List of Irrational Numbers

n arithmetic, an irrational number is some real number which cannot be expressed as a fraction p/q; anywhere p and q are integers, through q non-zero and is then not a rational number. Easily, this way that an irrational number cannot be represent as an easy fraction.Thus the introduction to list of irrational numbers.

Example proofs in list of irrational numbers

Let us see about list of irrational numbers,  

Square roots in irrational numbers
The square root of 2 was the initial digit to be proving irrational and to item haves a number of proofs. The golden ratio is the next mainly well-known quadratic irrational and present is an easy proof of its irrationality in its object. The square root of every non-square natural digit is irrational and a proof might be established in quadratic irrationals.


General roots in irrational numbers
This asserts that each integer has a single factorization into prime. With it we can list of  illustrate that if a rational number is not an integer next no integral power of it can be an integer, as in lowest words there must be a main in the denominator which do not divide into the numerator anything power every is raise to. Then if a digit is not an exact k^th power of a different list of integer then its k^th root is irrational.

Logarithms in irrational numbers
Possibly the list of numbers mainly simply prove to be irrational are definite logarithms. Now is a proof by reductio ad absurdum that log₂ 3 is irrational. Observe that log₂ 3 ≈ 1.58 > 0.
Suppose log₂ 3 is rational. For several positive integers’ m and n, we have

It follows that

The decimal representation of irrationals

Let us see about list of irrational numbers,          
 When we state a rational number as a decimal, then also the decimal will be correct, as ¼ =.25, or it will not, as 1/3 ≈.3333. However, there will be a expected pattern of digits.  But if we attempt to state an irrational number as a correct decimal, then, clearly, we power not, because if we possibly will then the number would be rational.

Examples of list of irrational number

Example 1

• 22/7

 = 3.1428571428571….
Example 2

• 31/7

= 4.428571428571….
Example 3

• 47/9

= 5.2222222222222…..

Between, if you have problem on these topics Median Define, please browse expert math related websites for more help on gate ece 2013.

List of irrational numbers

Here some examples of list of irrational numbers are given below:
π ≈ 3.14159265358979323846264338327950...
 2   ≈ 1.4142135623730950488016887242096...
 3   ≈ 1.7320508075688772935274463415058...

Normal Distribution Data Set

Gaussian distribution is called as normal distribution. The normal distribution of the data is defined by two parameters mean (m) or average and standard deviation (s). A theoretical frequency distribution of the data is a set of variable; normally the data mean values are represented by bell-shaped curve symmetrical. In this article we shall discuss about normal distribution of data set example problem.

Normal distribution data set example problem:

Normal distribution:


X value < mean value = 0.5 - Z value

X value > mean value = 0.5+Z value

X value = mean value = 0.5

Z value = (X-m) / s

where,

m = Mean.

s = Standard Deviation.

X = Normal Random Variable

Example:

Let X be a normal random variable with mean value (m) 118 and standard deviation (s) is 5 find the P(X<108 p="">
Step 1:

For the given X value =108

Z = (108-118)/5

= 1

Step 2:

Find out the value of 1 in Z table (The value of Z table 1 = 0.3413)

Z = 1 = 0.3413

Step 3:

Here the value of X is less than mean so

P(X) = 0.5 - 0.3413 = 0.1587

Normal distribution is 0.1587

Example:

Let X be a normal random variable with mean value (m) 108 and standard deviation (s) is 6 find the P(X<112 p="">
Step 1:

For the given X value =112

Z = `(112-106)/6 ` = `4/6`

= 0.6

Step 2:

Find out the value of 0.6 in Z table (The value of Z table 0.6 = 0.2257)

Z = 0.6 = 0.2257
Step 3:

Here the value of X is greater than mean so

P(X) = 0.5 + 0.2257= 0.7257

Normal distribution is 0.7257

Normal distribution data set practice problem

Problem:

Let X be a normal random variable with mean value (m) 112 and standard deviation (s) is 6 find the P(X<106 p="">
Answer:

Normal distribution is 0.1587

Problem:

Let X be a normal random variable with mean value (m) 106 and standard deviation (s) is 5 find the P(X<110 p="">
Answer:

Normal distribution is 0.78819

Precalculus and Discrete Mathematics

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. Precalculus an advanced form of secondary school algebra, is a foundational mathematical discipline. Precalculus also called Introduction to Analysis. In this article we shall discuss about precalculus and discrete mathematics problem.(Source: wikipedia)


I like to share this Precalculus Problem Solver with you all through my article.

Discrete mathematics problem

Example:

Prove that (C, +) is an infinite abelian group.

Solution:

(i) Closure axiom: Sum of two complex numbers is always a complex number.

i.e., z1, z2 ∈ C ⇒ z1 + z2 ∈ C

Closure axiom is true.

(ii) Associative axiom: Addition is always associative in C

i.e., (z1 + z2) + z3 = z1 + (z2 + z3) ∀ z1, z2, z3 ∈ C

Therefore Associative axiom is true.

(iii) Identity axiom:

The identity element o = o + io ∈ C and o + z = z + o = z ∀ z ∈ C

Therefore Identity axiom is true.

(iv) Inverse axiom: For every z ∈ C there exists a unique − z ∈ C such that

z + (− z) = − z + z = 0. Inverse is true. ∴ (C, +) is a group.

(v) Commutative property:

∀ z1, z2 ∈ C, z1 + z2 = z2 + z1

Therefore the commutative property is true. Hence (C, +) is an abelian group.

Since C is an infinite set (C, +) is an infinite abelian group.

