Classifying Math Diagrams

Introduction to classifying math diagrams:

In mathematics we use different type of diagrams. In math, diagrams are classified as per the shape size of the figure. There are different types of diagrams are used in mathematics. In mathematics, diagrams are mainly used in geometrical problem. The diagrams in mathematics gives the clear explanation for particular problem. Classification of diagrams are given below,

• Circle
• Square
• Rectangle
• Rhombus
• Triangle
• Ellipse
• Parallelogram

Classifying math diagrams - Explanation
Classifying math diagrams - Circle:

• Circle has the radius (R), and diameter (d).
• The origin of the circle is indicated as O.
• The Interior angle of the circle is 360°.
• Area of the circle is pr2.
• Circumference of the circle is 2pr.

Classifying math diagrams - Square:


        

• Square has four equal sides.
• The side length of the square is denoted as S.
• Total angle of the square is 360°
• All the four sides have the same angle 90°
• Square has two diagonals and both the diagonals are equal in length
• Area of the square is s2

Classifying math diagrams - Rectangle:

• Rectangle have the four side lengths
• Length and width of the rectangle is denoted as L and W
• Opposite side of the rectangles are equal in length and also have same angle.
• Rectangle has two diagonal length
• Total angle of the rectangle is 360°
• Each side length of the rectangle  has 90°
• Area of the rectangle is (length * width)

Classifying math diagrams - Triangle

• A triangle has three side lengths.
• The sum of angle of the triangle is 180°
• Triangles are classified based on its side length and angle
• In right triangle, one angle should be 90°
• In equilateral triangle, all the three side lengths are equal
• Area of the triangle is (1 / 2) * b * h

Classifying math diagrams - Rhombus:

         

• Rhombus has four equal side length.
• It is also called as diamond.
• It is one of the quadrilateral diagram
• Opposite angles of the rhombus is equal
• Diagonals are perpendicular to each other
• Area of the rhombus is (base * height)

Surd Online

Introduction to surd online

Expressions like `sqrt(16)` ,`root(3)(27)` ,etc. have exact numerical value.They are terminating and rational numbers.These numbers are perfect roots (square root and cube root). But expressions such as `sqrt(2)`,`root(3)(3)` , etc.cannot be written as exact numeric value. Such numbers are called irrational and it is convenient to leave them in the form as in Decimal form  they would go on non-terminating. These are called surds. Surds are numbers of the form `root(n)(k)` ,where k is not a perfect nth power of any number.`root(n)(k)` is a surd of nth order.

Ex 1: `sqrt(2)` is a surd of 2nd order.

`root(3)(3)`is a surd of 3rd order.

`root(4)(5)`is a surd of 4th order

A surd has infinite number of non-terminating decimals.Surds are always irrational.

Ex 2: Which of the following numbers are surds?

`sqrt(7)`
`sqrt(9)`
`root(3)(27)`
`sqrt(5)`
`sqrt(100)`
`root(4)(16)`
Solution :

`sqrt(7)` is a surd.
`sqrt(9)` is not a surd because `sqrt(9)` = 3
`root(3)(27)` is not a surd because `root(3)(27)` = 3
`sqrt(5)` is a surd.
`sqrt(100)` is not a surd because `sqrt(100)` = 10
`root(4)(16)` is not a surd because `root(4)(16)` = 2


Simplifying a surd:

Consider the entire surd `sqrt(18)`  , it has 9 as one of its factors which is a perfect square.So `sqrt(18)` can be expressed as a product of rational number and a surd.

`sqrt(18)`  = `sqrt( 9 * 2)`

= `sqrt(9)` *`sqrt(2)`

= 3 *`sqrt(2)`

= 3`sqrt(2)`  This is called a mixed surd.

Ex 3: Express the mixed surd 4`sqrt(2)` as an entire surd.

Solution:  4`sqrt(2)` =`sqrt(16)`  * `sqrt(2)`       because 4 = `sqrt(16)`

= `sqrt( 16 * 2)`

= `sqrt(32)`


Addition , Subtraction and Multiplication on surd online


Addition and subtraction of surds is simple, however operation can be performed only on surds of same order and the radical must be same.That is number inside the root must be the same.Lets consider an example:

Ex 1: 5`sqrt(11)` + 2`sqrt(11)`  = `sqrt(11)` ( 5 + 2)  = 7`sqrt(11)`

Ex 2: 6`sqrt(5)` - 3`sqrt(5)` = `sqrt(5)` ( 6 - 3) = 3`sqrt(5)`

Ex 3: Simplify 6`sqrt(3)` + `sqrt(75)`

Solution : First simplify `sqrt(75)` .Express it as mixed surd.

