<![CDATA[Math Is Fun Forum / LaTeX - A Crash Course]]> 2023-01-29T09:28:32Z FluxBB https://www.mathisfunforum.com/viewtopic.php?id=4397 <![CDATA[Re: LaTeX - A Crash Course]]> Triangular Prism

A prism is a solid figure with flat faces, two identical bases, and with the same cross-section all along its length. The name of a particular prism depends on the two bases of the prism which can be triangles, rectangles, or any polygon. For example, a prism with triangular bases is called a triangular prism and a prism with a square base is called a square prism, and so on. A triangular prism has two triangular bases and three rectangular lateral faces. Let us learn more about the triangular prism in this article.

What is a Triangular Prism?

A triangular prism is a 3D shape with two identical faces in the shape of a triangle connected by three rectangular faces. The rectangular faces are referred to as the lateral faces, while the triangular faces are called bases. The bases are also called the top and the bottom (faces) of the prism.

Triangular Prism Meaning: A triangular prism is a 3D polyhedron with three rectangular faces and two triangular faces. The 2 triangular faces are congruent to each other, and the 3 lateral faces which are in the shape of rectangles are also congruent to each other. Thus, a triangular prism has 5 faces, 9 edges, and 6 vertices. Observe the following of a triangular prism in which l represents the length of the prism, h represents the height of the base triangle, and b represents the bottom edge of the base triangle.

Triangular Prism Properties

The properties of a triangular prism help us to identify it easily. Listed below are a few properties of a triangular prism:

* A triangular prism has 5 faces, 9 edges, and 6 vertices.
* It is a polyhedron with 3 rectangular faces and 2 triangular faces.
* The two triangular bases are congruent to each other.
* Any cross-section of a triangular prism is in the shape of a triangle.

Right Triangular Prism

A right triangular prism is a prism in which the triangular faces are perpendicular to the three rectangular faces. In other words, the angle formed at the intersection of triangle and rectangle faces should be 90 degrees, therefore, the triangular faces are perpendicular to the lateral rectangular faces. A right triangular prism has 6 vertices, 9 edges, and 5 faces.

Triangular Prism Faces Edges Vertices

As mentioned above, a triangular prism has 5 faces including 3 lateral rectangular faces and 2 triangular bases, 9 edges, and 6 vertices. The vertices of the triangular prism are the vertices of the two triangular bases connected by lines that form rectangles. Its edges include 6 edges of two triangular bases (3 + 3) and 3 sides that join the bases.

Faces  :  Edges  :  Vertices
Triangular Prism  :  5     :  9  :  6

Triangular Prism Formulas

There are two important formulae of a triangular prism which are surface area and volume. A brief explanation of both is given below along with the formula.

Surface Area of a Triangular Prism

The surface area of a triangular prism is the area that is occupied by its surface. It is the sum of the areas of all the faces of the prism. Hence, the formula to calculate the surface area is:

Surface \ area = (Perimeter \ of \ the \ base \times Length) + (2 \times Base \ Area) = (a + b + c)L + bh

gives

where,

b is the bottom edge of the base triangle,
h is the height of the base triangle,
L is the length of the prism,
a, b, and c are the three edges (sides) of the base triangle
(bh) is the combined area of the two triangular faces, because

[2 \times (1/2 \times bh)] = bh

gives

.

Volume of a Triangular Prism

The volume of a triangular prism is the product of its triangular base area and the length of the prism. As we already know that the triangular prism base is in the shape of a triangle, the area of the base will be the same as that of a triangle. Hence, the

Volume \ of \ a \ Triangular \ Prism = Area \ of \ base \ triangle \times \ length

gives

or it can also be written as

Volume \ of \ Triangular \ Prism = 1/2 \times b \times h \times l

gives

,
where b is the base length of the triangle, h is the height of the triangle, and l is the length of the prism.

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<![CDATA[Re: LaTeX - A Crash Course]]> Tetrahedron

A tetrahedron is a three-dimensional shape that has four triangular faces. One of the triangles is considered as the base and the other three triangles together form the pyramid. The tetrahedron is a type of pyramid, which is a polyhedron with triangular faces connecting the base to a common point and a flat polygon base. It has a triangular base and thus it is also referred to as a triangular pyramid.