Example 2:

Find the order of each element of the group (Z4, +4)

Solution: Z4 = {[0], [1], [2], [3]} is an abelian group under the addition modulo 4. The identity element is [0] and notes that [4] = [8] = [12] = [0]

Therefore 0([0]) = 1

0([1]) = 4 [we have to add [1] four times to get [4] or [0]]

0 ([2]) = 2 [we have to add [2] two times to get [4] or [0]]

0 ([3]) = 4 therefore we have to add [3] four times to get [12] or [0]

Precalculus example problem

Find the H.C.F. of the polynomials 2x3 + 2x2 + 2x + 2 and 6x3 + 12x2 + 6x + 12

Solution: Let f(x) = 2x3 + 2x2 + 2x + 2 = 2 (x3 + x2 + x + 1) and

g(x) = 6x3 + 12x2 + 6x + 12 = 6(x3 + 2x2 + x + 2)

x3 + x2 + x + 1 )      x3 + 2x2 + x + 2                 (   1
x3 + x2    + x + 1
___________________

x2 + 1                                       ≠ 0
___________________

Dividing x3 + x2 + x + 1 by x2 + 1, we have

x2 + 1  )           x3 + x 2+ x + 1                          (   x + 1

x3 + x
__________________

x2 + 1

x2 + 1
___________________

0

H.C.F. of the polynomials = 2 (x2 + 1) (since the G.C.D. of 2 and 6 is 2).

Algebra is widely used in day to day activities watch out for my forthcoming posts on how to add and simplify fractions and iit jee 2013 syllabus. I am sure they will be helpful.

Volume of a Rectangular Prism Formula

Rectangular prism is a combination of six parallelograms and it is a three dimensional figure. The shape of the rectangular prism is same as the rectangular box. The three dimensions of the rectangular prism are length, width and height. With the rectangular prism the volume, later surface area and surface area can be determined.

Example diagram and formula – volume of a rectangular Prism formula:

Rectangular prism

The formula to find the volume of rectangular Prism is length * height* width

Example problems – Volume of a rectangular prism formula:

Example 1 - Volume of a rectangular prism formula:

Find the volume of the Rectangular Prism whose length is 10cm, width is 12cm and height is 14 cm.

Solution:

Given, Length =10cm, width = 12cm and height = 14cm.

The volume of rectangular Prism is length * height *width

= 10 * 12* 14

=1680cm3 .

Example 2 - Volume of a rectangular prism formula:

Find the volume of the Rectangular Prism whose length is 18cm, width is 15cm and height is 13cm.

Solution:

Given, Length =18cm, width = 15cm and height = 13cm.

The volume of the rectangular Prism is length * height *width

= 18*15*13

= 3150cm3

Example 3 - Volume of a rectangular prism formula:

Find the volume of the Rectangular Prism whose length is 11cm, width is 12cm and height is 13cm.

Solution:

Given, Length =11cm, width = 12cm and height = 13cm.

The volume of the rectangular Prism is length * height *width

= 11*13*12

= 716cm3

= 718cm2

Example 4 - Volume of a rectangular prism formula:

Find the volume of the Rectangular Prism whose length is 21cm, width is 19cm and height is 16cm.

Solution:

Given, length=21cm, width =19cm and height =16cm.

The volume of rectangular Prism is length *height *width

= 21*16*19

= 6384cm3

Practice problems - Volume of a rectangular prism formula:

Problem 1:

Find the volume of the Rectangular Prism whose length is 21cm, width is 19cm and height is 16cm.

[Ans: 6384cm3]

Between, if you have problem on these topics math help online tutor free, please browse expert math related websites for more help on iseet syllabus 2013.

Problem 2:

Find the volume of the Rectangular Prism whose length is 30cm, width is 25cm and height is 20cm.

[Ans: 15000cm3]

Problem 3:

Find the volume of the Rectangular Prism whose length is 17cm, width is 18cm and height is 19cm.

[Ans: 5814cm3]

Problem 4:

Find the volume of the Rectangular Prism whose length is 24cm, width is 20cm and height is 23cm.

[Ans:11040cm3]

Exponents and Power

Here in this article we are going to discuss about exponents and power. Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a:

Having problem with Sample Size Power keep reading my upcoming posts, i will try to help you.

Just as multiplication by a positive integer corresponds to repeated addition:

A power is an exponent to which a certain quantity is increased. The expression x a is therefore called as "x to the power." The power can be an integer, real number, or complex number. But, the power of an actual number to a non-integer power is not fundamentally itself a real number. For example, x12 is real only for x>=0.

Properties of exponents

The most important identity fulfilled by integer exponentiation is xm+n = xm. xn. This identity has the consequence

xm-n =`x^m/x^n`

• for x ≠ 0, and

(xm)n=xm.n

• Another basic identity is (x. y)n = xn. yn

Raised to the positive power value:

A+2 = A * A.

A+3 = A * A * A

While the above term is given as A squared (A2) and A+3 is known as x cubed.

Examples:

3+2 = 3 * 3

The answer is same 9 .This is known as positive term.

Positive power in terms of cube:

3+3 = 3 * 3* 3

The answer is same 27.

Raised to the negative power value:

The term xm-1 = (`x^m/x` ). While m =1, we get x0 = 1 .so the term as written as

`x^n/x^m`  = xn-m

In the special case when n and m are equal, so

1 = `x^n/x^n`  = xn-n = x0.

The number which raised to the power 1 is the number itself in it

The value of the any power 0 is 1.

