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.

Derivative Math Problems

In math, differentiation is the process of finding the derivative which means measuring how a function changes with respect to its input. The derivative of y with respect to x is given by`(dy)/(dx)` . The reverse process of derivative is antiderivative. Following is the example and practice problems of math derivative which helps you for learning derivative in math.

Example Problems of Math Derivative:

Learn math derivative with Example problem 1:

Find the derivative of the function y = 7x8 + 2x6 + 5x5

Solution:

Step 1: Given function

y = 7x8 + 2x6 + 5x5

Step 2: Differentiate the given function y = 7x8 + 2x6 + 5x5 with respect to ' x ', to get `(dy)/dx`

`(dy)/dx` = 56x7 + 12x5 + 25x4

Learn math derivative with example problem 2:

Find the derivative of the function y = sin (7x4 + 1)

Solution:

Step 1: Given function

y = sin (7x4 + 1)

Step 2: Differentiate the given function y = sin (7x4 + 1) with respect to ' x ', to get `(dy)/dx`

`(dy)/dx` = [cos(7x4 + 1)] (28x3)

= 28x3 cos (7x4 + 1)

Learn math derivative with example problem 3:

Find the derivative of the function y = ecos(9x - 14)

Solution:

Step 1: Given function

y = ecos (9x - 14)

Step 2: Differentiate the given function y =  ecos (9x - 14) with respect to ' x ', to get `(dy)/dx`

`(dy)/dx` = ecos (9x - 14) (- 9sin (9x - 14))

= - 9 ecos (9x - 14) sin (9x - 14)

Practice Problems of Math Derivative:

A few practice problems are given below with solutions which gives you idea how to differentiate the function.

1) Find the derivative of the function y = 16x2 + 11x + 5

2) Find the derivative of the function y = - 5tan (7x2 + 2)

3) Find the derivative of the function y = `6/5` e sin (5x - 9)

Solutions:

1) 32x + 11

2) - 70x sec2 (7x2 + 2)

3) 6e sin (5x - 9) cos (5x - 9)

Solving Equations with Exponents

Introduction:

Exponent equations are the equations in which variable appear as an exponent.

To solve these equations rules and laws of exponents are used. Exponent equations are of two types
(1) Exponent equations in which bases are same
(2) Exponent equations in which bases are different.

Steps to Solve Equations with Exponent

Solving Exponential Equations of the same base

1) Ignore the bases, and simply set the exponents equal to each other
2) Solve for the variable

 When the bases of the terms are different

1) Ignore the exponents; rewrite both of the bases as powers of same number.
For example if there are 2 and 4 in the bases, then convert base 4, in to base 2
by writing it again as (2)^2
2) once the bases are same , ignore them
3) Equalize the exponents
4) Solve for variable

Simple Problems of Equations with Exponents

Solve for variable Answer 1. 3m  =  35 Since the bases are the same, set the exponents equal to one another: m = 5 2. 5t   = 125 125can be expressed as a power of 5: 5t  = 53 t = 3 3.  493y=343 49 and 343 can be expressed as a power of 7: [(7)2]3y = 73 76y = 73 6y = 3 y = 1/2

More Problems of Equations with Exponents

Solve for x. Answer 1.  52x+1  =  53x-2 Since the bases are the same, set the exponents equal to one another: 2x + 1 = 3x - 2 3 = x 2.  32x-1  = 27x 27 can be expressed as a power of 3: 32x-1  = 33x 2x - 1 = 3x -1 = x 3.   43x-8  = 162x 16 can be expressed as a power of 4: 43x-8= [(4)2]2x 3x - 8 = 4x   -8 = x

Units of Meters

In this article discuss about standard units (international system) of meters. The meter (or meter), symbol m, is the base unit of length in the International System of Units (SI).The meters is the length of path travelled by light in vacuum during a time interval of 1/299792458 of a seconds.

The Basic Units of Length is Meters (m):

Linear measures     Short form of writing

Millimeter                         mm

Centimeter                         cm

Decimeter                          dm

Meter                                m

Decameter                         dam

Hectometer                         hm

Kilometer                           km

Some examples of units values:

1 mm = 0.001 m

1 cm = 0.01 m

1 dc = 0.1 m

1 km = 1000 m

Units of Meters - Examples:

Units of meters - Example 1:

Convert 378.6 cm into meter

Solution:

Here the conversion is from centimeter to meter. i.e. from lower unit to higher unit

100cm = 1m.

