A

PROJECT OF

MATHEMATICS SUBMITTED TO

MR. GIRISH CHANDRA

SUBMITTED BY

SAURABH VERMA

Class = IX- ‘C’

Roll. No. 39

KENDRIYA VIDYALAYA No. 1.

AFS CHAKERI, KANPUR-208007

Pythagorean Theorem

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The Pythagorean Theorem: The sum of the areas of the

two squares on the legs (a and b) equals the area of the

square on the hypotenuse (c).

In mathematics, the Pythagorean theorem (in

American English) or Pythagoras' theorem (in British

English) is a relation in Euclidean geometry among the

three sides of a right triangle (right-angled triangle in

British English). It states:

In any right triangle, the area of the square whose side is

the hypotenuse (the side opposite the right angle) is

equal to the sum of the areas of the squares whose sides

are the two legs (the two sides that meet at a right

angle).

The theorem can be written as an equation:

where c representsL the length of the hypotenuse, and a

and b represent the lengths of the other two sides.

The Pythagorean theorem is named after the Greek

mathematician Pythagoras, who by tradition is credited

with its discovery and proof,[1] although it is often argued

that knowledge of the theory predates him. (There is

much evidence that Babylonian mathematicians understood the

principle, if not the mathematical significance.)

Contents

1 In formulae

2 Proofs

o 2.1 Proof using similar triangles

o 2.2 Euclid's proof

o 2.3 Garfield's proof

o 2.4 Proof by subtraction

o 2.5 Similarity proof

o 2.6 Proof by rearrangement

o 2.7 Algebraic proof

o 2.8 Proof by differential equations

3 Converse

4 Consequences and uses of the theorem

o 4.1 Pythagorean triples

o 4.2 List of primitive Pythagorean triples up to

100

o 4.3 The existence of irrational numbers

o 4.4 Distance in Cartesian coordinates

5 Generalizations

o 5.1 The Pythagorean theorem in non-

Euclidean geometry

o 5.2 In complex arithmetic

6 History

7 Cultural references to the Pythagorean theorem

1. In formulae

If we let c be the length of the hypotenuse and a and b be

the lengths of the other two sides, the theorem can be

expressed as the equation:

or, solved for c:

If c is already given, and the length of one of the legs

must be found, the following equations (which are

corollaries of the first) can be used:

This equation provides a simple relation among the three

sides of a right triangle so that if the lengths of any two

sides are known, the length of the third side can be

found. A generalization of this theorem is the law of

cosines, which allows the computation of the length of

the third side of any triangle, given the lengths of two

sides and the size of the angle between them. If the angle

between the sides is a right angle it reduces to the

Pythagorean theorem.

2. Proofs

This is a theorem that may have more known

proofs than any other (the law of quadratic reciprocity

being also a contender for that distinction); the book

Pythagorean Proposition, by Elisha Scott Loomis, contains

367 proofs.

Proof using similar triangles

Proof using similar triangles

Like most of the proofs of the Pythagorean theorem, this

one is based on the proportionality of the sides of two

similar triangles.

Let ABC represent a right triangle, with the right angle

located at C, as shown on the figure. We draw the

altitude from point C, and call H its intersection with the

side AB. The new triangle ACH is similar to our triangle

ABC, because they both have a right angle (by definition

of the altitude), and they share the angle at A, meaning

that the third angle will be the same in both triangles as

well. By a similar reasoning, the triangle CBH is also

similar to ABC. The similarities lead to the two ratios:

Euclid's proof

Proof in Euclid's Elements

In Euclid's Elements, Proposition 47 of Book 1, the

Pythagorean theorem is proved by an argument along the

following lines. Let A, B, C be the vertices of a right

triangle, with a right angle at A. Drop a perpendicular

from A to the side opposite the hypotenuse in the square

on the hypotenuse. That line divides the square on the

hypotenuse into two rectangles, each having the same

area as one of the two squares on the legs.

