Basic Scientific Method

In this article shall we discuss about basic scientific method. Basic scientific method explains to the organization of system to examine fact, attain new information, or precise and integrate preceding information. To be appearance scientific, a method of question have to be based on gathering visible, experimental and computable proof subject to precise principles of analysis. Scientific methods include the set of numbers through study and testing, and the formulation and testing of theory.



Steps for Basic Scientific Method:

A step for basic scientific method is given below that:

The basic scientific method is an approach to ask and answer scientific questions through creation comments and doing experiment.

The steps for basic scientific method are:

Inquire a Question

Perform Background Research

Build a Hypothesis

Observe Your Hypothesis through Doing an Experiment

Observe Your Data and sketch a Conclusion

Communicate your outcome

It is significant for your explore to be a fair test. A "fair test" occurs when you modify just one factor (variable) and stay all other situation the same.

Basic scientific method
Overview for basic scientific method:

The basic scientific method for testing that is used to find out comments and answer questions. Scientists use the scientific method to seem for reason and result relations in nature. In other words, they plan a research so that changes to one thing cause something else to vary in an expected way.

Now as it does for a specialist scientist, the scientific method will help you to center your science fair scheme question, construct a hypothesis, plan, perform, and estimate your experiment.




Application of Basic Scientific Method:

The application of basic scientific method is given below that:

How many facts do you know about the sun?

Mass: 3.28 x 1027 tons

Temperature: 2.7 x 106 degrees Fahrenheit

Energy generated per minute: 3.5x 104 horsepower

Application of solving basic scientific method:

All the numbers now are writing as products of a number between 1 and 10 and a suitable power of 10. This is also denoted as scientific method. When written in standard method, these numbers are

3,280,000,000,000,000,000,000,000,000

2,700,000

35,000

Midpoint in Geometry

The midpoint in geometry is defined as the centre point equally divides any line. The distance formula can be used to verify or derive a formula for finding the coordinates of the midpoint of a segment when the coordinates of the endpoints are known. These midpoint are used in the applications of finding some terms in the ellipse, triangles etc,.

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Formula for Midpoint in Geometry:

The midpoint formula is given by: M=( `(x1+x2)/(2)`  , `(y1+y2)/(2)`   )

Where:

x1 and y1 - coordinates of first vertices.

x2 and y2 - coordinates of second vertices.

`(x1+x2)/(2)`  = mid-point of x-co-ordinates

`(y1+y2)/(2)`   = mid-point of y-co-ordinates.
More Usage on Midpoint in Geometry:

Find the equation of the ellipse whose foci are (1, 0) and (− 1, 0) and eccentricity is 1/2 .
Solution:
The centre of the ellipse is the midpoint of FF′ where F is (1, 0) and F′ is (− 1, 0).

∴ Centre C is (`(1-1)/(2)` , `(0+0)/(2)` ) = (0, 0)
But F1F2 = 2ae = 2 and e = 1/2

2a ×1/2 = 2

a = 2
b2 = a2 (1 − e2) = 4 (1- `(1)/(4)` ) = 3
From the given data the major axis is along x-axis.
∴ the equation of the ellipse is of the form

`(x^(2))/(4)` +  `(y^(2))/(3)` = 1

This is how the midpoint of the ellipse is used in finding the equation of the ellipse in geometry.

My forthcoming post is on Find the Probability and free algebra 2 solver will give you more understanding about Algebra.

Example:

For a line segment one endpoint is (-18, 10) and the midpoint of (- 2, 6). Find the coordinates of the other endpoint (x1, y1).

(-2 , 6) = ((x1 + (-18))/2 , (y1 + 10)/2)

(-2, 6) = ((x1 - 18)/2 , (y1 + 10)/2)

To Solve for X1                                  To Solve for y1

-2 = (x1 - 18)/2                                     6 = (y1 + 10)/2

-4 = x1 - 18                                          12 = y1 + 10

x1 = 14                                                  y1 = 2

The coordinates for the other endpoint are (14, 2).

This is how the midpoint of the ellipse is used in finding the end point in geometry.

Largest Number

Let us study about the largest number. When we enter into the world of mathematics we can come to know that the numbers rule that world.
The various kinds of numbers with their different combinations form this math world. These numbers can be classified into two types basically as smallest and largest numbers.
The largest numbers are nothing but the largest values among the given ones. Some of the examples are given below.