Hence 378.6cm = 378.6/100 m

= 3.786 m (shifting the decimal two digits to the left)

Units of meters - Example 2:

Convert 40.1735 km into meter.

Solution:

Here the conversion is from higher to lower. Hence we have to shift the decimal point to the right.

1km = 1000 m

40.1735 km = 40.1735 × 1000 m

= 40173.5 m

Units of meters - Example 3:

Convert 6m into millimeter.

Solution:

Note: Here we are converting the unit meter into millimeter. i.e. from higher unit to lower unit. So the operation should be multiplication.

1 m = 1000 mm

Hence 6 m = 6 × 1000 mm

= 6000 mm

Units of meters - Example 4:

Convert 778.6 cm into meter

Solution:

Here the conversion is from centimeter to meter. i.e. from lower unit to higher unit

100cm = 1m.

Hence 778.6cm = 778.6/100 m

= 7.786 m (shifting the decimal two digits to the left)

Units of meters - Example 5:

Convert 90.1736 km into meter.

Solution:

Here the conversion is from higher to lower. Hence we have to shift the decimal point to the right.

1km = 1000 m

90.1736 km = 90.1736 × 1000 m

= 90173.6 m

Units of meters - Example 6:

Convert 9m into millimeter.

Solution:

Note: Here we are converting the unit meter into millimeter. i.e. from higher unit to lower unit. So the operation should be multiplication.

1 m = 1000 mm

Hence 9 m = 9 × 1000 mm

= 9000 mm.

Convert Fractions to Decimals

How to convert fractions to decimals with calculator ?

In this article, let us learn what is fraction and what is decimal and how to convert fractions to decimals using calculator.

Fractions: Let us divide a circle into four parts and take a part away and ask yourself a question,"How many parts are taken away from the whole ?" It is one part from 4. we could represent this as 1/4. So, fraction is a number composed of two parts say top part which is called as numerator and the bottom part is called as denominator.So, numerator gives the required part whereas denominator part gives the number of parts in the whole.

Example: `(15)/(17)`

Decimals: Decimals are numbers which has a dot which we say it as decimal point.

Example: 0.234, 67.45

Convert Fractions to Decimal( Relation)

We could observe that 0.5 is in 1/10th position. It means that 0.5 = 5/10

We could observe 9 in 0.59 is in 1/100th position. It means that 0.59 = 59/100

We could observe 1 in 0.591  is in 1/1000th position. It means that 0.591= 591/1000

If any fraction has the denominator has 10, 100, 1000or any multiples of 10  then the fraction is called as decimal fraction.

Convert Fractions to Decimal( Examples)

Consider a fraction `(1)/(4)`

Let us see how to convert the above fraction to decimal using calculator.

Choose a number such that the number when multiplied with denominator becomes a number which is a multiple of 10.So, i am choosing the number 25 since 25 x 4 = 100

Multiply the number 25 with the numerator and denomiantor.

So, 1 x 25 = 25 and 4 x 25 = 100

1/4 = 25/100

Since 100 has two zeros keep decimal point after counting 2 numbers from the right.

So, 25/100 = 0.25

In some case, we could not make the denominator as multiples of 10. For eample, 2/3. We could not find a number such that when denominator multiplied by the number gives denominator as multiples of 10.But 3 x 333 = 999

Since 999 is nearest to 1000, multiply the numerator and denomiantor with 333. So,

2/3 = 2 x333 / 3 x 333 = 666/999 = 0.666 (approximately)

Otherwise, to convert fractions to decimals with calculator, enter the numerator in the calculator and press the division sign and then the denominator. The resulting number in the calculator is the decimal.

Solve Rational Equation Problems

In algebra solving rational equation is very simple,we have few rules to solve any type of equations. Whatever we do on  one side  of the equation, we must do to the other side also. If you have fractions, we can try to eliminate them by multiplying by the common denominator. If there are quadratics involved in our equations, we must get all the  terms to one side with zero on the other.The basic rational expression is in the form of fraction.where there is at least one variable in the denominator.