Solved Examples

Below are the examples on exponents and power-

Example 1:

Solve equation on exponents : (a3)(a4)

Solution:

Terms of what those exponents mean. "To the 3rd" means "multiplying three copies" and "to the 4th" means "multiplying four copies". By the simplification method the factors are then multiplied. It is of the form

(a3)(a4) = (aaa) (aaaa)

= aaaaaaa

= a7

Example 2:

Solve equation on exponents: x3 x8

Solution:

Terms of what those exponents mean. "To the 3rd" means "multiplying three copies" and "to the 8th" means "multiplying eight copies". By the simplification method the factors are then multiplied. It is of the form

(x3)(x8)= (xxx)(xxxxxxxx)

= xxxxxxxxxxx

= x11

Between, if you have problem on these topics Derivatives Calculus, please browse expert math related websites for more help on cbse sample papers for class 11/commerce.

Example 3:

Solve: `(ab^3)/a^2`

Solution:

Terms of what those exponents mean. "To the 3rd" means "multiplying three copies" and "to the 2th" means "multiplying two copies". By the simplification method the factors are then multiplied. It is of the form

`(ab^3)/a^2`  = b3a1-2

= `b^3/a`

Word Problem Using Decimals

The word problems using decimals are first used to recognize the variables present in the given problem. Also the word problem using decimal are used in favor of the unit numbers and unit variables. Word problems using decimals are used for finding the phrases. The contemplation includes in this is, we have to analyze the problem present in the statements. And also a number that includes the decimal point is called as decimal. For every place in a decimal number has a dissimilar place value.

Understanding Rounding with Decimals is always challenging for me but thanks to all math help websites to help me out.

Examples for word problem using decimals:

Example 1:

1. Vasu bought 3.5 kg of flour, 7.34 kg of sugar and 0.25 kg of rice. What is the total weight of the things bought by vasu?

Solution:

Vasu bought,

Flour = 3.5 kg

Sugar = 7.34 Kg

Rice = 0.25 Kg

Total = 3.5+7.34+0.25

=11.09 kg

The result is 11.09 kg

Example 2:

The mass of a banana is 2.54 kg. A watermelon is 5.69 kg heavier than the banana. What is the mass of the watermelon?

Solution:

Mass of a banana is 2.54 kg

Watermelon is 5.69 kg heavier than the banana

So that the mass of watermelon is 5.69 – 2.54

= 3.15

So the final result is 3.15.

Example 3:

Kate has 8.8 m of cloth material. Kate used 2.95 m of the cloth to make a blouse. What is the length of cloth left?

Solution:

Kate has 8.8 m of cloth

Used cloth is 2.95 m

So the length of cloth left in original meter = 8.8 – 2.95

= 5.85

So the final result is 5.85


Practice problem for word problem using decimals:

Problem 1:

The variation of twice a number 4.44 is 8.54. What is that number?

Solution:

Step 1: First varying the above assumed sentence as variation of twice a number 4.44 equals 8.54.

Step 2: Then the step is to inscribe the above assumed equation in the equation form.

2D-4.44=8.54.

The final result is 6.49 for a word problem using decimals.



Problem 2:

The variation of twice a number 2.32 is 4.7. What is that number?

Solution:

Step 1: First altering the above said sentence as variation of twice a number 2.32 equals 4.7.

Step 2: Then the next step is to inscribe the above assumed equation in the equation form.

2I-2.32=4.7

The final result is 3.51 for a word problem using decimals.

Math Radicals Answer Key

Math radicals deal with solving square root problems with detailed answer keys. The radical of a given number or given expression is a number time itself is equal to the stated number or expression. The linear algebraic equation with radicals is discussed here with detailed answer keys. The radicals can be easily eliminated by squaring the given terms. The following are the example problem which deals with math radicals with proper answer keys.

Math radicals example problems with answer keys:

Math radical example problems are discussed below.

Example 1

Solve the math radical function.

√ (y 2 – 11y+28) = 2

Solution:

Given function is
√(y 2 – 11y+28) = 2

Squaring on both sides, then the above equation becomes
[√ (y 2 – 11y+28)] 2 = (2) 2

And simplify.
y 2 – 11y+28= 4

Make the above equation in factor form.
y 2 – 11y + 24 =0

The above equation is in quadratic form with two solutions.
y = 3 and y = 8

Example 2

Solve the math radical function.

√ ( y + 2) = y - 4

Solution:

Given function is
√( y + 2) = y - 4

Squaring on both sides, then the above equation becomes
[√ ( y + 2)] 2 = (y - 4) 2

And simplify.
y + 2 = y 2 - 8 y + 16

Make the above equation in factor form.
y 2 - 9 y + 14 = 0

The above equation is in quadratic form with two solutions.
y = 2 and y = 7

Example 3:

Solve the math radical function.

√ (y + 2) = 5

Solution:

Given function is
√(y + 2) = 5

Squaring on both sides, then the above equation becomes
[√ (y + 2)] 2 = 5 2

And simplify
y + 2 = 25

Solve for y.
y = 23

Math radicals practice problems with answer keys:

The following are practice problems for math radicals with answer keys for self doing.

1) Solve the math radical function.

√ (y + 9) = 9

Answer key: y = 72 is the solution

2) Solve the math radical function.

√ (14y –33 ) = y

Answer key: y = 11 and y = 3 is the solution.

Real Life Expressions

An algebraic expression is formed by combination from numbers, letters, and four basic arithmetic operations.

The four basic arithmetic operations are:

1)     Addition: a + b. The operating symbol is +.

2)     Subtraction: a – b. The operating symbol is - .

3)     Multiplication: a * b. the operating symbol is *.

4)     Division: a/b. The operating symbol is /.

The numbers exposed in algebraic expression are called constants.

I like to share this Rational Algebraic Expressions with you all through my article.

Example problems for expression in real life:

Yesterday I bought 5 apples. Totally I have 10 apples. So find the expression how many more apples to buy today?