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Largest number – example 1:

Arrange the following number series and find out the largest number: 524, 32, 43, 254, 234, 74, 930, 157, 338, 99, 86, 50, 290, 98 and 78.



Solution:

The given number series is as follows: 524, 32, 43, 254, 234, 74, 930, 157, 338, 99, 86, 50, 290, 98 and 78.
Do comparing each and every number of the series with the other numbers and arrange them from the largest numbers to the smallest numbers order as follows:
32 < 43 < 50 < 74 < 78 < 86 < 98 < 99 < 157 < 234 < 254 < 290 < 338 < 524 < 930
32, 43, 50, 74, 78, 86, 98, 99, 157, 234, 254, 290, 338, 524 and 930
Therefore the largest number in the series is found to be as ‘930’ among all other numbers.



Largest number – example 2:

By solving the following number series (4 * 4), (10 - 1), `(9/3)` , (16 + 7), (11 + 9), (12 – 8), (25 – 11) and find out the large number among them.



Solution:

The given number series is (4 * 4), (10 - 1), `(9/3)` , (16 + 7), (11 + 9), (12 – 8), (25 – 11)
The steps that are carried out to solve the given number series and to find the largest number among all others are as follows:

(4 * 4), (10 - 1), `(9/3)` , (16 + 7), (11 + 9), (12 – 8), (25 – 11)

(16), (9), (3), (23), (20), (4), (14)

Arrange all the numbers after solving in the series as from the largest numbers to the smallest numbers as follows:

23, 20, 16, 14, 9, 4, and 3

So by solving the given number series the largest number is found to be as ‘23’ among all other numbers.

I am planning to write more post on Rounding Calculator and Radical Calculator. Keep checking my blog.

Largest number – exercises:

Give the large number that gets divisible by ‘13’ between 100 and 200. (Answer: 195)
Give the large number that gets divisible by ‘21’ between 200 and 300. (Answer: 294)
Give the large prime number between 900 and 1000. (Answer: 997)

Axis of Symmetry Equation

The most interesting part of High-school mathematics is probably the visualization of the problem. The intrinsic part of learning mathematics is knowing the Function! Functions can be visualized by putting them in a graph in the form of equations. The axis of symmetry equation is such a graph in which there is a line which is the axis of symmetry of the given function. The line of symmetry acts as a mirror to one part of the graph. Thus we can call the axis of symmetry equation function, the mirror function!
Visualization of Axis of Symmetry Equation

Let us now visualize the simplest axis of symmetry equation: y = |x|;

For this function,

y = IxI

So, our graph would be: mod

So, we can see that the y-axis acts just like a mirror in this function!

Now, let us take the Quadratic equations. At a glance, this wouldn't have anything to do with the Absolute Value function but if we take a little deeper look, we will find that all quadratic equations can be expressed by the modulus function, i.e. it is again an axis of symmetry equation.

Let us take the example y = x2 (a parabola). This equation is the same as y = | x2 | because any squared number is always positive!

The graph of this equation would be: axis of symmetry

So, once again, we can see that the y-axis acts just like a mirror in this function. So the equation is an axis of symmetry equation!

Now, if the vertex is shifted to (2, 1) (say), the equation would be : y - 1 = (x - 2)2

The graph of this equation would be: axis of symmetry

Having dealt with the axis of symmetry equations where axis is either y-axis or a line parallel to y-axis, we can easily understand that the same thing will happen if the axis is x-axis or a line parallel to the x-axis if the form of the equation becomes : x = | f(y) |;

For example the parabola, y2 = x(i.e. x = |y2|) will have the x-axis as the "mirror"!

The graph of this equation would be:axis of symmetry

Here, we can see that the x-axis acting just like a mirror in this function. So the equation is an axis of symmetry equation!

There can be other axis of symmetry equations where the axis of symmetry can be inclined to the x and y axes at different angles. Let's take the case of the functions, y = ln x and y = ex. If we plot these curves, the line y = x acts as the axis of symmetry to these equations.

Here we see the axis of symmetry equationinclined axiswith inclined symmetry
Axis of Symmetry Equation-real Life Examples

There can be numerous examples of the axis of symmetry equations, in fact if you stand in front of the mirror and your shape can be defined by any function, your image and you, together can be described by the axis of symmetry equation or the modulus of that function!!!! If we look around, we'd just have to appreciate mathematics for giving us wonderful analogies of the real world! That is the beauty of Mathematics!!!

Solve Precalculus Inequalities

Precalculus inequalities are  statement that one algebraic expression is greater than (or less than) another algebraic expression is called as precalculus inequalities.

There are three rules for preparation algebraic inequality:

The same quantity can be added or subtracted from each side of inequalities.

Each side of inequalities can be multiplied or divided by the same positive quantity.

If each side of an inequality is multiplied or divided by the same, negative quantity is equivalent to the first.

Properties for Solve Precalculus Inequalities:

Trichotomy property for solve precalculus inequalities:

For any real number, a and b, exactly one of the following statements is true:

pq

The trichotomy property indicates that exactly one of the following statements is true about any two real numbers. Either

The first is less than the second

The first is equal to the second, or

The first is greater than the second one.

Transitive property for solve precalculus inequalities:

If p,q and s are real numbers with p
If p,q and s are real numbers with a>b and b>c, then a>c.

The first part of the transitive property indicates that:

If a first number is less than a second number and the second is less than a third, then the first number is less than the third.

The second part is similar, with the word “is greater than” substituted for “is less than”.

Addition property for solve precalculus inequalities:

Any real number can be added to (or subtracted from) mutually sides of an inequality to produce another inequality with the same direction.

To solve the addition property, we add 3 to both sides of the inequality 1<2 get="get" p="p" to="to">
3+1<3 p="p">
4<5 p="p">
We note that the < sign is unmovable (has the similar direction).

Subtracting 5 from both sides of 15<5 change="change" direction="direction" does="does" either.="either." inequality="inequality" not="not" of="of" p="p" the="the">
15-5<5-5 p="p">
10<0 p="p">
Algebra is widely used in day to day activities watch out for my forthcoming posts on Simplify Fractions and Volume of a Cube. I am sure they will be helpful.

Multiplication property for solve precalculus inequalities:

If both sides of an inequality are multiple (or divided) by a positive number, another inequality results wilt the same direction as the original inequality.

To solve the multiplication property, we multiply both sides of the inequality 3<5 2="2" by="by" get="get" p="p" to="to">
3(2)<5 p="p">
2<10 p="p">
The < symbol is unaffected.

Dividing property for solve precalculus inequalities:

Dividing both sides by 2 does not change the direction of the inequality either.

`-4/2`<`8/2`

-2<4 p="p">Example for Solve Precalculus Inequalities:

To solve the  precalculus inequality 25≤12y-3<17 p="p">
Solution:

This inequality means that 12y-3 is between 25 and 17. We can solve it by isolating a between the inequality symbols.

25≤12y-3<17 p="p">
28≤12y<20 p="p">
14≤6y<10 p="p">
7≤3y<5 p="p">
The solution set is {y|7≤3y<5 5="5" in="in" interval="interval" notation="notation" or="or" p="p">

Types of Coordinate Systems

The coordinate systems use values or numbers for the location of a point or a shape in the system. And the values or the numbers are called as the coordinates of the point and are related between the various systems either in the two dimensions or three dimensions. In the following article we will see in detail about the topic types of coordinate systems.
More about Types of Coordinate Systems:

There are basically two types of systems like the two dimensional coordinate systems and the three dimensional coordinate systems. The two dimensional coordinate systems are,

1. Cartesian coordinate systems

2. Polar coordinate systems

Some of the three dimensional coordinate systems are,

1. Cartesian coordinate system.

2. Spherical coordinate system.

3. Cylindrical coordinate systems.

Cartesian coordinate system:

cartesian coordinate system

A point in the cartesian coordinate system has the coordinates (x,y,z). Other three dimensional coordinates in terms of Cartesian coordinates,

Spherical coordinates: `r = sqrt (x^2+y^2+z^2)` ; `theta = arccos (z/r)` ; `phi = atan2(y,x)`

Cylindrical coordinates: `rho = sqrt (x^2+y^2)` ; ` phi = arcsin (y/rho)` ; `z = z`

Spherical coordinate system:

Spherical coordinate system

A point in the spherical coordinate system has the coordinates (r, theta, phi). The other three dimensional coordinates in terms of the spherical coordinates,

Cartesian coordinates: `x = r *sin theta*cos phi` ; `y = r *sin theta*sin phi` ; `z = r cos theta`

Cylindrical coordinates: `rho = r*sin theta` ; `phi = phi` ; `z = r*cos theta`

Cylindrical coordinate system:

My forthcoming post is on solving algebraic proportions and how to solve proportion problems will give you more understanding about Algebra.

Points in the cylindrical coordinates have coordinate values (rho, phi, z). The relation between the cylindrical coordinate system and the other three dimensional coordinate systems,

Cartesian coordinates: `x = rho*cos phi` ; `y = rho*sin phi` ; `z = z`

Spherical coordinates: `r = sqrt (rho^2+z^2)` ; ` theta = arccos (z/r)` ; ` phi = phi`

Polar coordinate system:

polar coordinates

Points in the polar coordinate system has the coordinates (r,theta). The relation between the polar coordinates and the two dimensional Cartesian coordinates are,

Cartesian coordinates: `x = r* cos theta` ; `y = r* sin theta`
Example Problem on Types of Coordinate Systems:

1. Find the equivalent Cartesian coordinates for the point with the polar coordinates (6, 50).

Solution:

`x = r* cos theta`

`= 6*cos 50`

`= 6*0.64`

`= 3.8`

`y = r* sin theta`

`= 6*sin 50`

`= 4.6`

My previous blog post was on General Equation of an Ellipse please express your views on the post by commenting.

Practice problem on types of coordinate systems:

1. Find the equivalent spherical coordinates for the location of the point (7.8, 50, 5) in the cylindrical coordinates.

Answer: (9.3, 50, 50)

fundamental laws of algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics and number theory, algebra is one of the main branches of pure mathematics.Source: Wikipedia

I like to share this Limit Laws with you all through my article.

Fundamental Laws of Algebra:

We list the fundamental rules and properties for useful algebra and give examples on they may be used.

Suppose that a, b and c are variables or applied algebraic expressions.

Commutative laws for addition.

a + b = b + a

Example:

5+2=2+5

Commutative laws for multiplication.

a * b = b * a

Example:

6*5=5*6

Associative laws for addition.

(a + b) + c = a + (b + c)

Example:

(6+2)+5=6+(2+5)

Associative laws for multiplication.

(a * b) * c = a * (b * c)

Example:


(5*4)*2=5*(4*2)

Distributive laws of addition over multiplication.

a * (b + c) = a * b + a * c


And


(a + b) * c = a * c + b * c

Example:
2*(5+3)=2*3+2*5

Fundamental Laws of addition in algebra:

For the addition of positive and negative number, the follow rules, established in the First Course, apply make easy to solve algebra equation:

A number represented by a letter is called a literal number, and any number expression in which more than numbers of symbols are variable is called a literal expression.

Laws for fundamental addition in algebra:

1)  To add two numbers with similar signs, find the sum of their total values, and prefix the sign common to both.

2)  To add two numbers with different signs, find the diversity of their total values, and prefix the sign of the one with the larger total value.
Order of Operations for Fundamental Laws of Algebra:

In long math problems with +,-,x,%,(), and exponents in them, you have to know what to do first. Without follow the same rules, you may get unlike answers. You can easily keep in mind the silly sentence, Big Elephants Destroy Mice And Snails, you can commit to memory the order of operations, and you must follow.

Big               “B” means Brackets. You need to do operation in side parentheses first.

Elephants    “E” means an exponent, you must calculate exponents next.

Destroy        “D” means division. Start on the left of the equation and perform all multiplication and divisions in the order in which they appear.

Mice             “M” means multiply

And               “A” means addition.

Snails           “S” means subtract. Always on the left hand side and perform all additions and subtractions.


Example for fundamental laws of algebra:

Solve 12%2(10-6)-2+4 using order of operations.

Solution:

Big                 reduce brackets. 10-6=4                                                                          12%2*4-2+4

Elephants       reduce exponents. We do not have any exponents.                             12%2*4-2+4

Destroy           make divide operation in order from left to right     12%2=6                 6*4-2+4

Mice                carry out multiply operation in order from left to right 6*4=24                24-2+4

And                 do addition operation in order from left to right 24-2+4=24+2=26        26



Answer is 26.