For the formal proof, we require four elementary

lemmata:

1. If two triangles have two sides of the one equal to

two sides of the other, each to each, and the angles

included by those sides equal, then the triangles are

congruent. (Side - Angle - Side Theorem)

2. The area of a triangle is half the area of any

parallelogram on the same base and having the

same altitude.

3. The area of any square is equal to the product of two

of its sides.

4. The area of any rectangle is equal to the product of

two adjacent sides (follows from Lemma 3).

The intuitive idea behind this proof, which can make it

easier to follow, is that the top squares are morphed into

parallelograms with the same size, then turned and

morphed into the left and right rectangles in the lower

square, again at constant area.[2]

Illustration including the new lines

The proof is as follows:

1. Let ACB be a right-angled triangle with right angle

CAB.

2. On each of the sides BC, AB, and CA, squares are

drawn, CBDE, BAGF, and ACIH, in that order.

3. From A, draw a line parallel to BD and CE. It will

perpendicularly intersect BC and DE at K and L,

respectively.

4. Join CF and AD, to form the triangles BCF and BDA.

5. Angles CAB and BAG are both right angles; therefore

C, A, and G are collinear. Similarly for B, A, and H.

6. Angles CBD and FBA are both right angles; therefore

angle ABD equals angle FBC, since both are the sum

of a right angle and angle ABC.

7. Since AB and BD are equal to FB and BC,

respectively, triangle ABD must be equal to triangle

FBC.

8. Since A is collinear with K and L, rectangle BDLK

must be twice in area to triangle ABD.

9. Since C is collinear with A and G, square BAGF must

be twice in area to triangle FBC.

10. Therefore rectangle BDLK must have the same

area as square BAGF = AB2.

11. Similarly, it can be shown that rectangle CKLE

must have the same area as square ACIH = AC2.

12. Adding these two results, AB2 + AC2 = BD × BK

+ KL × KC

13. Since BD = KL, BD* BK + KL × KC = BD(BK +

KC) = BD × BC

14. Therefore AB2 + AC2 = BC2, since CBDE is a

square.

This proof appears in Euclid's Elements as that of

Proposition 1.47.[3]

Garfield's proof

James A. Garfield (later President of the United States) is

credited with a novel algebraic proof:

The area of a trapezoid is

where h is the height, and s1 and s2 are lengths of the

parallel sides.

So the area of the trapezoid in the figure is

While Triangle 1 and triangle 2 each have area .

And triangle 3 has area , and it is half of the square on

the hypotenuse.

Then the Area of trapezoid is

The two areas must be equal, so

Therefore the square on the hypotenuse = the sum of the

squares on the other two sides.

Proof by subtraction

In this proof, the square on the hypotenuse plus four

copies of the triangle can be assembled into the same

shape as the squares on the other two sides plus four

copies of the triangle. This proof is recorded from China.[

Similarity proof

From the same diagram as that in Euclid's proof above,

we can see three similar figures, each being "a square

with a triangle on top". Since the large triangle is made of

the two smaller triangles, its area is the sum of areas of

the two smaller ones. By similarity, the three squares are

in the same proportions relative to each other as the

three triangles, and so likewise the area of the larger

square is the sum of the areas of the two smaller

squares.

Proof by rearrangement

Proof of Pythagorean Theorem by rearrangement of

4 identical right triangles: Since the total area and the

areas of the triangles are all constant, the total black area

is constant. But this can be divided into squares

delineated by the triangle sides a, b, c, demonstrating

that a2 + b2 = c2.

A proof by rearrangement is given by the illustration

and the animation. In the illustration, the area of each

large square is (a + b)2. In both, the area of four identical

triangles is removed. The remaining areas, a2 + b2 and c2,

are equal.

Proof using rearrangement

Algebraic proof

A square created by aligning four right angle

triangles and a large square

This proof is indeed very simple, but it is not elementary,

in the sense that it does not depend solely upon the most

basic axioms and theorems of Euclidean geometry. In

particular, while it is quite easy to give a formula for area

of triangles and squares, it is not as easy to prove that

the area of a square is the sum of areas of its pieces. In

fact, proving the necessary properties is harder than

proving the Pythagorean Theorem itself (see Lebesgue

measure and Banach-Tarski paradox). Actually, this

difficulty affects all simple Euclidean proofs involving

area; for instance, deriving the area of a right triangle

involves the assumption that it is half the area of a

rectangle with the same height and base. For this reason,

axiomatic introductions to geometry usually employ

another proof based on the similarity of triangles (see

above).