Solved Example Based on Rational Equation :

Ex 1:Solve `3/x+6=2/(4x)`

Sol:

Step 1: The given equation is

`3/x+6=2/(4x)`

subtracting both sides by `2/(4x)`

`3/x+6-(2/(4x))=(2/(4x))-(2/(4x))`

Step 2: Rearrange the equation.

`3/x-2/(4x)+6=0`

subtracting both sides by 6.

`3/x-2/(4x)+6-6=0-6`

`3/x-2/(4x)=-6`

Step 3: Find the l.c.d(least common denominator) x and 4x

L.C.D=4x

`(12-2)/(4x)=-6`

`10/(4x)=-6`

Step 4:Multiply both sides by 4x.

`10/(4x)xx4x=-6xx4x`

10=-24x

Step 5:Divide both sides by -24

`10/(-24)=(-24x)/(-24)`

`x=-5/12`

The solution is  [x=-5/12]

Example Based on Rational Equation:

Ex 2: Solve `x/(x-2)+1/(x-4)=2/(x^2-6x+8)`

Sol:

Step 1:first factor the `x^2-6x+8`

factors=(x-4)(x-2)

Step 2:convert common denominator to all

`(x/(x-2))((x-4)/(x-4))+(1/(x-4))((x-2)/(x-2))=2/((x-2)(x-4))`

`(x^2-4x)/((x-2)(x-4))+(x-2)/((x-2)(x-4))=2/((x-2)(x-4))`

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

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

`x^(2)-3x-4=0`

`(x-4)(x+1)=0`

`x=4 or x=-1`

If we submit the x=4 in the denominator,it is division by zero,so x=4 is not considerable.

x=-1   is the answer

Positive Integers Tutoring

Tutoring means a tutor taking an interactive session to individual student or a group of student in a class through online. The learning integer is defined the equal to the whole number. The negative numbers are including in the learning integer tutor. The integers do not used the fraction number. Each positive integer contains same negative integers. Let us discuss about the positive integers tutoring.

Positive Integer Tutoring

The general structure of integer is {…, -4, -3, -2, -1, 0, 1, 2, 3, 4…}. The integer is commonly representing the three types of numbers.

The first type is positive counting number. The second type is negative counting number. The zero value is third type.

The positive integer is equal to the whole number. The general structure of the positive integer is {1, 2, 3, 4, 5…}. The positive integer is also called as the positive number counting. This number is specifying positive value only. The + is the sign of the positive integers.

The value +86 and 86 is representing the same value. The positive integers do not necessary for the sign representation. The value 55 is automatically representing the positive integer.

The non negative integer is equal representation of the positive integer. The main difference of the positive integer and non negative integer is the positive integer starts with 1 but the non negative integer starts with 0. Many operations are performing the positive integer tutoring.

Example Problem of Positive Integer Tutoring

Problem 1:

Calculate the following positive integer.

2415 + 1564 =?

Solution

2415
1564(+)
--------
3979
---------

Problem 2:

Calculate the following positive integer.

4698 - 2456 =?

Solution

4698
2456 (-)
--------
2242
---------

Practice problem of positive integer tutoring

1. Add the positive integers 568 + 637.

2. Subtract the integers 634 – 542.

Answer

1. 1205

2. 92

Theory of Proportions

Introduction to theory of proportions:

       The theory of proportion is one of the basic topic in mathematic. In our usual life, there are a lot of occasions as we compare two quantities by means of their measurements. As soon as we compare two quantities of the same type by division, we contain a ratio of those two quantities.

Definition of Ratio: Ratio means similarity of two like quantities by division.

Definition of Theory Proportions:

      Proportions is a correspondence of two ratios.

      Consider the proportions

                a: b = c: d

      The first and fourth terms (a and d) are knoen as the extreme terms or extremes.

      The second and third terms (b and c) are known as the middle terms or means.

Important property:

      Product of extremes = Product of means.

Examples of Theory Proportions:

Let us see some examples of theory of proportions.

Example 1:

      Verify 5: 6 = 10: 12 is a proportion or not.

Solution:

      Product of extremes = 5*12 = 60

      Product of means = 6*10 = 60.

             `:.`   60 = 60

       These two products are equal.

            `:.` 5:6 = 8: 6 is a proportion.