Tetrahedron Definition

A tetrahedron is a polyhedron with 4 faces, 6 edges, and 4 vertices, in which all the faces are triangles. It is also known as a triangular pyramid whose base is also a triangle. A regular tetrahedron has equilateral triangles, therefore, all its interior angles measure 60°. The interior angles of a tetrahedron in each plane add up to 180° as they are triangular.

Properties of Tetrahedron

A tetrahedron is a three-dimensional shape that is characterized by some distinct properties.

A tetrahedron has :

* 4 Faces
* 4 Vertices
* 6 Edges

The following are the properties of a tetrahedron which help us identify the shape easily.

* It has 4 faces, 6 edges, and 4 vertices (corners).
* In a regular tetrahedron, all four vertices are equidistant from each other.
* It has 6 planes of symmetry.
* Unlike other platonic solids, it has no parallel faces.
* A regular tetrahedron has four equilateral triangles as its faces.

Surface Area of Tetrahedron

The surface area of a tetrahedron is defined as the total area or region covered by all the faces of the shape. It is expressed in square units, like

m^2, {cm}^2, {in}^2, {ft}^2, {yd}^2,

gives

etc. A tetrahedron can have two types of surface areas:

* Lateral Surface Area of Tetrahedron
* Total Surface Area of Tetrahedron

Lateral Surface Area of a Tetrahedron

The lateral surface area of a tetrahedron is defined as the surface area of the lateral or the slant faces of a tetrahedron. The formula to calculate the lateral surface area of a regular tetrahedron is given as,

LSA of Regular Tetrahedron = Sum of 3 congruent equilateral triangles (i.e. lateral faces)

= 3 \times (\sqrt{3})/4 \  a^2

gives

square units
where a is the side length of a regular tetrahedron.

Total Surface Area of a Tetrahedron

The total surface area of a tetrahedron is defined as the surface area of all the faces of a tetrahedron. The formula to calculate the total surface area of a regular tetrahedron is given as,

TSA of Regular Tetrahedron = Sum of 4 congruent equilateral triangles (i.e. lateral faces)

= 4 \times (\sqrt{3})/4 \ a^2 = \sqrt{3} \ a^2

gives

square units
where a is the side length of the regular tetrahedron.

Volume of Tetrahedron

The volume of a tetrahedron is defined as the total space occupied by it in a three-dimensional plane. The formula to calculate the tetrahedron volume is given as,

The volume of regular tetrahedron

= (1/3) \times area \ of \ the \ base \times height = (1/3) \cdot (\sqrt{3})/4 \cdot a^2 \times (\sqrt{2})/(\sqrt{3}) \times a

gives

= (\sqrt{2})/12 \ a^3

gives

cubic units
where a is the side length of the regular tetrahedron.

]]>
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<![CDATA[Re: LaTeX - A Crash Course]]> Cuboid

A cuboid is a three-dimensional solid shape that has 6 faces, 8 vertices, and 12 edges. It is one of the most commonly seen shapes around us which has three dimensions: length, width, and height. Sometimes the cuboid shape is confused with a cube since it shares some properties of a cube, however, they are different from each other.

What is a Cuboid?

We know that a rectangle is a two-dimensional shape that has 4 sides. Now, imagine a shape that is formed when many congruent rectangles are placed one on top of the other. The shape thus formed is called a cuboid. Observe the following cuboid which shows its three dimensions: length, width, and height.

Dimensions of a Cuboid

It should be noted that there is no strict rule according to which an edge of a cuboid shape should be named as its length, width (breadth), or height. However, it is understood that if a cuboid is placed flat on a table, then the height represents the length of any vertical edge; the length is taken to be the larger of the two dimensions of the horizontal face of the cuboid, and the width is the smaller of the two dimensions. These dimensions of a cuboid are denoted by 'l' for length, 'w' for width (breadth), and 'h' for height. Apart from these, the face of a cuboid is the flat surface; the edge is the line segment connecting two adjacent vertices, and the vertex is a point at which two or more edges meet.

Cuboid Faces Edges Vertices

Every 3D shape has a definite number of faces, edges, and vertices. A cuboid shape has 6 faces, 12 edges, and 8 vertices. A cuboid has 4 lateral faces and 2 faces of top and bottom. All are in the shape of rectangles. It has 12 edges that include 8 edges of the top and bottom faces and 4 edges that connect them. And, it has 8 vertices which are the vertices of the top and bottom faces. At each vertex, three segments meet from all three dimensions.

Cuboid Formulas

Considering the three main dimensions of a cuboid to be the length (l), width (w), and height (h), observe the basic formulas of a cuboid shape in the following table.

i) Face Diagonals :

\sqrt{l^2 + w^2}

gives

units
ii) Space Diagonals  :

\sqrt{(l^2+ w^2 + h^2)}

gives

units
iii) Perimeter  :

4(l + w + h)

gives

units
iv) Volume  :

(l \times w \times h)

gives

cubic units
v) Surface Area  :

2(lw + wh + lh)

gives

square units

Diagonals of a Cuboid

Since a cuboid is a 3D shape, there are two types of diagonals in it:

* Face Diagonals
* Space Diagonals

Diagonals of a Cuboid:

Face Diagonal

Face diagonals can be drawn by connecting the opposite vertices on a particular face of a cuboid and we know that only two diagonals can be drawn on one face of a cuboid. Since a cuboid has 6 faces, a total of 12 face diagonals can be drawn in a cuboid.

Space diagonal

A space diagonal is a line segment that joins the opposite vertices of a cuboid. The space diagonals pass through the interior of the cuboid. Therefore, 4 space diagonals can be drawn inside it.

Surface Area of Cuboid

The total area occupied by a cuboid shape is considered the surface area of a cuboid. Since a cuboid is a 3D figure, the surface area will depend on the length, breadth, and height. It can have two kinds of surface areas - Total surface area and lateral surface area. Hence, the formulae to find the surface area of a cuboid are given below:

i) Total Surface Area of Cuboid,

S = 2 (lb + bh + lh)

gives

square units
ii) Lateral Surface Area of Cuboid,

L = 2h (l + b)

gives

square units

where,

l = Length,
h = Height,
S = Total surface area, and
L = Lateral surface area

Volume of Cuboid

The volume of a cuboid is considered the space occupied inside a cuboid. A cuboid's volume depends on its length, breadth, and height. Hence, changing any one of these quantities changes the volume of the shape. The unit of cuboid's volume is given as the cubic units. Therefore, the formula to calculate the volume of a cuboid is:

Volume of a Cuboid =

Base \ Area \times \ Height

gives

The base area for cuboid =

l \times b

gives

Hence, the volume of a cuboid,

V = l \times b \times h = lbh

gives

cubic units.

where,

l = Length
h = Height

Cuboid Properties

The important properties of a cuboid help us to identify a cuboid shape easily. They are as follows:

* A cuboid has 6 faces, 8 vertices, and 12 edges.
* All the angles formed at the vertices of a cuboid are right angles.
* All the faces are rectangular in shape.
* Two diagonals can be drawn on each face of a cuboid.
* The opposite edges are parallel to each other.
* The dimensions of a cuboid are length, width, and height.

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<![CDATA[Re: LaTeX - A Crash Course]]> Cube - II

A cube is a three-dimensional object that has 6 congruent square faces. Dimensions of all the 6 square faces of the cube are the same. A cube is sometimes also referred to as a regular hexahedron or as a square prism. It is one of the 5 platonic solids. Some real-life examples of a cube are an ice cube, a Rubik's cube, a regular dice, etc.

Cube Definition

A cube is a 3D solid object with six square faces and all the sides of a cube are of the same length. It is also known as a regular hexahedron and is one of the five platonic solids. The shape consists of six square faces, eight vertices, and twelve edges. The length, breadth, and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length. In a cube, the faces share a common boundary called the edge that is considered as the bounding line of the edge. The structure is defined with each face being connected to four vertices and four edges, vertex connected with three edges and three faces, and edges are in touch with two faces and two vertices.

Cube Definition

A cube is a 3D solid object with six square faces and all the sides of a cube are of the same length. It is also known as a regular hexahedron and is one of the five platonic solids. The shape consists of six square faces, eight vertices, and twelve edges. The length, breadth, and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length. In a cube, the faces share a common boundary called the edge that is considered as the bounding line of the edge. The structure is defined with each face being connected to four vertices and four edges, vertex connected with three edges and three faces, and edges are in touch with two faces and two vertices.