A third graphic illustration of the Pythagorean

Theorem (in yellow and blue to the right) fits parts of the

sides' squares into the hypotenuse's square. A related

proof would show that the repositioned parts are identical

with the originals and, since the sum of equals are equal,

that the corresponding areas are equal. To show that a

square is the result one must show that the length of the

new sides equals c. Note that for this proof to work, one

must provide a way to handle cutting the small square in

more and more slices as the corresponding side gets

smaller and smaller.[5]

Algebraic proof

An algebraic variant of this proof is provided by the

following reasoning. Looking at the illustration which is a

large square with identical right triangles in its corners,

the area of each of these four triangles is given by an

angle corresponding with the side of length C.

The A-side angle and B-side angle of each of these

triangles are complementary angles, so each of the

angles of the blue area in the middle is a right angle,

making this area a square with side length C. The area of

this square is C2. Thus the area of everything together is

given by:

However, as the large square has sides of length A + B,

we can also calculate its area as (A + B) 2, which expands

to A2 + 2AB + B2.

Proof by differential equations

One can arrive at the Pythagorean Theorem by studying

how changes in a side produce a change in the

hypotenuse in the following diagram and employing a

little

Proof using differential equations

As a result of a change da in side a,

By similarity of triangles and for differential changes. So

Upon separation of variables.

Which results from adding a second term for changes in

side b.

Integrating gives

When a = 0 then c = b, so the "constant" is b2.

As can be seen, the squares are due to the particular

proportion between the changes and the sides while the

sum is a result of the independent contributions of the

changes in the sides which is not evident from the

geometric proofs. From the proportion given it can be

shown that the changes in the sides are inversely

proportional to the sides. The differential equation

suggests that the theorem is due to relative changes and

its derivation is nearly equivalent to computing a line

integral.

These quantities da and dc are respectively infinitely

small changes in a and c. But we use instead real

numbers Δa and Δc, then the limit of their ratio as their

sizes approach zero is da/dc, the derivative, and also

approaches c/a, the ratio of lengths of sides of triangles,

and the differential equation results.

3. Converse

The converse of the theorem is also true: For any three

positive numbers a, b, and c such that a2 + b2 = c2, there

exists a triangle with sides a, b and c, and every such

triangle has a right angle between the sides of lengths a

and b.

This converse also appears in Euclid's Elements. It can be

proven using the law of cosines (see below under

Generalizations), or by the following proof:

Let ABC be a triangle with side lengths a, b, and c, with a2

+ b2 = c2. We need to prove that the angle between the a

and b sides is a right angle. We construct another triangle

with a right angle between sides of lengths a and b. By

the Pythagorean Theorem, it follows that the hypotenuse

of this triangle also has length c. Since both triangles

have the same side lengths a, b and c, they are

congruent, and so they must have the same angles.

Therefore, the angle between the side of lengths a and b

in our original triangle is a right angle. A corollary of the

Pythagorean Theorem’s converse is a simple means of

determining whether a triangle is right, obtuse, or acute,

as follows. Where c is chosen to be the longest of the

three sides:

If a2 + b2 = c2, then the triangle is right.

If a2 + b2 > c2, then the triangle is acute.

If a2 + b2 < c2, then the triangle is obtuse.

4. Consequences and uses of the

theorem

Pythagorean triples

Main article: Pythagorean triple

A Pythagorean triple has three positive integers a, b, and

c, such that a2 + b2 = c2. In other words, a Pythagorean

triple represents the lengths of the sides of a right

triangle where all three sides have integer lengths.

Evidence from megalithic monuments on the Northern

Europe shows that such triples were known before the

discovery of writing. Such a triple is commonly written (a,

b, c). Some well-known examples are (3, 4, 5) and (5, 12,

13).