`sqrt(75)` = `sqrt(25 * 3)` = 5`sqrt(3)`

Then,    6`sqrt(3)` + 5`sqrt(3)` = 11`sqrt(3)`

Multiplication of similar surds gives rational number.  i.e.   `sqrt(x)` * `sqrt(x)` = x

Ex 4: Simplify (`sqrt(6)`)2

Solution: (`sqrt(6)` )2 = `sqrt(6)` * `sqrt(6)`

= `sqrt(6 * 6)`

=  6

Multiplication of unlike surds give irrational number.

Ex 5: Simplify `sqrt(8)` * `sqrt(3)`

Solution: `sqrt(8)` * `sqrt(3)` = `sqrt(8 * 3)`

= `sqrt(24)`

= `sqrt(4 * 6)`

= 2`sqrt(6)`

Ex 6: Simplify 3`sqrt(5)` * 5`sqrt(3)`

Solution:

Method 1:    3`sqrt(5)` * 5`sqrt(3)` = `sqrt(3 * 3 * 5)` * `sqrt(5 * 5 * 3)`

= `sqrt(3 * 3 * 5 * 5 * 5 * 3)`

= 3 * 5 `sqrt(15)`

= 15`sqrt(15)`

Method 2:  3`sqrt(5)` * 5`sqrt(3)` = 3 * 5 *`sqrt(5)` * `sqrt(3)`

= 15`sqrt(5 * 3)`

= 15`sqrt(15)`

Ex 7: Simplify 2`sqrt(18)` * 3`sqrt(20)`

Solution: 2`sqrt(18)` * 3`sqrt(20)` = 2`sqrt(9 * 2)` * 3`sqrt(4 * 5)`

= 2*3`sqrt(2)` * 3*2`sqrt(5)`

= 6*6*`sqrt(2)` * `sqrt(5)`

= 36`sqrt(2*5)`

= 36`sqrt(10)`


Rationalisation of the denomiator on surd online


We express a radical fraction, such as `sqrt(5)` /`sqrt(2)` in a form that has rational denominator.

Ex 1: Simplify `sqrt(5)` / `sqrt(2)`

Solution: In order to rationalise we multiply numerator and denominator by `sqrt(2)`

`(sqrt(5))/(sqrt(2))`  * `(sqrt(2))/(sqrt(2))`= `(sqrt(5 * 2))/(2)`  = `(sqrt(10))/(2)`

Ex 2: simplify `(sqrt(35))/(sqrt(15))`

Solution: `(sqrt(35))/(sqrt(15))` * `(sqrt(15))/(sqrt(15))`    = `(sqrt(7 * 5 * 5 * 3))/(sqrt(15 * 15))`   = `(5sqrt(21))/(15)`   = `(sqrt(21))/(3)`

Consider (2 + `sqrt(3)`) * ( 2 - `sqrt(3)`)  = 4 - 3 = 1. Their product is a rational number. Hence 2 - `sqrt(3)` is called the conjugate of the surd 2+`sqrt(3)` or vice verse.`sqrt(a)` + `sqrt(b)` is called the conjugate of `sqrt(a)` - `sqrt(b)` if their product is a rational number

Ex 3 : Simplify `(1)/(sqrt(2) + sqrt(3))`

Solution: Here we have to rationalise the denominator.Hence multiply and divide numerator and denominator by `sqrt(2)`-`sqrt(3)`(the opposite sign of the denominator)

We have, `(1)/(sqrt(2) + sqrt(3))` *  `(sqrt(2) - sqrt(3))/(sqrt(2) - sqrt(3))` = `(sqrt(2)- sqrt(3))/(2-3)` = `sqrt(3)` - `sqrt(2)`

Study Online Dilation

Study Online Dilation Introduction:

Dilation is a change (notation Dk) that produces a picture that is the same shape as the original, but is a different size. Dilation stretches or shrinks the original diagram.

The explanation of dilation contains the ratio or scale factor and the middle of the dilation. The middle of dilation is a set point in the plane about which every point are expanded or contracted.  It is the just invariant point under dilation.

A dilation of scalar factor k whose middle of dilation is the basis  written by:  Dk (x, y) = (kx, ky). If the scale factor, k, is larger than 1, the picture is an enlargement.

If the scale factor is 0 to 1, the picture is a reduction.


Study Online Dilation - Definition:


A dilation is a vary of the plane, Dk, such that if O is a set point, k is a non-zero real number, and P' is the picture of point P, then O, P and P' are collinear and `(OP ' )/(OP)` = k.
Notation:  Dk(x, y) = (kx, ky )

Examples for Study online Dilation:


Study online Dilation - Example 1:

PROBLEM:

Sketch the dilation picture of triangle ABC with the middle of dilation at the origin and a scale factor of 2.