Properties of Cube

A cube is considered a special kind of square prism since all the faces are in the shape of a square and are platonic solid. There are many different properties of a cube just like any other 3D or 2D shape. The properties are:

* A cube has 12 edges, 6 faces, and 8 vertices.
* All the faces of a cube are shaped as a square hence the length, breadth, and height are the same.
* The angles between any two faces or surfaces are 90°.
* The opposite planes or faces in a cube are parallel to each other.
* The opposite edges in a cube are parallel to each other.
* Each of the faces in a cube meets the other four faces.
* Each of the vertices in a cube meets the three faces and three edges.

Cube Formula

The cube formula helps us to find the surface area, diagonals, and volume of a cube. Let us discuss the different formulas of a cube.

Surface Area of a Cube

There are two types of surface areas of a cube - Lateral surface area and Total surface area

Lateral Surface Area of a Cube

The lateral area of a cube is the sum of areas of all side faces of the cube. There are 4 side faces so the sum of areas of all 4 side faces of a cube is its lateral area. The lateral area of a cube is also known as its lateral surface area (LSA), and it is measured in square units.

LSA of a Cube =

4a^2

gives

where a is the side length.

Total Surface Area of a Cube
The total surface area of the cube will be the sum of the area of the base and the area of vertical surfaces of the cube. Since all the faces of the cube are made up of squares of the same dimensions then the total surface area of the cube will be the surface area of one face added five times to itself. It is measured as the "number of square units" (square centimeters, square inches, square feet, etc.). Therefore, the formula to find the surface area of a cube is:

Total Surface Area (TSA) of a Cube =

6a^2

gives

where a is the side length.

Volume of a Cube

The volume of a cube is the space occupied by the cube. The volume of a cube can be found out by finding the cube of the side length of the cube. To determine the volume of a cube, there are different formulas based on different parameters. It can be calculated using the side length or the measure of the cube's diagonal and it is expressed in cubic units of length. Hence, the two different formulas to find the volume of the cube are:

i) The Volume of a Cube (based on side length) =

a^3

gives

where a is the length of the side of a cube
ii) The volume of a Cube (based on diagonal) =

\dfrac{\sqrt{3} \times d^3}{9}

gives

where d is the length of the diagonal of a cube.

Diagonal of a Cube

The diagonal of a cube is a line segment that joins two opposite vertices of a cube. The length of the diagonal of a cube can be determined using the diagonal of a cube formula. It helps in finding the length of the face diagonals and the main diagonals. Each face diagonal forms the hypotenuse of the right-angled triangle formed. A cube has six faces (square-shaped). On each face, there are two diagonals joining the non-adjacent vertices. Therefore, we have twelve face diagonals and four main diagonals that connect the opposite vertices of the cube. The diagonal of a cube formula to calculate the length of a face diagonal and the main body diagonal of a cube is given as,

i) Length of face diagonal of cube =

\sqrt{2}a

gives

units, where a = Length of each side of a cube.
ii) Length of main diagonal of a cube =

\sqrt{3}a

gives

units, where a = Length of each side of a cube.

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<![CDATA[Re: LaTeX - A Crash Course]]> Cube - I

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.

The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.

The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.

The cube is dual to the octahedron. It has cubical or octahedral symmetry.

The cube is the only convex polyhedron whose faces are all squares.

Formulas

For a cube of edge length a:

Surface  area :

6a^{2}

gives

,
Volume :

a^{3}

gives

,
Face  diagonal :

\sqrt{2}a

gives

,
Space  diagonal :

\sqrt{3}a

gives

\frac{\sqrt {3}}{2}}a

gives

Radius  of  sphere  tangent  to  edges :

\frac{a}{\sqrt {2}}

gives

{\frac {a}{2}}

gives

Angles  between faces (in radians) :

{\frac {\pi }{2}}

gives

.

As the volume of a cube is the third power of its sides

a\times a\times a

gives

, third powers are called cubes, by analogy with squares and second powers.

A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length+width+height).

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<![CDATA[Re: LaTeX - A Crash Course]]> Octahedron

In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.

An octahedron is the three-dimensional case of the more general concept of a cross polytope.

Area and volume

The surface area A and the volume V of a regular octahedron of edge length a are:

A=2{\sqrt {3}}a^{2}\approx 3.464a^{2}

gives

V={\frac {1}{3}}{\sqrt {2}}a^{3}\approx 0.471a^{3}

gives

Octahedron

An octahedron is a shape that is formed by joining by two pyramids at its bases. Once joined the shape converts to 8 faced, 12 edges, and 6 vertices. An octahedron is most commonly known as the regular octahedron i.e. when all the faces are of the same shape and size. But in most cases, it is not necessary that all faces have to be the same to be called an octahedron, with the same size also, the shape is still called octahedron. Let us learn more about the meaning, properties, area, and volume formulas of an octahedron.

Meaning of Octahedron

The word octahedron is derived from the Greek word 'Oktaedron' which means 8 faced. An octahedron is a polyhedron with 8 faces, 12 edges, and 6 vertices and at each vertex 4 edges meet. It is one of the five platonic solids with faces that are shaped like an equilateral triangle.

Properties of an Octahedron

Mentioned below are a few properties of an octahedron:

* An octahedron has 6 vertices and at each vertex 4 edges meet.
* An octahedron has 8 faces shaped like an equilateral triangle, in the case of a regular octahedron.
* An octahedron has 12 edges.
* In a regular octahedron, the angles between the edges are measured at 60° but the dihedral angle is measured at 109.28°.
* The formula to calculate the surface area of an octahedron is

2 \times \sqrt{3}\times a^2

gives

.
* The formula to calculate the volume of an octahedron is

\sqrt{2}/3 \times a^3

gives

.

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<![CDATA[Re: LaTeX - A Crash Course]]> Dodecahedron

Surface area and volume

The surface area A and the volume V of a regular dodecahedron of edge length a are:

{\displaystyle {\displaystyle A=3{\sqrt {25+10{\sqrt {5}}}}a^{2}\approx 20.645\,728\,807a^{2}}}

gives

{\displaystyle V={\frac {1}{4}}(15+7{\sqrt {5}})a^{3}\approx 7.663\,118\,9606a^{3}}

gives

Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with an edge length of one unit has the properties:

{\displaystyle {\displaystyle A={\frac {15\phi }{\sqrt {3-\phi }}}}}

gives

{\displaystyle {\displaystyle V={\frac {5\phi ^{3}}{6-2\phi }}}}

gives

.

A dodecahedron is a three-dimensional figure having twelve faces that are pentagonal in shape. All the faces are flat 2-D shapes. There are five platonic solids and dodecahedron is one of them. Platonic solids are convex polyhedra in which the faces are made up of congruent regular polygons with the same number of faces meeting at each of their vertices. A dodecahedron is made up of 12 congruent pentagons with 3 pentagonal faces meeting at each of its 20 vertices. There are two types of dodecahedrons - regular and irregular dodecahedrons.

What is a Dodecahedron?

Dodecahedron is derived from the Greek word "dōdeka" means "12" and "hédra" means "face or seat" that shows that it is a polyhedron with 12 sides or 12 faces. Hence, any polyhedra with 12 sides can be called a dodecahedron. It is made up of 12 pentagonal faces.

Regular Dodecahedron

A regular dodecahedron has 12 regular pentagonal sides. You can see in the image of the dodecahedron net shown above that there are 12 pentagonal sides on a dodecahedron.

The surface area of a Dodecahedron

\approx 20.64 \times a^2

gives

square units (where a is the length of one side).

The Volume of a Dodecahedron

\approx 7.66 \times  a^3

gives

cubic units (where a is the length of one side)

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<![CDATA[Re: LaTeX - A Crash Course]]> Icosahedron

An icosahedron is a three-dimensional shape with twenty faces which makes it a polyhedron. It is one of the few platonic solids. In this lesson, we will discuss more about the shape of Icosahedron and formulas related to it with the help of solved examples.

What is Icosahedron?

The word "Icosahedron" is made of two Greek words "Icos" which means twenty and "hedra" which means seat. The icosahedron's definition is derived from the ancient Greek words Icos (eíkosi) meaning 'twenty' and hedra (hédra) meaning 'seat'. It is one of the five platonic solids with equilateral triangular faces. Icosahedron has 20 faces, 30 edges, and 12 vertices. It is a shape with the largest volume among all platonic solids for its surface area. It has the most number of faces among all platonic solids.