List of primitive Pythagorean triples

up to 100

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41),

(11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20,

21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80,

89), (48, 55, 73), (65, 72, 97)

The existence of irrational numbers

One of the consequences of the Pythagorean Theorem is

that incommensurable lengths (ie. their ratio is irrational

number), such as the square root of 2, can be

constructed. A right triangle with legs both equal to one

unit has hypotenuse length square root of 2. The proof

that the square root of 2 is irrational was contrary to the

long-held belief that everything was rational. According to

legend, Hippasus, who first proved the irrationality of the

square root of two, was drowned at sea as a

consequence.[7]

Distance in Cartesian coordinates

The distance formula in Cartesian coordinates is derived

from the Pythagorean Theorem. If (x0, y0) and (x1, y1) are

points in the plane, then the distance between them, also

called the Euclidean distance, is given by

More generally, in Euclidean n -space , the Euclidean

distance between two points, , is defined, using the

Pythagorean theorem, as:

5. Generalizations

Generalization for similar triangles, green area = red area

The Pythagorean Theorem was generalized by Euclid in

his Elements:

If one erects similar figures (see Euclidean geometry) on

the sides of a right triangle, then the sum of the areas of

the two smaller ones equals the area of the larger one.

The Pythagorean Theorem is a special case of the more

general theorem relating the lengths of sides in any

triangle, the law of cosines:

Where θ is the angle between sides a and b.

When θ is 90 degrees, then cos(θ) = 0, so the

formula reduces to the usual Pythagorean theorem.

Given two vectors v and w in a complex inner product

space, the Pythagorean Theorem takes the following

form:

In particular, ||v + w||2 = ||v||2 + ||w||2 if v and w are

orthogonal, although the converse is not necessarily true.

Using mathematical induction, the previous result can be

extended to any finite number of pair wise orthogonal

vectors. Let v1, v2, …, vn be vectors in an inner product

space such that <vi, vj> = 0 for 1 ≤ i < j ≤ n. Then

The generalization of this result to infinite-dimensional

real inner product spaces is known as Parseval's identity.

When the theorem above about vectors is rewritten in

terms of solid geometry, it becomes the following

theorem. If lines AB and BC form a right angle at B, and

lines BC and CD form a right angle at C, and if CD is

perpendicular to the plane containing lines AB and BC,

then the sum of the squares of the lengths of AB, BC, and

CD is equal to the square of AD. The proof is trivial.

Another generalization of the Pythagorean theorem to

three dimensions is de Gua's theorem, named for Jean

Paul de Gua de Malves: If a tetrahedron has a right angle

corner (a corner like a cube), then the square of the area

of the face opposite the right angle corner is the sum of

the squares of the areas of the other three faces.

There are also analogs of these theorems in dimensions

four and higher.

In a triangle with three acute angles, α + β > γ holds.

Therefore, a2 + b2 > c2.

In a triangle with an obtuse angle, α + β < γ holds.

Therefore, a2 + b2 < c2.

Edsger Dijkstra has stated this proposition about acute,

right, and obtuse triangles in this language:

sgn(α + β − γ) = sgn(a2 + b2 − c2)

Where α is the angle opposite to side a, β is the angle

opposite to side b and γ is the angle opposite to side c.[8]

The Pythagorean Theorem in non-

Euclidean geometry

The Pythagorean Theorem is derived from the axioms of

Euclidean geometry, and in fact, the Euclidean form of

the Pythagorean Theorem given above does not hold in

non-Euclidean geometry. (It has been shown in fact to be

equivalent to Euclid's Parallel (Fifth) Postulate.) For

example, in spherical geometry, all three sides of the

right triangle bounding an octant of the unit sphere have

length equal to 3.14 ; this violates the Euclidean

Pythagorean theorem because .

This means that in non-Euclidean geometry, the

Pythagorean theorem must necessarily take a different

form from the Euclidean theorem. There are two cases to

consider — spherical geometry and hyperbolic plane

geometry; in each case, as in the Euclidean case, the

result follows from the appropriate law of cosines:

For any right triangle on a sphere of radius R, the

Pythagorean Theorem takes the form

This equation can be derived as a special case of the

spherical law of cosines. By using the Maclaurin series for

the cosine function, it can be shown that as the radius R

approaches infinity, the spherical form of the

Pythagorean Theorem approaches the Euclidean form.

For any triangle in the hyperbolic plane (with Gaussian

curvature −1), the Pythagorean Theorem takes the form

where cosh is the hyperbolic cosine.

By using the Maclaurin series for this function, it can be

shown that as a hyperbolic triangle becomes very small

(i.e., as a, b, and c all approach zero), the hyperbolic form

of the Pythagorean Theorem approaches the Euclidean

form.

In hyperbolic geometry, for a right triangle one can also

write,

where is the angle of parallelism of the line segment AB

where μ is the multiplicative distance function (see

Hilbert's arithmetic of ends).

In hyperbolic trigonometry, the sine of the angle of

parallelism satisfies

Thus, the equation takes the form

Where a, b, and c are multiplicative distances of the sides

of the right triangle (Hartshorne, 2000).

In complex arithmetic

The Pythagoras formula is used to find the distance

between two points in the Cartesian coordinate plane,

and is valid if all coordinates are real: the distance

between the points (a, b) and (c, d) is

√((a − c)2 + (b − d)2). With complex coordinates, this

formula breaks down, e.g. the distance between the

points {0,1} and {i,0} would work out as 0, resulting in a

reductio ad absurdum. This is because this formula

depends on Pythagoras's theorem, which in all its proofs

depends on areas, and areas of triangles and other

geometrical figures depend on the edge lines of these

figures separating an inside from an outside, which does

not happen if the coordinates can be complex.

Instead, for the distance between the points (a, b) and

(c, d) it is usual to use:

(p and q are the real and imaginary parts of (a − c))

(r and s are the real and imaginary parts of (b − d))

Where is the complex conjugate of z. For example, the

distance between the points (0, 1) and (i, 0) would work

out as 0 if complex conjugates were not taken. But the

distance is

6. History

This section needs additional citations for

verification.

Please help improve this article by adding

reliable references. Unsourced material may be

challenged and removed. (April 2008)

Visual proof for the (3, 4, 5) triangle as in the Chou

Pei Suan Ching 500–200 BC

The history of the theorem can be divided into

four parts: knowledge of Pythagorean triples, knowledge

of the relationship among the sides of a right triangle,

knowledge of the relationships among adjacent angles,

and proofs of the theorem.

Megalithic monuments from circa 2500 BC in

Egypt, and in Northern Europe, incorporate right triangles

with integer sides.[9] Bartel Leendert van der Waerden

conjectures that these Pythagorean triples were

discovered algebraically.[10]

Written between 2000 and 1786 BC, the Middle

Kingdom Egyptian papyrus Berlin 6619 includes a

problem whose solution is a Pythagorean triple.

The Mesopotamian tablet Plimpton 322, written

between 1790 and 1750 BC during the reign of

Hammurabi the Great, contains many entries closely

related to Pythagorean triples.

The Baudhayana Sulba Sutra, the dates of which

are given variously as between the 8th century BC and

the 2nd century BC, in India, contains a list of

Pythagorean triples discovered algebraically, a statement

of the Pythagorean theorem, and a geometrical proof of

the Pythagorean theorem for an isosceles right triangle.

The Apastamba Sulba Sutra (circa 600 BC)

contains a numerical proof of the general Pythagorean

Theorem, using an area computation. Van der Waerden

believes that "it was certainly based on earlier traditions".

According to Albert Bŭrk, this is the original proof of the

theorem; he further theorizes that Pythagoras visited

Arakonam, India, and copied it.

Pythagoras, whose dates are commonly given as

569–475 BC, used algebraic methods to construct

Pythagorean triples, according to Proklos's commentary

on Euclid. Proklos, however, wrote between 410 and 485

AD. According to Sir Thomas L. