Thursday, December 27, 2012

Polynomials Chart

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source Wikipedia)

In this article polynomial chart we see about basic concepts of polynomial, its types of polynomial ,some example problems

Polynomial Types:


Basic concepts of polynomials:
Polynomial is nothing but algebraic expression and also concept of algebras. More types of polynomials are available in the algebra depends on the number of terms. Based on the number of terms polynomial was classified four types

Different types of polynomial:

Polynomial chart

Types of polynomial

Monomial:

If the expression having one term mean it was called as monomial

Example: 7x ,8x2

Binomial:

If the expression having two terms mean it was called as binomial

Example: 6x+4x

Trinomial

If the expression having three terms mean it was called as trinomial

Example: 3x+8x2+9

Polynomial:

If the expression having more than three terms mean it was called as polynomial

Example: 5x2+12x3+9x+10

Polynomial operations are addition of polynomial, subtraction of polynomial, multiplication of polynomial, division of polynomial.

Example Problems in Polynomial:

Example problems in Polynomial degree chart:

Polynomial addition chart:

Example 1:

Add the polynomial 3x2+5x+2 and 5x+6

Given polynomials: 3x2+5x+2,5x+6

Now we have to arrange the  terms for addition

After than add the terms one by one.

This is a polynomial addition chart

Polynomial addition

Example 2:

Polynomial multiplication chart:

(2x+5)(3x+1)

Now we have to multiply the one terms with another terms

And then add the terms

Polynomial multiplication

Example 3:

Degree of polynomial:

(9z9 +8 z4 − 6z5 + 8) Find the degree of polynomial for each term?

Degree of polynomial for first term=9

Degree of polynomial for second term =4

Degree of polynomial for third term=5

Degree of polynomial for fourth term=0

Highest degree of polynomial is 9


Tuesday, December 25, 2012

Ratio to Fraction Converter

Ratio :

In mathematics, The ratio can be used to relate two quantities by using the symbol : Also it can be expressed as follows,

  • x is to y
  • the ratio of x to y
  • x : y

Fraction :

In mathematics , Part of the whole can be expressed as fraction. There are three kinds of fraction
  • Proper fraction
  • Improper fraction
  • Mixed fraction

In this article we are going to see about how to simplify the ration as fraction by using the ratio to fraction converter.

Ratio to Fraction Converter :

Converter:

The electronic or software device that can perform the operations Quickly. The ratio to fraction converter can be used to convert fraction for the given ratio.

Ratio to fraction converter

Fig(i) Ratio to fraction converter

Let us see some problems on ratio to fraction convertor.

Problems on Ratio to Fraction Converter :

Problem 1:

Convert the ratio 45 : 180 into simplified fraction

Solution:

Given,The ratio 45 : 180

We need to convert the given ratio into fraction .

we know that 45 : 180 = ` 45/ 180`

Divided by  45 on both numerator and denominator,

`45/180` = `( 45 / 45 ) / ( 180 / 180 )`

= `1 / 4`

Answer: The simplified fraction of the given fraction is  `1/4` .


Problem 2:

In a bag, there is Blue  and Green balls, the ratio of Blue balls to Green balls is 5:6. If the bag contains 180 Blue balls, how many green balls are there?

Solution:

Given The ratio of the Blue and green balls = 5 : 6

Number of blue balls = 180

Let us take x = green balls

To find the green balls we need to convert the given ratio into fraction,

Write the items in the ratio as a fraction.

`(blue) / (green)` = `5/6` = `x / 180`

`5/6` = `x / 180`

Multiply by 6 on both sides,

5 = `x / 180 `

5 = `x / 30`

Now multiply by 30 on both sides,

5 * 30 = x

150 = x

x = 150

Total number of black balls = 150

Answer: Green balls = 150

Sunday, December 23, 2012

Square Root of Complex Number

A square root of an x is a numeral r such to r2 = x, or a numeral r whose square is x. each non-negative real number x have a exclusive non-negative square root, recognized the principal square root, signify by a essential symbol as . For optimistic x, the main square root knows how to as well be written in exponent information, as x1/2.  (Source: Wikipedia)

Square Root of Complex Number:


A complex number is a number regarding of a real also imaginary part. It know how to be written in the structure a + bi, where a and b are real numbers, also i is the ordinary imaginary unit among the property i 2 = −1. The complex numbers include the ordinary factual numbers, however expand them by adding in extra numbers and equally increasing the considerate of addition also multiplication.

Every positive numbers x contain two square roots. One of them is` sqrt(x)` , to be constructive, with the other `-sqrt(x)` that is negative. Collectively, these two roots are signifying `+-sqrt(x)` . Square roots of unconstructive numbers know how to be converse in the structure of complex numbers. More commonly, square roots know how to be thinks in any context in that a notion of square of several arithmetical objects is definite.

Example for Square Root of Complex Number:


Example 1:

Solve square root of compel numbers `sqrt(3) - isqrt(6)`

Solution:

Step 1: the given square root complex number is `sqrt(3) - isqrt(6)`

Step 2: `x = sqrt(3) and y = -sqrt(6)`



Step 3:    `r =sqrt( x^2+y^2)`

Step 4: substitute x and y values

`r =sqrt( (sqrt(3)^2)+(-sqrt(6)^2))`

Step 5:   r = 3

Example 2:

Solve square root of comple numbers 4+13i

Solution:


Step 1: the given square root complex number is   4+13i

Step 2: the given problem of the form is a+bi

Step 3: `x = 4 and y = 13`

Step 4:    `r =sqrt( x^2+y^2)`

Step 5: substitute x and y values

`r =sqrt( (4^2)+(13^2))`

Step 6:    `sqrt(377)`

Step 7:   r = 19

Step 8: finding b value

` b =sqrt((r-x)/2)) `

Step 9:    ` b =sqrt(((19)-4)/2) `

Step 10:    ` b =sqrt((15)/2) `

Step 11:  b = 2

substitute y and b in a

Step 12:  a = `y/(2b)`

a = `13/(2(2))`

a = 3

Step 13: find square root of  r1 and r2

r1 =a + bi = 3 + 2i

r2 = -a-bi = -3-2i

so the square root of complex numbers are 3+2i and -3-2i

Wednesday, December 19, 2012

Simplifying Radicals Activity

Simplifying radicals activity involves the process of solving radicals equation with step by step solution. Activity is the process of solving equations with radicals symbol. The square root symbol is also represented as radicals. Simplifying radicals is easily carried out by performing squaring operations on the given equation is known as simplifying radicals activity. The following are the example problems which explain the radicals activity.

In logarithm, the radical pictogram is represented by √. The symbol n√x , n, x and √ are represented by index , radical and and radical symbol. Steps to multiplying radicals is the similar way of ordinary multiplications, additionally it has the following conditions. When multiplying two or more radicals, we must multiply the numbers exterior radicals and then multiply the numbers in the interior radicals.

Steps to multiplying radicals:

The following steps to multiplying radicals, when the radicals have the indistinguishable key, n:

Step 1:  Utilize the multiplying radicals rule for nth roots to find the product of the radicals.

Step 2:  Simplify the product by factoring and taking the nth root of the factors that are ideal nth powers.

Rules on multiplying radicals:

The following rules for steps to multiplying radicals helps the easy way of understanding and simplifying the radical expression.

Rule 1:

n√x . n√y  =  n√(x . y)

Rule 2:

n√( x + y) . n√(x – y) = n√(x2 – y2)

Rule 3:

(x. n √y)n  =  xn . y

Simplifying Radicals Activity Example Problems:

Ex:1 Solve the radicals.


`sqrt(u^2-5u+7) = 1`

Sol:

Given equation is
`sqrt(u^2-5u+7) = 1`

To solve the radical symbol perform squaring operation on both sides
`[sqrt(u^2-5u+7)]^2 = (1)^2`

And simplify.
u 2 – 5u+7= 1

Make the above equation in factor form.
u 2 - 5 u + 6 = 0

The above equation is in quadratic equation form with two solutions
u = 3 and u = 2 is the answer.

Ex:2 Solve the radicals.

`sqrt(3u-5) = u - 1`

Sol:


Given equation is
`sqrt(3u-5) = u - 1`

To solve the radical symbol perform squaring operation on both sides

`[sqrt( 3u-5)]^2 = (u-1)^2`

Simplify the above equation
3 u - 5 = u 2 - 2 u + 1

Change the above equation in factor form.
u 2 - 5 u + 6 = 0

The above equation is in quadratic equation form with two solutions
u = 2 and u = 3 is the answer.

