Two Perfect Squares

Definition:

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number is that its square root is again an integer. Having problem with Solid Geometry keep reading my upcoming posts, i will try to help you.


Examples


For example 1,

(x + 2)(x + 2)

You get:

x^2 + 4x + 4

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Therefore, the quadratic expression x^2 + 4x + 4 is a perfect square since it factors into two identical binomials which are (x + 2) and (x + 2).

Notice that (x + 2) (x + 2) can be written (x + 2)2. So:

x^2 + 4x + 4 = (x + 2)2

For example 2,

(x + 3)(x + 3)

You get:

x^2 + 6x + 9

Therefore, the quadratic expression x^2 + 6x + 9 is a perfect square since it factors into two identical binomials which are (x + 3) and (x + 3).

Notice that (x + 3) (x + 3) can be written (x + 3)2. So:

x^2 + 6x + 9 = (x + 3)2

For example 3,

(x + 4)(x + 4)

You get:

x^2 + 8x + 16

Therefore, the quadratic expression x^2 + 8x + 16 is a perfect square since it factors into two identical binomials which are (x + 4) and (x + 4).

Notice that (x + 4) (x + 4) can be written (x + 4)2. So:

x^2 + 8x + 16 = (x + 4)2

For example 4,

(x + 5)(x + 5)

You get:

x^2 + 10x + 25

Therefore, the quadratic expression x^2 + 10x + 25 is a perfect square since it factors into two identical binomials which are (x + 5) and (x + 5).

Notice that (x + 5) (x + 5) can be written (x + 5)2. So:

x^2 + 10x + 25 = (x + 5)2

For example 5,

(x + 6)(x + 6)

You get:

x^2 + 12x + 30

Therefore, the quadratic expression x^2 + 12x + 30 is a perfect square since it factors into two identical binomials which are (x +6) and (x + 6).

Notice that (x + 6) (x + 6) can be written (x + 6)2. So:

x^2 + 12x + 30 = (x + 6)2

For example 6,

(x -5)(x -5)

You get:

x^2 - 10x + 25

Therefore, the quadratic expression x^2 - 10x + 25 is a perfect square since it factors into two identical binomials which are (x - 5) and (x - 5).

Notice that (x - 5) (x - 5) can be written (x - 5)2. So:

x^2 - 10x + 25 = (x - 5)2

Discrete Mathematics Notes

Introduction to discrete mathematics notes:

Discrete Mathematics deals with several selected topics in Mathematics that are essential to the study of many Computer Science areas. Since it is very difficult to cover all the topics, only two topics, namely “Mathematical Logic” and “Groups” have been introduced. These notes will be very much helpful to the students in certain practical applications related to Computer Science. In this article we shall discuss about discrete mathematics notes.

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Discrete mathematics notes:


Logical statement or Proposition:

A statement or a proposition is a sentence which is either true or false but not both.

A sentence which is both true and false simultaneously is not a statement, rather it is a paradox.

Example 1:

(a) Consider the following sentences:

(i) The earth is a planet.

(ii) Rose is a flower.

Truth value of a statement:

The truth of a statement is called its truth value. If a statement is true, we say that its truth value is TRUE or T and if it is false, we say that its truth value is FALSE or F.

Simple statements:

A statement is said to be simple if it cannot be broken into two or more statements. All the statements in (a) and (b) of Example 1 are simple statements.

Compound statements:

If a statement is the combination of two or more simple statements, then it is said to be a compound statement.

Conjunction:

If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement ‘p and q’ is called the conjunction of p and q and is written in the symbolic form as ‘p ? q’.

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Discrete mathematics notes problems:

Example 1:

(i) Show that ((~ p) ? (~ q)) ? p is a tautology.

Solution:

(i) Truth table for ((~ p) ? (~ q)) ? p

p          q         ~ p      ~ q      (~ p) ? (~ q)     ((~ p) ? (~ q))? p

T          T          F            F                      F                      T

T          F          F           T                      T                      T

F          T          T          F                      T                      T

F          F          T          T                      T                      T

The last column contains only T. Therefore ((~ p) ? (~ q)) ? p is a tautology.

Example 2:

Show that ((~ q) ? p) ? q is a contradiction.

Truth table for ((~ q) ? p) ? q

p          q          ~ q      (~ q) ? p          ((~ q) ? p) ? q

T          T          F          F                                  F

T          F          T          T                                  F

F          T          F          F                                  F

F          F          T          F                                  F

The last column contains only F. ? ((~ q) ? p) ? q is a contradiction.

Discrete Mathematics Sample

Introduction discrete mathematics sample

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Having problem with Graph Ordered Pairs keep reading my upcoming posts, i will try to help you.

Source Wikipedia


Discrete mathematics sample explanations:


Here will study about the discrete mathematics problems.

The discrete mathematics contains the set of topics. These topics are cover in Tautologies and Logical Equivalence

Sentential Functions and Sets are in logic and sets. Relation and functions,   Equivalence Relations Equivalence Classes. Natural numbers, division and factorization.  Division  ,Factorization ,

Greatest Common Divisor. these are the some of the discrete mathematics

Here we will see some of the samples in the discrete mathematics

Example for sentence

1.The sentence \1 + 2 = 3 and 2 + 2 = 4" is true.

2.The sentence \3 + 3 = 6 and _ is rational" is false.

Example for relations and functions

Definition. Let A and B be sets. By a relative R on A and B, we mean a subset of the Cartesian

Product a x b.

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Discrete mathematics sample problems:


Here we will learn about the discrete mathematics sample problems

The natural numbers

Example1:

The set of natural numbers is usually given by

N = {1, 2, 3 …}

Division

Examples

Division we divide 24 by 4

Solution:

24/4 =6

We divide 24 by 4 we get answer is 6

Factorization

(x² +9 ) to factorize the given problem

Solution:

The general form of the given equation is

(x² +a²) =(x+a) (x+a)

So. We factorize given problem

(x² +9 ) =(x+3) (x+3)

Greatest Common Divisor

Example:

12,4,36

We find the greatest common divisor of the given problems?

Solution:

We form the given problem

We divide by 4 all the numbers

12=2x2x 3

4=2x2 x 1

36=2x2x 9

We get the greatest common divisor of the given problems

Final answer is 2

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