Icosahedron's vertices  :  12
Icosahedron's faces  :  20
Icosahedron's edges  :  30

Icosahedron's angles

(a)Angle between edges: 60°
(b)Dihedral angel: 138.19°

Icosahedron's volume formula

(5/12) \times (3+\sqrt{5}) \times a^3

gives

Icosahedron's surface area formula

(5\sqrt{3} × a^2)

gives

.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:

Tetrahedron  :  Cube  :  Octahedron  :  Dodecahedron  :  Icosahedron
Four faces  :  Six faces  :  Eight faces  :  Twelve faces  :  Twenty faces.

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<![CDATA[Re: LaTeX - A Crash Course]]> Transcendental Numbers - III

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi }

gives

where the Greek letter phi

\varphi

gives

or

\phi

gives

denotes the golden ratio. The constant

\varphi

gives

{\displaystyle \varphi ^{2}=\varphi +1}

gives

and is an irrational number with a value of

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=}1.618033988749....

gives

.

The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of

\varphi

gives

-may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.

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<![CDATA[Re: LaTeX - A Crash Course]]> Transcendental Numbers - II

In mathematics, we deal with several terminologies with fixed numeric values. These are called constants, and they help in solving mathematical problems with ease.

For example,

\pi \approx 3.14

gives

Likewise, there is a constant e, known as Euler’s number. It was discovered in the 18th century. It is quite interesting to learn about the history of Euler’s number. It is the base of the natural logarithm. The Euler's number digit are e is approximately 2.712818284

The number was first coined by Swiss mathematician Leonhard Euler in the 1720s. John Napier, the inventor of logarithms, made significant contributions to developing the number, while Sebastian Wedeniwski calculated it to 869,894,101 decimal places.

e is called the Euler Number, the Eulerian number, or Napier's Constant. Euler's number proof was first given as that e  is an irrational number, so its decimal expansion never terminates.

This mathematical constant is not only used in math but equally important and applicable in physics.

In math, the term e is called Euler's number after the Swiss mathematician Leonhard Euler.

It is mathematically known as Euler's (pronounced as oiler) constant or Napier's constant.

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of

(1 + 1/n)^n

gives

as n approaches infinity, an expression that arises in the study of compound interest.

{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}

gives

It is also the unique positive number a such that the graph of the function y = ax has a slope of 1 at x = 0.

The (natural) exponential function f(x) = ex is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one. There are various other characterizations.

The number e is of great importance in mathematics, alongside

0, 1, \pi, \ and \ i

gives

. All five appear in one formulation of Euler's identity

{\displaystyle e^{i\pi }+1=0}

gives

and play important and recurring roles across mathematics. Like the constant

\pi

gives

, e is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients). To 50 decimal places, the value of e is:

2.7182818284590452353602874713526624977572470

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base e. It is assumed that the table was written by William Oughtred.

The discovery of the constant itself is credited to Jacob Bernoulli in 1683, the following expression (which is equal to e):

{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}

gives

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, while the first appearance of e in publication was in Euler's Mechanica (1736).

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<![CDATA[Re: LaTeX - A Crash Course]]> Transcendental Numbers - I

Value of Pi

The value of pi is the ratio of the circumference of a circle to its diameter. The pi symbol is denoted as

\pi

gives

It is also called Archimedes' constant which was named after the Greek mathematician, Archimedes, who created an algorithm to approximate the pi value. The value of pi is irrational, which means that the count of digits after the decimal point is infinite. It is used as either 3.1415929 or 22/7. In geometry, it is used in calculating the surface area, volume, and circumference of various three-dimensional shapes.

What is the Value of Pi?

The value of 'pi' is constant, which means it cannot be changed. It is an irrational number usually approximated to 3.14. It is used in various formulas for the measurement of surface area and volume of various solid shapes. 'Pi' is defined as the ratio of the circumference of the circle to the diameter of the circle. We know that the diameter of a circle is the longest line segment that passes through the center of the circle. Imagine the line of diameter is bent such that it covers a part of the circumference of the circle. Now, π is defined as the total number of times the diameter is wrapped around the circumference of the circle which is 3.14 times approximately.

If we divide the circumference by the diameter, we get approximately 3.14. It should be noted that no matter what size of a circle we draw, the ratio of the circumference to its diameter will always be the same.

Symbol for Pi

The symbol used to denote pi in math is π, which is used in Greek letters. It is known as a mathematical constant as its value does not change. The pi symbol represents 22/7 in fractions or 3.14 (approximately) in decimals.