Ex:3 Solve the radicals.

`sqrt(u^2-16u+37) = 3`

Sol:

Given equation is
`sqrt(u^2-16u+37) = 3`

To solve the radical symbol perform squaring operation on both sides
`[sqrt(u^2-16u+37)]^2 = (3)^2`

And simplify.
u 2 – 16u+37= 9

Make the above equation in factor form.
u 2 - 16 u + 28 = 0

The above equation is in quadratic equation form with two solutions
u = 2 and u = 14 is the answer.

Practice Problems on Simplifying Radicals:

Q:1 Solve the radicals.

` sqrt(u^2-12u+29) = 3`

Answer: u = 2 and u = 10.

Q:2 Solve the radicals.

`sqrt(4u-3) = u -2`

Answer: u = 7 and u = 1

Multiplying Radicals:


Ex:1 (a)     Steps to multiplying radicals √2 . √36.

Sol:

1.   Multiply the i nside radical numbers  :          = √(2 .36)   = √72

2.   Simplify radicals if possible             :          =  √(2 . 62 )

3.   Answer                                           :          = 6 √2

(b) Steps to multiplying radicals  √5 . √7

Sol:

1.    Multiply the inside radical numbers = √(5 . 7)

2.    Simplify radicals if possible     =  √35

(c)   Steps to multiplying radicals                                      4√7 . 7√5

Sol:

1.   Multiply the outside of radical numbers first      (4 . 7) = 28

2.   Multiply the inside radical  numbers              √(7 . 5)  = √35

3. . Put steps 1 and 2 together                                   28√35

4.  Answer                                                                 28√35

Ex:2 (a) Steps to multiplying radicals √(4 + 6) . √(4 - 6)

Sol:

√(4 + 6) . √(4 - 6)       (given)

= √(42 – 62)

= √(16 – 36)

= √-20                  (Simplify radicals if possible )

= 2√-5                           (Answer)

(b) Steps to multiplying radicals √(3 + 2) . √(3 - 2)

Sol:

√(3 + 2) . √(3 - 2)           (given)

= √(32 – 22)

= √(9 -4)                  (Simplify radicals if possible )

= √5                                  (Answer)

Examples on rule:

(a) Steps to multiplying radicals  (7√3)2

Sol:

(7√3)2          (given)

= 72 . 3

= 49 . 3

= 147                  (Answer)

(b) Steps to multiplying radicals  (3√2)2

Sol:

(3√2)2            (given)

=  32 . 2

=  9 . 2

=  18               (Answer)

Sunday, December 16, 2012

Standard Deviation Percentage

 The standard deviation is  most commonly used term in statistics. The relative standard deviations to consider the accuracy of compute the standard deviation of given analytical data. The Standard deviation is the square root of average squared deviation from the mean.

Standard Deviation  :

Standard  deviation is the arithmetic mean of all the deviation of observations taken about their mean

Standard equation When each of the given terms has frequency 1  
Let x1, x2, …, xn be the n given observations and let M be their mean. Then, the variance σ2 is given by
σ2 =[ (x1  M)2 + (x2  M)2 + … + (xn – M)2 ] / n = Σ d2i /n,
where the deviation from the mean, di = (xi  M)
And, therefore, the standard deviation σ is given by
σ = + √{Σ(xi  M)2/n} = √Σ di2/n, where di = (xi – M)

Formula to Calculate Percentage of Standard Deviation:


When frequencies of the variable are given
In this case, the variance is given by
σ2 = (Σ fi di2 fi) & S.D.= σ = √(Σ fi di2)/n
we proceed in same way  as we have done earlier
but here each di is multiplied by correponding fi
and apply the above formula
and we get standard deviation for frequency distribution

Ex :  Find the variance and standard deviation from the following frequency distribution table:

Variable (xi)246810121416
Frequency (fi)445158545

Sol :  We have

Variable
xi
Frequency
fi
fi xi_ 
di = (xi – M)
di2fi di2
248– 749196
44`16–     525100
6530–  3945
815120–  1115
10880118
125603945
14456525100
16580749245
Σ fi = 50Σ fi xi = 450Σ fi di2 = 754

... M =
450/50
= 9

... Variance, σ2 = Σ fi di2/ Σ fi = 754/50 = 15.08
And, standard deviation, σ = √15.08 = 3.88

Percentage of standard deviation:

Percentage of standard deviation or relative standard deviation = (standard deviation / mean)  x 100.