Solution:

Let us take the number of apples buy today = x

Totally number of apples =10

Yesterday I bought 5 apples, so remaining apples x=10-5

In this expression here we are using subtraction operation

X=5 apples. Today I buy 5 apples and the totally I have 10 apples

Problems on expression in real life:
On this year birthday my height is 163 centimeters. One year before my height is x centimeter lesser than this year. Which expression gives my height one year before?

Solution:

This year birthday my height is 163 centimeters

Last year my height is x centimeters.

Expression = 163-x

Problems on expression in real life:

The cost of a new television is y dollars. The television is on sale for 25% off. What expression will you write for this situation?

Solution:

The television cost is Y dollars

Discount is 25%

New price = (Original price – 25% * original price)

Rewrite the expression as follows,

New Price = Original price (1-0.25)

Original Price = New price / (1-0.25)

Original Price = New price/0.75

=y/0.75

Answer: y/0.75



Problems on expression in real life:
Ryan and her three school friends are going to be sharing the cost of a 3 bedroom apartment. The cost of bedroom rent is n dollars. What expression can you write that will tell you what Jane's share is?

Solution:

Ryan and three friends, so total number of person = 4

They are sharing 3 bed rooms. Let us take room rent as n

Total expenses for bedrooms = 3* n

The total expenses is shares by 4 persons

So Ryan share = 3*n/4

Answer:
3n/4

Normal Approximation Binomial Distribution

In this page we are going to discuss about  normal approximation to the binomial distribution concept .Normal distribution is fine approximation to the binomial distribution, in a binomial distribution one can easily confirm that the mean for a single binomial trial, where Success is scored as 1 and Failure is scored as 0, is p; where p is the probability of S. Where s is a sample space hence the mean for the binomial distribution with n trials is np. Condition to the normal approximation is good for the binomial distribution. Condition of failure for normal approximation is p (1-p), standard deviation for normal approximation (np(1-p))^5 .

Understanding R Binomial Distribution is always challenging for me but thanks to all math help websites to help me out.

Normal approximation binomial distribution examples

If the number of trials in a sample space, the n is large, the binomial distribution is just about equal to the normal distribution. This is fine, since we actually do not feel like to plainly calculate binomial probabilities when n > 100.

Example 1:

The diameter of an telephone cable is in general distributed with mean 0.7 cm and variance 0.0002 cm2 .what is the probability that the diameter will exceed 0.71 cm, the cable is measured imperfect if the diameter differs from the mean by more than 0.015 cm. what is the probability of obtaining a defective cable?

X is N( 0.7, 0.0002)

a)      P(x> 0.71) = p(z> ((0.71-0.7)/0.01))

= p(z>0.5) = 1 – p(0
= 1 – 0.381 = 0.62

The result is 0.62

b)      P[(x> 0.825) U (x< 0.785)] = 2P(x> 0.825)

= 2P(z> 0.025/0.02)

= 2P(z> 1.25)

= 2[-0.3944 + 0.5 ]

= 0.2112

Example 2:

A cricket match has 45 multiple teams, in each  team with choices of a to e. One player did not play and must guess on each team, Using normal approximation, estimate the probability that the player:
a) failure in the team or gets less than 50%
b) gets a score of at least 10%
c) gets a score between 20% and 40% inclusively

a) a 50% would be getting 22.5 success in the team. Getting less than that is player getting 22 in the team.

P(X ≤ 22 ) =

22
∑ P(X = x) = 0.9999967
x = 0

≈ Probability ( Xn < 22.5 )
= P( y < ( 22.5 - 9 ) / 2.683282 )
= P( y < 5.031153 )
= 0.9999998

b) 10% of 45 is 4.5 so we are looking to get 5 or more teams.

P( X ≥ 5 ) =

45
∑ P(X = x) = 0.961764
x = 5
≈ P( Xn ≥ 4.5 )
= P( y ≥ ( 4.5 - 9 ) / 2.683282 )
= P( y ≥ -1.677051 )
= 0.9532337

c) 20% is 9 teams, 40% is 18 teams.

P( 9 ≤ X ≤ 18 ) =

18
∑ P(X = x) = 0.5587208
x = 9
≈ P( 8.5 < Xn < 18.5 )
= P( ( 8.5 - 9 ) / 2.683282 < y < ( 18.5 - 9 ) / 2.683282 )
= P( -0.186339 < y < 3.540441 )
= P( y < 3.540441 ) - P( y < -0.186339 )
= 0.9998003 - 0.4260895
= 0.5737108

Example 3 :

The FOARD Company manufactures cars. They claim that only for .08 of MNW cars are defective. What is the probability of finding 3 defective cars in a random sample of 45 FOARD cars?

Solution : Formula for cumulative binomial distribution

P(X = r) = nCr *p r *(1-p) n-r

Given r=3, n=45, p=0.08

Now we can substitute this values for this formula

= 45C3 *(.08)3* (.92)45-3

= 45C3 *(.08)3* (.92)42

= 0.219
Simulation with a binomial experiment

The authority of the Normal approximation to the binomial distribution. Simulation with a binomial experiment is one way

to produce a normal distribution. Either entire the calculations with calculate data, or convert everything including sample

space and the standard deviation to proportions.

Integers Algebra

The integers (from the Latin integer, literally "untouched", hence "whole": the word entire comes from the same origin, but via French) are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}.

Source – Wikipedia.


Addition of Integers:

The followings are some of the examples for addition of integers in algebra.

Find two consecutive integers whose sum is equal 225.

Solution:

We can take in this as x and x + 1 are the two numbers. Then we can use the fact in their sum is equal to 225 to write the equation as
x + (x + 1) = 225

2x + 1 = 225

We can solve for x to obtain
x = 112

Therefore, two numbers are
x = 112 and x + 1 = 113 is the solution of the given sum of two consecutive integers.