Examine: Notice how EACH coordinate of the triangle has been multiplied by the scale factor (x2).

Study online Dilation - Example 2:

PROBLEM:

Sketch the dilation picture of pentagon ABCDE with the middle of dilation at the origin and a scale factor of 1/3.



Examine: Notice how EACH coordinate of the pentagon has been multiply the scale factor (1/3).

Note: Multiplying by 1/3 is the same as dividing by 3!

Study online Dilation - Example 3:

PROBLEM:

Sketch the dilation diagram of rectangle EFGH with the middle of dilation at point E and a scale factor of 1/2.



Examine:

E and its picture are the same.  It is main to observe the distance from the middle of the dilation, E, to the other points of the diagram.  Notice EF = 6 and E'F' = 3.

Note:

Be sure to measure distances for this problem.

Geometric Word Problems : 2

Introduction to geometric word problems:     

Geometry is a theoretical subject, but easy to learn, and it has many real practical applications. Finally, geometry has developed into a skillfully arranged and sensibly organized body of knowledge.

Geometry gives the planning of different geometrical shapes and figures in our daily life such as articles in the houses, wells, buildings, bridges etc. The term 'Geometry' means a study (learn) of properties of figures and shapes and the relationship between them.

Example problems of geometric word problems:


Geometric word problem 1:

Find the largest possible rectangular area we can enclose, assuming we have 144 centimeters of fencing. What is the implication of the dimensions of this largest possible enclosure?

Geometric word problem Solution:

Let the length be L and the width be W. We have 144 centimeters of fencing, so the perimeter equation is:

2L + 2W = 144

Dividing by 2 to make things simpler, then we get

L + W = 72

Area of the rectangle formula as,

A = L × W

We can substitute for either one of the above variables by solving the perimeter equation:

L + W = 72
L = 72 – W   

Then we get,

A = (72 – W) × W   

 = 72W – W 2

This equation is in the format of ax2+bx+c.



A = –W 2 + 72W

The vertex of a parabola is the point (h, k), where h = –b/2a.  In this case:

h = –(72)/(2×(–1)) = 36

To find the "k" part of the vertex, all we do is plug 36 in for W:

k = –(36)2 + 72(36) = 3888                                         

The largest possible area is 3888 square centimeters

Now w e can find the length by using the value of width. Then we get,

L = 72– W = 72 – 36 = 36

Then the length and width are the same: 36 centimeters.

Therefore, the largest possible rectangular area is in the shape of a square.

Geometric word problem 2:

A square has an area of twenty five square centimeters. What is the length of each of its sides?

Geometric word problem Solution:

The formula for the area A of a square with side-length ‘a’ is:

A = a2

Substitute the value of A in the above formula: Then we get,

25= a2

√25 = a
5 = a

After re-reading,

  a=5 cm

The length of each side is 5 centimeters.


Practice geometric word problems:


A circle has an area of 81π square units. What is the length of the circle's diameter?
 Answer: 9

A piece of 16-gauge copper wire 54 cm long is twisted into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle.
 Answer: L=9 cm and W=18 cm

Types Of Lines In Math

Introduction to Different types of Line:

 A line is the collection of points, along a straight line path and going to opposite direction, and also a line segment is part of a line, it’s having two end points. A ray also a part of line, one side is endpoint and other side goes to one direction.
A line doesn’t have starting point imagine it is endless from both directions. A continuous level of the length contains more than two points. Put the little arrows on both sides for understanding.
Different Types of lines:


In the following three different types of lines:

Parallel line,
Perpendicular line, and
Intersecting line.

These three are the most important types of line in math.

Explain the different types of line:

1. Parallel line

 It is one of the important line in math, It having two lines, both in the same plane, Lines are remain the same distance but never touching each other going to same direction.
Example :-

 1. Solve:-

   r is parallel to s

    angle 1 = 60 degrees

    To find the measures of seven angles in an accompanying figure (below).

 Solution:

  An Angle 2 = 120 degrees is supplementary to angle 1.These angles are any two angles whose sum is 180 degrees.

   An Angle 3 = 60 degrees since angle 1 and Angle 3 are vertical angles. The two nonadjacent angles are formed by two intersecting lines.

   An Angle 4 = 120 degrees is supplementary to angle 1.

   An Angle 5 = angle 1 by the transversal Postulate.