Calculating Percentage of Standard Deviation:

Calculate the variance as well as the standard deviation percentage of the given table of the data:

xi7101216182528
fi251310641

Solution:
Presenting the data in tabular form, we get

xififixi(xi - mean)(xi - mean)2fi(xi - mean)2
7214-864128
10550-525125
1213156-39117
16101601110
1861083954
25410010100400
2812813169169
416161003


Here, N= 41 and `sum_(i=1)^7` fixi  = 616.

Therefore, mean = (`sum_(i=1)^7` fixi ) `-:` N = (1/41) x 616   =  15



and                `sum_(i=1)^7`fi(xi - mean)2   = 1003.

Hence, Variance(σ2)  =  (1/ N) `sum_(i=1)^7` fi(xi - mean)2  =  (1/41) x 1003   =  24.46

and

Standard deviation (σ) = `sqrt(24.46)`   =  4.94

Relative standard deviation or standard deviation percentage = (σ / mean) x 100  =  (4.94 / 15 ) x 100     =   32.9%

Wednesday, December 12, 2012

Division of Decimal Numbers

There are some rules of dividing the decimal numbers. Different methods of division are applied in dividing the decimal numbers.

(1)   Division of a decimal by a whole number:

Rules: (a) Divide as in division of numbers.

(b) When you reach the tenth digit, place the decimal in the quotient.

When the number of digits in the dividend is less and the division is not complete, keep adding zero at every step till the division is complete.

Example:  (1) 100.4 ÷ 25

25)100.4(4.016

100_
40-------------- Put zero

25_
150------------ Put zero

150
X

(2)   1.2 ÷ 25                                    ‌‌‌

25)1.20(0.048

100_
200               Since 12 is not multiply of 25, so 0 is added.

200_
Xx

(2)   Division by decimal numbers by 10, 100, 1000 etc.



Rules: (a) While dividing a decimal by 10, 100 or 1000 etc, multiples of 10 the decimal shifts to the left by as many places as there are zeros in the divisor.

(b) If the number of places in the integral part is less, then put the required number of zeros to the left of the integral part, then shift the decimal point.



Example: (a) 71.6 ÷ 10



=       716    ÷10
          10

           716      x    1_
=        10           10



=      716
       100

= 7.16 Ans.



(b) 923.07 ÷ 100



92307 ÷ 100

100

=      92307      x     1__
         100              100

=         92307
           10000

=        9.2307 Ans.

Rules of Division of Decimal Numbers Continued

(3)   Division of decimal numbers by multiples of 10, 100, 1000 etc.

Example: 245.1 ÷30

3)245.1(81.7

24___
x  5

3__
21

21_
X                  so, 245.1 ÷ 30 =8.17



245.1

30           = 245.1 =    245.1     x     1_

3x10        3               10

=   81.7 x    1

10       =   8.17 Ans.

(4)   Division of a decimal by a decimal:

Example: 14.7 ÷ 2.1

147 ÷ 21

10    10

=   147   x   10

10         21

= 7 Ans.

Alternative method: (a) count the number of decimal digits in the divisor.

(b) Move the decimal in dividend that many places to right.

(c) Write the divisor without the decimal.

(d) Now divide the numbers as usual.

(5)   Division of a whole number by a decimal:

Rules: (a) Count the number of decimal digits in the divisor.

(b) Add as many zeros to the dividend.

(c) Remove the decimal in the divisor.

(d) Divide as usual.

Example: 42 ÷ 0.7

42.0 ÷ 0.7

= 420 ÷ 7 = 60 Ans.

Exercise of Decimal Number Fractions

Divide: (1) 234.65 by 25

(2) 193.92 by 800

(3) 8.16 by 0.24

(4) 68 by 4.25

(5) 44 by 176

Answer:  (1) 9.386 (2) 0.2424 (3) 34 (4) 16 (5) 0.25

Sunday, December 9, 2012

Trig Equation Solving Examples


Trigonometric equations is shortly called as trig equations.  An equation involving trigonometrical function is called a trigonometrical equation.

cosθ =`1/2` , tanθ = 0, cos2θ − 2sinθ =`1/2 `

There are some examples for trigonometrical equations. To solve these equations we find all replacements for the variable θ that make the equations true. A solution of a trigonometrical equation is the value of the unknown angle that satisfies the equation. A trigonometrical equation may have infinite number of solutions. The solution in which the absolute value of the angle is the least is called principal solution. In this article let us study trig equations solving examples.