Find two consecutive integers whose sum is equal 147.

Solution:

We can take in this as x and x + 1 are the two numbers. Then we can use the fact in their sum is equal to 147 to write the equation as
x + (x + 1) = 147

2x + 1 = 147

We can solve for x to obtain
x = 73

Therefore, two numbers are
x = 73 and x + 1 = 74 is the solution of the given sum of two consecutive integers.

Multiplication of Integers:

The followings are some of the examples for multiplication of integers in algebra.

Find the multiplication of integers 6 * 4.

Solution:

6 * 4 = six four

= 6 + 6 + 6 + 6

= 24

Likewise, we have 6 * (-4) = four minus six

= (-6) + (-6) + (-6) + (-6)

= -24 `rArr` (1)

By commutative property, you know that

6 * 4 = 6 * 4 and 6 * 4 = four six = 6 + 6 + 6 + 6

You can also write, 4 * 6 = 6 + 6 + 6 + 6

Likewise, (- 4) * 6 = (- 6) + (- 6) + (- 6) + (- 6)

= - 24 `rArr` (2)

Hence, from (1) and (2), 4 * (- 6) = (-4)* 6 + (-24) = (4 * 6) is the solution for the given multiplication of the integers.

Find the Multiplication of the given integers: 8 x 5.

Solution:

Given integers are (8) × (5)

= |8| × |5|

= 8 × 5

= 40 is the solution for the given integers.


Find the Multiplication of the integers: 7 x (-5).

Solution:

Given integers are 7 x (-5)

= | 7 | × | -5 |

= 7 × -5

= -35 is the solution for the given integers.

Logarithmic Mean

In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. In symbols:

M_lm(x,y) = lim(xi,eta)->(x,y) (xi-eta)/(Inxi-lneta)

= {(0 if x=0 vv y=0),(x if x=y),((y-x)/(lny-lnx) else):}

for the positive numbers x,y. This measure is useful in engineering problems involving heat and mass transfer. (Source: Wikipedia)

Please express your views of this topic Logarithmic Series by commenting on blog.

Theorem for logarithmic mean

Theorem 1:-

The inequalities,

 alphaA(a,b)+(1-alpha)G(a,b)
hold for all positive real numbers a and b with a≠b if and only if alpha<=2/3 and beta>=2/e=0.7357 .

Theorem 2:-

Let a and b be real numbers with a≠b. If 0
[G(a,b)]A(a,b)<[L(a,b)]I(a,b)<[A(a,b)]G(a,b).(1.9)

And if a,b>=e , then

[A(a,b)]G(a,b)<[I(a,b)]L(a,b)<[G(a,b)]A(a,b).(1.10)

Theorem 3:-

For all positive real numbers a and b with a≠b, we have

Mp(a,b)<12 a="" b="" p="">
with the best possible parameter p=log2/(1+log2)=0.40938.

for α,β,γin(0,1) with α+β+γ=1, what is the larger value of p and q is the smaller value of the double inequality



Lp(a,b)0 with a≠b.

Example problem for logarithmic mean:-

Problem 1:-

For α,β,γ in (0,1) in logarithmic mean α+β+γ=1,p is the larger value and q is the smaller value these types of value is double inequality. Lp(a,b)0 with a≠b?

Solution:-

In these equations Lp(a,b), A(a,b), G(a,b), and H(a,b) denoted as generalized logarithmic mean, arithmetic mean, geometric mean, and harmonic means of two positive numbers a and b.

For p inR the generalized logarithmic mean Lp(a,b) of two positive numbers a and b with a≠b is defined by



Lp(a,b)={([(a^p+1-b^p+1)/(p+1)(a-b)]^1/p,p!=0,p!=-1),(1/e(b^b/a^a)^1/(b-a),p=0),(b-a/logb-loga,p=-1):}



It is well known that Lp(a,b) is continuous and strictly increasing with respect to p inR for fixed a,b>0 with a≠b. The logarithmic mean of monotonicity  and the inequalities of the logarithmic mean.

Let A(a,b)=(a+b)/2 , I(a,b)=(1/e)(b^b/a^a)^1/(b-a) , L(a,b)=(b-a)/(logb-loga) , G(a,b)=sqrtab , and H(a,b)=(2ab)/(a+b) be the arithmetic mean, identric mean, logarithmic mean, geometric mean, and harmonic mean value of  a and b two positive numbers with a≠b,

Then

min{a,b}
For p inR , the pth power mean M_p(a,b) of two positive numbers a and b with a≠b is defined by

 M_p(a,b)=((a^p+b^p)/2)^1/p(p!=0), M_0(a,b)=(ab)^1/2.

M_log2/log_3(a,b)<2 a="" b="" p="">
for all a,b>0 with a≠b.


For α in(0,1), in p is the larger value and q is the smaller value such that,

M_p(a,b)
for all a,b>0 with a≠b.

Multiplying Fractions Chart

A fraction is a value that shows the number of equal parts taken of a whole quantity or unit. The denominator of a fraction is the number that shows how many equal parts are in the whole quantity. The numerator of a fraction is the number that shows how many equal parts of the, whose are taken.

The numerator and denominator are called the term of the fraction,

3 – (Numerator) / 4 – (Denominator)

(Source: Wikipedia)

Multiplying fractions chart definition and methods:

Definition for fraction multiplying chart:

To multiply algebraic fraction, follow these steps:


Understanding Like Fractions Definition is always challenging for me but thanks to all math help websites to help me out.

Write the given fraction and cancel any common factors.
Multiply the numerators.
Multiply the denominators.

Multiplying fractions chart:

Multiplying fractions chart

Multiplying fraction chart methods:

Multiplying common fractions:

To multiply common fractions, multiply the numerator by the numerator and the denominator by the denominator. Reduce to lowest terms when possible.

Example:

(2 / 7 ) * (5 / 4 )

= (2*5)/(7*4)

= 10 / 28

= 5 / 14

Reduce lowest terms:

(2 / 6 ) * (3 / 4 )

= (3 * 2) / (6 * 4)

= 6 / 24

= 1 / 4

Multiplying fraction chart and a whole number:

To multiply a fraction and a whole number, follow these steps:

Change the whole number to a fraction by placing the whole number over one.
Then multiply the numerator by the numerator and the denominator by the denominator.
Reduce to lowest terms when possible.

Example:

18 x (4/8 )

= (18 * 4) / (1* 8)

= 72 / 8

= 9

Multiplying fractions chart and a mixed numbers:

To multiply mixed number, follow these steps:

Change the mixed numbers to improper fractions

Then multiply the numerator by the numerator and denominator by the denominator

Reduce to lowest terms when possible

Example:

(2 9/8 ) * (4 / 5 )

= (25 / 8 ) * (3 / 5 )

= (25 * 3) / (8 * 5)

= 75 / 40 = 1 35/40

Practice problem for multiplying fraction chart:

Practice problem 1:

To find the multiplying common fraction (4 / 5 ) * (6 / 3 )

Answer: 24 / 15



Practice problem 2:

To find the multiplying fractions a whole number 24 * (5 / 6 )

Answer: 20

Practice problem 3:

To find the multiplying fractions a mixed number (1 3/4 ) * (3/5 )

Answer: 21 / 20 .

How to Graph Quadratic Inequalities

Learn here how to graph inequalities. Algebra is a subdivision in mathematics in which comprises of infinite number of operations on equations, polynomials, inequalities, radicals, rational numbers, logarithms, etc. Graphing algebra quadratic inequalities is also a part of algebra. It is similar to the graphing of ordinary equations, but in the final output of the graph, the inequality is indicated by shading the particular region depending on the inequality given. Help in graphing quadratic inequalities and their graph is shown in the following sections.


I like to share this Quadratic Inequalities Solver with you all through my article.

How to graph inequalities

Here is given the procedure how to graph inequalities. This procedure to help with graphing linear inequalities,

The steps involved in graphing quadratic inequalities are as follows, Consider the quadratic inequality y > ax2+ bx +c
Step 1: Convert the given equation as y = ax2+ bx +c
Step 2: Since the given inequality is a function of x, let y =f(x).
Step 3: Therefore f(x) = ax2+ bx +c
Step 4: Substitute various values for ‘x’ and find corresponding f(x).
Step 5: Tablulate the values for help as follows x and f(x), the values of x as2 ,3,4, 1. -1 and f9x) their corresponding values.
Step 6: The values in the table are the co-ordinates, graph them to form a line.
Step 7: Shade the inequality range above the line, since greater than symbol (>) is given.

Graphing inequalities

Here are few graphing inequalities examples. Following examples will help in learning how to graph quadratic inequality:
1. Graph the following inequality number y > x2- 2
Convert the given equation as
y = x2-2.
Since the given inequality is a function of x,
Let y =f(x).
Therefore
f(x) = x2-2.
Substitute various values for ‘x’ and find corresponding f(x).
When x= -3
f(-3) = (-3)2 -2,
9-2,
7, therefore the co-ordinates are (-3, 7)


When x= -2
f(-2) = (-2)2 -2,
4-2,
2, therefore the co-ordinates are (-2, 2)


When x= -1,
f(-1) = (-1)2 -2,
1-2,
-1, therefore the co-ordinates are (-1, -1)


When x= 0
f(0) = (0)2 -2,
0-2,
-2, therefore the co-ordinates are (0, -2)


When x= 1
f(1) = (1)2 -2,
1-2,
-1, therefore the co-ordinates are (1, -1)


When x= 2
f(2) = (2)2 -2,
4-2,
2, therefore the co-ordinates are (2, 2)

When x= 3
f(2) = (3)2 -2,
9-2,



7, therefore the co-ordinates are (3, 7)
The table for help is as follows,
The graph of the quadratic inequality is shown below,
graphing quadratic inequality (1)
The region shaded shows the inequalityann if the quadratic inequality is y< x2 -2 then the graph would have been,

graphing quadratic inequality (2)

What is a Complex System

Why makes a system complex? Fundamentally a system is complex, if its behavior cannot be easily described. One way this can arise is if the system consists of many components, with numerous relationships and interactions between these components. The presence, absence, or nature of these relationships may affect the behavior of the aggregate system, so a description of this behavior must take into account each of these relationships. (Notice that if the components of the system are identical and their interconnection is regular, then the aggregate behavior can be quite simple. This is well demonstrated by the behavior of a memory chip, which have the greatest density of transistors of any integrated circuit.


Another way in which a system can be complex, is if the inherent behavior of a component is non-linear. Such systems can exhibit chaotic behavior. Examples of such systems are certain non-linear oscillators, weather systems (at almost all levels of fidelity) and the 3n+1 (Collatz) process. Such systems may possess succinct descriptions, but highly complex behavior. Detailed behavior of non-linear systems will not be the focus of this study, although the difficulties they induce will arise in systems of the type discussed above.

Thus, to understand a type I complex system (consisting of many components), we must reduce the number of components that must be examined. There appear to be two basic approaches that one might take. One can partition the components into collections of components, where each collections has a relatively well defined behavior. If the number of components in a collection is two large, then the process may be repeated recursively. This hierarchical partitioning is characteristic of top down or structured design. Notice that the boundaries that arise at the first partitioning persist at every refinement of the partitioning.