Formula for Pi

The value of pi can be obtained by dividing the circumference of a circle by its diameter. Since pi is the ratio of the circumference of a circle to its diameter, the formula for calculating

\pi

gives

is:

\pi

gives

=

Circumference/Diameter

gives

One interesting statement that helps us to remember the value of pi is "How I wish I could calculate pi". By counting the number of letters in each word, we can easily write the value of pi. Since 'How' has 3 letters; 'I' - 1 letter, 'wish' - 4 letters, 'I' - 1 letter, 'could' - 5 letters, 'calculate' - 9 letters, 'pi' - 2 letters. Therefore, the value of pi approximated to 6 decimal places is 3.141592.

How to Calculate the Value of Pi?

The value of pi can be calculated with the help of a simple activity. Follow the steps given below to know why the value of pi is the ratio of the circumference to the diameter of the circle:

Step 1: Draw a circle of diameter 1 unit.
Step 2: After this step, take a thread and place it along the border of the circle (the circumference).
Step 3: Now, place the thread on the ruler and note the length.

Repeat the process with diameters of 2 units, 3 units, 4 units, 5 units, and record your observations in the table.

Diameter    Circumference    Circumference/Diameter
1 unit    3.1 units    3.1/1
2 units    6.2 units (approx.)    6.2/2 = 3.1
3 units    9.3 units (approx.)    9.3/3 = 3.1
4 units    12.4 units (approx.)    12.4/4 = 3.1
5 units    15.5 units (approx.)    15.5/5 = 3.1

We can observe that the ratio of circumference to diameter is always the same which is 3.1.

Value of Pi in Decimals

As discussed above, the value of pi is an irrational number, which means there are infinite decimal places after the number 3. The 100 decimal places of pi consist of all digits from 0 to 9. There are eight 0s, eight 1s, twelve 2s, eleven 3s, ten 4s, eight 5s, nine 6s, eight 7s, twelve 8s, and fourteen 9s. Observe the figure given below which shows that the value of pi in decimals starts with 3 and the 100th decimal digit holds the number 9.

Value of pi in decimals

The value of pi in decimals is non-terminating and non-recurring and is approximated to 100 decimal places as 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679. For ease of calculations, it is often approximated to 3.14.

Value of Pi in Fraction

The value of pi in fraction is 22/7. We know that the value of pi is an irrational number in which the digits after the decimal point are infinite. Therefore, to make calculations easier the value of pi is also defined in the form of a fraction and is expressed as 22/7.

Value of Pi in Degree

The value of pi in degree is 180 degrees, as π radians is equal to 180°. To understand this, we will be looking at the concept of radians which is the SI unit of measuring angles. We know that in a circle, one complete revolution makes the circumference of the circle. It means if we rotate by 360° around a point (known as the center of the circle) at a fixed distance (radius), we will get the circumference, which is equal to

2\pi{r}

gives

, where r denoted the radius of the circle.

So, we can write, Circumference =

2\pi{r} = {360}^{o}

gives

. And, for half a circle, the angle is 180° and the length of the arc will be

2\pi{r}/2 = \pi{r}

gives

. The formula of radians is arc length/radius. So, for half a circle, the arc length is πr and the radius is r.

*

\pi{r}/r \ radians = {180}^o

gives

.

*

\pi \ {radians} = {180}^o

gives

Therefore, the value of pi in degree is 180 degrees.

Important Notes:

Some important points about the pi symbol and value are given below:

* 'Pi' is a mathematical constant that is the ratio of the circumference of a circle to its diameter. It is an irrational number often approximated to 3.14159.
* Based on the problem, for ease of calculation, we use the value of pi as 22/7 or 3.14.

A few basic circle formulas using pi are,
a)

Circumference = \pi \times \ Diameter

gives

b)

{Area} = \pi{r}^2

gives

.

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<![CDATA[Re: LaTeX - A Crash Course]]> Golden Ratio - Part IV

What is Golden Rectangle?

In geometry, a golden rectangle is defined as a rectangle whose side lengths are in the golden ratio. The golden rectangle exhibits a very special form of self-similarity. All rectangles that are created by adding or removing a square are golden rectangles as well.

Constructing a Golden Rectangle

We can construct a golden rectangle using the following steps:

Step 1: First, we will draw a square of 1 unit. On one of its sides, draw a point midway. Now, we will draw a line from this point to a corner of the other side.

Step 2: Using this line as a radius and the point drawn midway as the center, draw an arc running along the square's side. The length of this arc can be calculated using Pythagoras Theorem:

\sqrt{(1/2)^2 + (1)^2} = \sqrt{5}/2 \ units.

gives

Step 3: Use the intersection of this arc and the square's side to draw a rectangle: This is a golden rectangle because its dimensions are in the golden ratio. i.e.,

\phi = (\sqrt{{5}/2 + 1/2)/1 \approx 1.61803

gives

.

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<![CDATA[Re: LaTeX - A Crash Course]]> Golden Ratio Part III

Golden Ratio Equation

To calculate the value of the golden ratio is by solving the golden ratio equation.

We know,

\phi = 1 + 1/\phi

gives

Multiplying both sides by

\phi

gives

,

{\phi}^2 = \phi + 1

gives

On rearranging, we get,

{\phi}^2 - \phi -1 = 0

gives

The above equation is a quadratic equation and can be solved using quadratic formula:

\phi = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

gives

Substituting the values of a = 1, b = -1 and c = -1, we get,

\phi = \dfrac{ 1 \pm (1 + 4)}{2}

gives

The solution can be simplified to a positive value giving:

\phi = 1/2 + \sqrt{5}/2

gives

Note that we are not considering the negative value, as

\phi

gives

is the ratio of lengths and it cannot be negative.

Therefore,

\phi = 1/2 + \sqrt{5}/2

gives

.

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<![CDATA[Re: LaTeX - A Crash Course]]> Golden Ratio Part II

From equation (1),

a/b = \phi

gives

a = b

Substitute this in equation (2),

(b\phi + b)/b\phi = \phi

gives

b(\phi + 1)/b\phi = \phi

gives

(\phi + 1)/\phi = \phi

gives

1 + 1/\phi = \phi

gives

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<![CDATA[Re: LaTeX - A Crash Course]]> Golden Ratio Part I

The golden ratio, which is often referred to as the golden mean, divine proportion, or golden section, is a special attribute, denoted by the symbol

\phi

gives

, and is approximately equal to 1.618. The study of many special formations can be done using special sequences like the Fibonacci sequence and attributes like the golden ratio.

This ratio is found in various arts, architecture, and designs. Many admirable pieces of architecture like The Great Pyramid of Egypt, Parthenon, have either been partially or completely designed to reflect the golden ratio in their structure. Great artists like Leonardo Da Vinci used the golden ratio in a few of his masterpieces and it was known as the "Divine Proportion" in the 1500s. Let us learn more about the golden ratio in this lesson.

What is the Golden Ratio?

The golden ratio, which is also referred to as the golden mean, divine proportion, or golden section, exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity between the two. With reference to this definition, if we divide a line into two parts, the parts will be in the golden ratio if:

The ratio of the length of the longer part, say "a" to the length of the shorter part, say "b" is equal to the ratio of their sum "(a + b)" to the longer length.

Refer to the following diagram for a better understanding of the above concept:

\dfrac{a}{b} = \dfrac{a + b}{a} =1.618.... = \phi

gives

Golden ratio definition

It is denoted using the Greek letter ϕ, pronounced as "phi". The approximate value of ϕ is equal to 1.61803398875... It finds application in geometry, art, architecture, and other areas. Thus, the following equation establishes the relationship for the calculation of golden ratio:

\phi = a/b = (a + b)/a = 1.61803398875...

gives

where a and b are the dimensions of two quantities and a is the larger among the two.

Golden Ratio Definition

When a line is divided into two parts, the long part that is divided by the short part is equal to the whole length divided by the long part is defined as the golden ratio. Mentioned below are the golden ratio in architecture and art examples.

There are many applications of the golden ratio in the field of architecture. Many architectural wonders like the Great Mosque of Kairouan have been built to reflect the golden ratio in their structure. Artists like Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this as an attribute in their artworks.

Golden Ratio Formula

The Golden ratio formula can be used to calculate the value of the golden ratio. The golden ratio equation is derived to find the general formula to calculate golden ratio.

Golden Ratio Equation

From the definition of the golden ratio,

a/b = (a + b)/a = \phi

gives

From this equation, we get two equations:

a/b = ϕ - (1)

gives

(a + b)/a = \phi - (2)

gives

From equation (1),

a/b = \phi

gives

(a + b)/a = \phi - (2)

gives

.

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