Trig Equation Solving Examples:

Let us see sample problems for trig equation solving examples.

General solutions of sin θ = 0 ; cosθ = 0 ; tan θ = 0

Find the principal value of the following:

(i) cosx =`sqrt3/2`

Solution: (i) cosx =`sqrt3/2` > 0

∴ x lies in the first or fourth quadrant. Principal value of x must be in[0, π]. Since cosx is positive the principal value is in the first quadrant

cosx =`sqrt3/2` = cos`pi/6` and `pi/6`

∈ [0, π]∴ The principal value of x is `pi/6` .

(ii) cosθ = −`sqrt3/2` < 0

Since cos θ is negative, θ lies in the second or third quadrant. But the  principal value must be in [0, π] i.e.  Within  1st  or  2nd  quadrant. The principal value is in the 2nd quadrant.

cosθ = −`sqrt3/2` = cos (180° − 30°) = cos150°.

The principal value is θ = 150° =`(5pi)/6` .

Trig Equation Solving Examples:

Projection formula

In any triangle ABC  a = b cos C + c cosB

True with usual notations and it is called projection formula.

Proof:

In triangle ABC, draw AD perpendicular to BC.From the right angled triangles ABD and ADC,

cosB =BD

AB ⇒ BD = AB × cosB

cosC =DC

AC ⇒ DC = AC × cosC

But BC = BD + DC = AB cosB + AC cosC

a = c cosB + b cosC

or a = b cosC + c cosB


Solve : sin2x + sin6x + sin4x = 0

Solution:

sin2x + sin6x + sin4x = 0 or (sin6x + sin2x) + sin4x = 0 or 2sin4x. cos2x + sin4x = 0

sin4x (2 cos2x + 1) = 0

when sin4x = 0 ⇒ 4x = nπ or x =`(npi)/4` ; n ∈ Z

When 2 cos2x + 1 = 0 ⇒ cos 2x =− `1/2`

= − cos`pi/3` = cos (π −π3)= cos`(2pi/3)`

∴ 2x = 2nπ ±`(2pi)/3` or x = n π ±`pi/3`

Hence x =`(npi)/4` or x = nπ ±`pi/3`

; n ∈ Z

Please visit this website and Know more on Derivative of Cosine .


Wednesday, December 5, 2012

The Number -‘One followed by Six zeros’



The number system started with the natural number system .Now we have huge numbers
being represented. They can be either positive or negative. They can be fractions. They can
be rational or irrational. The natural numbers start with 1 and extend up to infinity. One
million is also natural number. One million is nothing but 10 lakhs. It is the number of zeros
that follow ‘1’ that is important. The number of zeros decides the value of the number.
In ten lakhs, we have six zeros. In ten we have one zero. In hundred two zeros, thousand
three zeros, ten thousand four zeros, lakh five zeros and finally in ten lakhs six zeros. So,
the number of zeros is very important. Now we will try to write 1 million in numbers to
understand this. It comes after the number 999999 in the natural number system. It is also
the number which comes before the number 1000001 in the natural number system. So,
we now understand how to write 1 million after coming to know its position in the natural
number system. Expressing  million in numbers is quite easy and counting up to a million can
be a difficult task.

In the modern era we use the term a million rather than saying ten lakhs or thousand
thousands to represent the same number. Using this term has become very easy. So, we
need to know how to write 1 million in numbers as this term is very commonly used. The
word million is derived is one of the most beautiful languages in the world, Greek. This
shows how to write one million in numbers and can be clearer if we understand it to be just
a number on the natural number system number line.

Just studying a million is not enough we need to study its multiples too. If we do that
we can be clearer with the concepts. So, how to write millions in numbers if there are
numbers greater than a million. We now come to the concept of a billion. In a billion there
are nine zeros. One is followed by nine zeros to form a billion. This shows that there are
1000 millions in a billion. A billion is also known as a milliard. The natural number system
further increases to trillion, quadrillion and so on. A trillion consists of a one followed by 12
zeros.These are very large numbers.