An alternative approach to partitioning the system, is to approximate the behavior of the system by a simpler model containing fewer components and/or simpler interactions. These types of approximate models abound in physics and engineering. The most accurate model of a the behavior of a mechanical system can be obtained using a relativistic, quantum mechanical model of the system. However, this is often too complicated and a Newtonian or simple relativistic models are often used instead. These models approximate the behavior of the relativistic quantum mechanical model. The boundaries of the components and interactions in the each of the three modes are different.

Simple Percent Problems

In mathematics, a percent (%) is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. (Source: From Wikipedia).

Fraction or Decimal to percent conversion means, we have to multiply by 100. Now, we are going to see some of the simple percent problems.


Problems on simple percentages:

Example problem 1:

What is 40% of 90?

Solution:

The original number is 90. 40% means 40/100. So, 40/100 is multiplied with the original number.

40% of 90 = (40/100)*90

= 36.

So, 40% of 90 is 36.

Example problem 2:

What percent of 28 is 3?

Solution:

Let us take x percent of 28 is 3.

x percent means x/100. So, x/100 is multiplied with the original number (28) is equal to 3.

So, the equation is

(x /100) * 28 = 3

x = 300 / 28

x = 10.7%

So, 10.7% of 28 is 3.

Example problem 3:

Express the decimal number 0.17 in percentage:

Solution:

Multiply 100 with the given decimal number.

0.17 = 0.17*100 %

= 17%

So, the answer is 17%.

Few more simple percent problems:

Example problem 4:

Express the fraction number (2/5) in percentage:

Solution:

Multiply 100 with the given fraction number.

2/5 = (2/5) *100 %

= 40%

So, the answer is 40%

Example problem 5:

25% of a certain number is 200.  Find the number.

Solution:

Let us take a number be ‘x’.

25% of a number means (25/100) is multiplied with x.

So, it can be written as

(25/100)*x = 200

(1/4)*x = 200

x = 800

So, the number is 800.



Example problem 6:

What is 70% of 10?

Solution:

The original number is 10. 70% means 70/100. So, 70/100 is multiplied with the original number.

70% of 10 = (70/100)*10

= 7.

So, 70% of 10 is 7.

Practice simple percent problems:

1)     What is 50% of 60? (Answer: 30)

2)     What percent of 40 is 4? (Answer: 10%)

3)     Express the fraction number (2/4) in percentage. (Answer: 50%)

4)    50% of a certain number is 200.  Find the number. (Answer: 400)

Solve for Slope of Parallel Lines

Slope of a line :

Slope or Gradient of a line is the value of the angle that a straight line makes with the positive direction of x-axis in the anticlockwise sense.

For any given straight line y = mx + b , m is referred to as slope.

Parallel lines always have the value of the angle made with the x axis  same.. they  have the same slope.

Method to solve for slope : parallel lines

Relation between slopes of parallel lines :

Slopes of any two parallel lines are equal.

Let the equation of two parallel lines be y=(m1)x + c1 and y=(m2)x + c2 where m1 and m2 are slopes of the respective lines. The two lines are considered parallel if and only if m1 = m2.

Examples to solve for slope parallel lines

Example 1 : The equations of two lines 2y - 4x = 12 and y = 2x +8.  Given the lines are parallel. Check the slope for both the lines . Are they equal or not ?

Solution : Given the equation of the two lines and also they are parallel lines,now we have to calculate slope. First convert the equation of lines into its standard form y = mx+b to determine the slope.

the equation of the first line is 2y - 4x = 12. It will be 2y = 4x+12 now dividing both sides by 2 we have y =2x+ 6. Hence slope of the first line lets say m1 = 2.

the equation of the second line is y=2x + 8. It is already in the standard form. Hence slope of the second line say  m2 = 2.

So the value of the slopes m1 = m2 = 2 are equal (Answer)

Example 2 : Given the equation of  line  y = 3x +8. What is the slope of the line parallel to given line ?

Solution : The equation of the line  is y=3x+8 . It is present in the standard form y = mx+b. Hence slope of the line is 3.

we have to find the slope of the line parallel to the line y=3x + 8. Now the two lines are given parallel then the slopes  must be the same. So, the slope of the line =3 (Answer)

Example 3: Given the equation of the line 2x – 3y = 9 and the point (4, –1), find line through the point that is parallel to the given line ?

Solution : We have to find the line passing through (4, –1) that is parallel to (that has same slope as) 2x – 3y = 9.

First of all, we have to convert the given equation into the standard form to determinethe slope.

2x – 3y = 9

–3y = –2x + 9

y = (2)/(3) x – 3

So the slope of the reference line is m = (2)/(3).

Since a parallel line has the same slope, then the parallel line passing through (4, –1) will have slope m = (2)/(3). Hence,  we have a point and a slope! So we have to use the point-slope form to find the line:

y – (–1) = ( (2)/(3) )(x – 4)

y + 1 = (2)/(3) x – (8)/(3)

y = ( (2)/(3) ) x – (8)/(3) – (3)/(3)
y = ( (2)/(3) ) x – (11)/(3) (Ans)

I am planning to write more post on prime factorization problems and cbse 11th maths. Keep checking my blog.

Unit Converter Meter

Meter defined as base unit of length. Meter is one of the measurement scales. It is used to calculate length of some things. Measurement is the method of calculating or resolving the magnitude of an amount. Converter is used to change a form to another form. In this article, we are going to discuss about unit converter meter with suitable example problems.

Example problem for unit converter meter:

Convert meters to centimeters:

1 m = 100 centimeter

Example 1:

Convert 100 meters into centimeters using converter

Solution:

First enter the meter value in first box.