    An Angle 6 = angle 2, Angle 7 = angle 3, and Angle 8 = angle 4 by the transversal Postulate

2.      Perpendicular line

 It is the another type of line in math, it is having 2 lines (Perpendicular) that lines intersect and form right angles are called perpendicular lines and other lines otherwise intersect the right angles.                                                                                                                                                                                                                                                                                                                    
Ex:1  Determine if y = 2x+5 and x+2y = -2 are perpendicular then graph the equations to check.

Equation

1) y = 2x+5  
2) x+2y = -2

Slope Intercept Form

y = 2x+5
y = (-1/2)x-2

Slope

2(-1/2)

The slope of the line is [2*(-1/2)]= -1, the two lines are perpendicular.


3. Intersecting lines


   This is the last one of the types of lines in math, Two different lines or more than two different lines that meet at a point is called as intersecting lines.Here in diagram line l and line m intersect at point Q

A line contains both ends having an end points that is known as line segments.

Ex:1. The Intersecting lines are lines that meet at a point. When two intersect lines, they define angles point of intersection.



                                           at point C.

How Do You Multiply Radicals

Introduction to multiply radicals:

The opposite operation to the exponent is known as radical. A radical is an expression which contains the square roots, cube roots etc. For example the expression v100 can also be called as square root of 100 or root of 100. Radicals have the same property of the numbers. The simplification is to reduce the numbers and reduce the power of variables inside the roots. By multiplying the radical with another radical. The radical symbol will gets canceled.


Rule for Multiplying radicals:


when a positive radical  is multiplied with another positive radical then the result will be a positive whole number

example:   v 5 x v 5

v 5 x v 5 = 5

So answer will be 5

when a negative radical  is multiplied with another negative radical then the result will be a positive whole number

example:  - v 4 x -v 4

- x - = +  ( by multiplying two negative symbols we will get positive )

v 4 x v 4 = 4

So answer will be +4


Example problems for Multiplying radicals:


1) Simplify  v2  x v2

Solution:

v2  x v2 = 2

So the answer is 2

2) Simplify v ( 6 x 6 )

Solution:

v (36) = 6

So the answer is 6

3) Simplify -v( 4 ) x -v( 4 )

Solution:

-v( 4 ) x -v( 4 )

Therefore the answer is 4

4) Simplify v ( -9 x -9 )

Solution:

v (+81) = 9

So the answer is 9

5) Simplify  v( 11 ) x  v( 11 )

Solution:

v( 11 ) x  v( 11 ) = 11

Therefore the answer is 11

6) Simplify  v100  x v100

Solution:

v100  x v100 = 100



So the answer is 100

7) Simplify  - v 256  x - v 256

Solution:

- v 256 x - v 256

-  x  -  = +

v256  x v256 = 256

So the answer is +256

Ancient Egyptian Math

Introduction to Ancient Egyptian Math:

       In ancient Egyptian math was probably the first civilization to observe the scientific arts. The ancient Egyptian math was talented in medicine and applied math. But there is huge bodies of papyrus literature describe their achievements in medicine, no records of how they reached their mathematical conclusions. The advance grasping power of the subjects since their exploits in astronomy and administration would not have been possible without it. Let us see about ancient Egyptian math.

Decimal System in Ancient Egyptian Math :

• 1 - represented by a single stroke.
• 10 - represented by a drawing of a hobble for cattle.
• 100 - represented by a coil of rope.
• 1,000- represented by a lotus plant.
• 10,000- represented by a finger.
• 100,000- represented by a tadpole or frog.
• 1,000,000- represented by the following figure of a god with arms raised above his head

For example:         

Numeral Hieroglyphs in Ancient Egyptian Math:

• Hieroglyphs are developed in 3200 BC and carved in stone.
• There are 7 Symbols.

     

• For example 5120 is written as: 
• The highest value number is written to the left of the minor number.
• Where there is above one row of numbers reading starts at the top.

Hieroglyphic signs are divided into four categories:

                                 

• Alphabetic signs are represented a single sound. An ancient Egyptian math took most vowels for determined and did not represent such as 'e' or 'v'.
• Syllabic signs are represented by a combination of 2 or 3 consonants.
• They are followed by an upright stroke, to indicate that the word is complete in one sign.
• A picture of an object are helps the reader. For example; if a word expressed an abstract idea, a picture of a roll of papyrus tied up and sealed was included to show that the meaning of the word could be expressed in writing although not pictorially.

Example problem for Ancient Egyptian Math:

Problem 1:-   

Adding Ancient Egyptian Math 208 +4

Solution:-

=    +  

Trade         for 

and you get

  

Problem 2:-

Subtracting Ancient Egyptian Math

56 - 9

Solution:-

=  - 

trade one   for    and you get   

+             minus          =