic compounds.

Elements and compounds are known as pure substances because their composition and properties are uniform throughout in any given sample. However, most of the naturally occurring substances are neither pure elements nor pure compounds but generally mixtures of both.

Default Probability

Probability is a method of expressing knowledge or attitude that an occurrence will happen. In mathematics the idea has been given a right sense in probability theory, that is used broadly in such areas of learn as mathematics, finance, statistics etc. Now we will learn the default examples of probability.

Examples- Laws of Probability

Example 1:

There are 50 things are available in a shop. In those things, 15 are the dolls, 20 are the balloons and 15 are the boxes. What is the probability for the following outcomes?

i) Select the dolls

ii) Select the balloons

iii) Select the boxes.

Solution:

Total number of things n(S) =50

Number of dolls n (A) =15

Number of balloons n (B) =20

Number of boxes n(C) =15

i) Assume P(A) is the probability for select the dolls.

P(A)=`(n(A))/(n(S))`

=`(15)/(50)`

=`(3)/(10)` .

ii) Assume P(B) is the probability for select the balloons.

P(B) =`(n(B))/(n(S))`

= `(20)/(50)`

=`(2)/(5)` .

iii) Assume P(C) is the probability for select the boxes.

P(C)=`(n(C))/(n(S))`

=`(15)/(50)`

=`(3)/(10)` .
Example 2:

The shop has 52 cars. In those cars, there are 16 are the Innova, 20 are the Scorpio and remaining is the Swift.

i) Select the Innova.

ii) Select the Scorpio.

Solution

First arrange the given information.

Number of cars n(S)=52

Number of Innova cars n(A)=16

Number of Scorpio cars n(B )=20

We don’t know the Swift car quantity. Now find that one with the help of the given information.

Number of swift cars n(C)=n(S)-(n(A)+n(B))

=52-(16+20)

n(C)=16.


I am planning to write more post on Decimal to Fraction Converter and how to solve word problems in algebra 2. Keep checking my blog.

i)Assume P(A) is the probability for select the Innova cars

P(A)=`(n(A))/(n(S))`

= `(16)/(52)`

=`(4)/(13)` .

ii)Assume P(B) is the probability for select the Scorpio cars .

P(B)=`(n(B))/(n(S))`

=`(20)/(52)`

=`(5)/(13)` .
Practice Problems for Default Probability

1) Alex has the 25 rings and Mercy has 20 rings. What is the probability for choose the Mercy’s rings?

Answer:

Probability=`(20)/(45)`

=`(4)/(9)` .

2) What is the probability for getting the head if a coin is flipped?

Answer:

Probability=`(1)/(2)` .

These default problems are used to learn the probability.