Then we get the answer in centimeter with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 m = 100 centimeter

Step 2: We need to find 100 meter into centimeters.

Step 3: Multiply 100 with 100 = 10000 centimeters.

Step 4: Therefore, 100 meters = 10000 centimeters

Answer is 10000 centimeters.

Convert meters to centimeters:

1 m = 1000 centimeter

Having problem with Hexadecimal to Octal Converter keep reading my upcoming posts, i will try to help you.

Example 2:

Convert 2500 meters into centimeters.

Solution:

First enter the meter value in first box.

Then we get the answer in centimeter with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 m = 1000 centimeter

Step 2: We need to find 2500 meter into centimeters.

Step 3: Multiply 2500 with 100 = 10000 centimeters.

Step 4: Therefore, 2500 meters = 10000 centimeters

Answer is 10000 centimeters.

Convert meters to kilometers:

1 m = 0.001 km

Example 3:

Convert 1000 meters into kilometers.

Solution:

First enter the meter value in first box.

Then we get the answer in centimeter with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 m = 0.001 km

Step 2: We need to find 1000 meter into kilometers.

Step 3: Multiply 1000 with 0.001 = 1 kilometer.

Step 4: Therefore, 1000 meters = 1 kilometer

Answer is 1 kilometer.

Convert meters to kilometers:

1 m = 0.001 km

Example 4:

Convert 25000 meters into kilometers.

Solution:

First enter the meter value in first box.

Then we get the answer in centimeter with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 m = 0.001 km

Step 2: We need to find 25000 meter into kilometers.

Step 3: Multiply 25000 with 0.001 = 25 kilometers.

Step 4: Therefore, 25000 meters = 25 kilometers

Answer is 25 kilometers.

More example problem for unit converter meter:

Convert foots to meter:

1 ft = 0.3048 meter

Example 1:

Convert 15 foots into meter.

Solution:

First enter the foot value in first box.

Then we get the answer in meter value with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 ft = 0.3048 meter

Step 2: We need to find 15 foot into meter.

Step 3: Multiply 15 with 0.3048 = 4.572 meter.

Step 4: Therefore, 15 foots = 4.572 meter

Answer is 4.572 meter.

Convert foots to meter:

1 ft = 0.3048 meter

Example 2:

Convert 50 foots into meter.

Solution:

First enter the foot value in first box.

Then we get the answer in meter value with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 ft = 0.3048 meter

Step 2: We need to find 50 foot into meter.

Step 3: Multiply 50 with 0.3048 = 15.24 meter.

Step 4: Therefore, 50 foots = 15.24 meter

Answer is 15.24 meter.

Convert meters to foot:

1 m = 3.280839895 foot

Between, if you have problem on these topics Definition of a Acute Angle, please browse expert math related websites for more help on cbse guides for class 10.

Example 3:

Convert 50 meters into foot using converter

Solution:

First enter the meter value in first box.

Then we get the answer in foot value with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 m = 3.280839895 foot

Step 2: We need to find 50 meters into foot.

Step 3: Multiply 50 with 3.280839895 = 164.041994751 foot.

Step 4: Therefore, 50 meters = 164.041994751 foot

Answer is 164.041994751 foot.

Convert meters to foot:

1 m = 3.280839895 foot

Example 4:

Convert 120 meters into foot.

Solution:

First enter the meter value in first box.

Then we get the answer in foot value with in a second. This is the converter process.

converter

Manual calculation:

Step 1: We knew that 1 m = 3.280839895 foot

Step 2: We need to find 120 meter into foot.

Step 3: Multiply 120 with 3.280839895 = 393.700787402 foot.

Step 4: Therefore, 120 meters = 393.700787402 foot

Answer is 393.700787402 foot.

Unreal Numbers

Unreal number is also called as imaginary number. An unreal number is a number in the form yi where y is a non-zero, real number and i, defined by i2 = − 1, is known as the imaginary unit. An imaginary number or unreal number yi can be added to a real number x to form a complex number of the form x+ yi, where x and y are called respectively, the "real part" and the "imaginary or unreal part" of the complex number x + yi.

Example problems of unreal numbers:

Unreal numbers example 1:

Simplify 7i + 6i

Solution:

We can find the addition value of given imaginary number.

Given problem is 7i + 6i.

Here both 7i and 6i are the imaginary number.

In the above problem, we can take i for common. Then we get,

7i + 6i= i (7+6)

Add the values for inside the parenthesis.

7+6=13

Therefore, i(13)= 13i

Answer: 13i

I like to share this Ascending Numbers with you all through my article.

Unreal numbers example 2:

Simplify (2i+ 5i) 2

Solution:

We can find the square value of given unreal numbers.

Given problem is (2i+ 5i) 2

Here both 2i and 5i are the imaginary numbers.

Now we can square the above problem.

i* i= i2

i2=-1

That is, ((2i+ 5i) 2= (7i) 2

=49(i) 2

Here substitute the value of i2 into the above.

Then we get,

49i2=49(-1)

=-49

Answer: -49

Unreal numbers example 3:

Simplify 5i- 8i

Solution:

We can find the subtraction value of given imaginary number.

Given problem 5i- 8i

Here both 5i and 8i are the imaginary numbers.

In the above problem, we can take i for common. Then we get,

5i- 8i = i (5-8)

Subtract the values for inside the parenthesis.

5-8= -3

That is, i (-3)

= -3i

Answer: -3i

Algebra is widely used in day to day activities watch out for my forthcoming posts on Histogram Analysis and cbse class 10 syllabus 2011. I am sure they will be helpful.

Practice problems of unreal numbers:

1. Simplify (8i) 2              Answer: -64

2. Simplify i+ 7i          Answer: 8i

3. Simplify 12i-9i        Answer: 3i