Example 2:

       Verify 6: 7 = 12: 15 is a proportion or not.

Solution:

       Product of extremes = 6*15 = 90

       Product of means = 7*12 = 84

                          `:.`    90 = 84

       These two products are not equal.

         So, 6: 7 =12: 15 is not a proportion.

Example 3:

       If 2: 3= 6:_ is a proportion, find the missing term.

Solution:

       Let us assume the missing value is x

       Product of extremes = 2*x

       Product of means = 3*6 =18

       Since it is a proportion, 2*x =18

                                          2x = 18

       Divide both sides by 2 on both sides we get,

                                        2x/2 = 18/2

                                             x = 18/2 = 9

       `:. ` The missing term is 9

       So the proportion is 2:3 =6: 9

Example 4:

       The income and Savings of a family are into the ratio 8: 3 If the income of the family is Rs. 3,300.Find how much is being saved.

Solution:

       Let us savings be Rs. x.

          `:.` The proportion is 8: 3 = 3300: x

                     (Income: saving) = (Income: saving)

                                       11x = 9900

                                   11x/11= 9900/11

                                           x= 900

      Therefore, the savings = Rs. 900.

Step by Step Adding Fractions

Introduction to step by step adding fractions:
                  A fraction is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

                                                                                                                                                 

Step by Step Adding Fractions:

1. Add the two fractions `20/3` and `22/4`

         Solution:

                                   The  two fractions are  `20/3` and `22/4`

                 Step1:             Add two fraction

                                              =`20/3` +`22/4`

                 Step2:          There the denominator are different  so we need to take lcm  to preform addition

                                                           =`(80+66)/12`

                  Step3:           By adding 80 and 66 the answer is 146

                                                         =`146/12`

                   Step4:                This can be simplified has

                                                           =12.1

 2. Add the two fractions `40/4` and `50/5`

            Solution:

                                  The  two fractions are `40/4` and `50/5`

                   Step1:                 Add two fraction

                                                     =`40/4` +`50/5`

                   Step2:     There the denominator is sameso we need to take lcm  to preform addition

                                                            =`(200+200)/20`

                   Step3:       By adding 200 and 200 the answer is 400

                                                                =`400/20`

                   Step4:           This can be simplified has

                                                       = 1

 3. Add the two fractions `60/5` and `70/6`

          Solution:

                                  The  two fractions are `60/5` and` 70/6`

                    Step1:          Add two fraction

                                                  =`60/5` +`70/6`

                    Step2:   There the denominator is same so we need to take lcm  to preform addition

                                                        =`(360+350)/30`

                    Step3:     By adding 360 and350 the answer is 710

                                                              =`710/30`

                    Step4:        This can be simplified has

                                                              = 23.66

Step by Step Adding Fractions:

4. Add the fractions `5/6` and `25/30`

      Solution:

                                        The two fractions are `5/6` and `25/30`

                 Step1:         The given Two fractions are equivalent fractions

                                         By simplifying `25/30` we get 5/6

                  Step2:

                                          Now we need to find the sum of `5/6` and `5/6`

                                                                   = `5/6` +`5/6`

                   Step3 :            There the denominator are same so add the numerators in the fractions together

                                                                   =`(5+5)/6`

                                              The sum of 5 and 5 is 10

                                               So `(5+5)/6` can be written as `10/6`

                  Step4:

                                             This can be reduced further as `5/3`

5. Add the two fractions `80/6` and `90/6`

        Solution:

                                     The  two fractions are `80/6` and `90/6`

                  Step1:             Add two fraction

                                                =`80/6` +`90/6`

                   Step2:       There the denominator is same so add the numerators in the fractions together

                                                            =`(80+90)/6`

                      Step3:        By adding 80 and 90 the answer is 170

                                                          =`170/6`

                      Step4:          This can be simplified has

                                                        = 28.33

 6. Add the two fractions `1000/3` and `120/4`

         Solution:

                                  The  two fractions are `100/3` and `120/4`

                Step1:           Add two fraction

                                         =`100/3` +`120/4`

                Step2:     There the denominator is same so we need to take lcm  to preform addition

                                           =`(400+3690)/12`

                Step3:      By adding 400 and 360 the answer is 760

                                                  =`760/12`

                Step4:        This can be simplified has

                                                  = 63.33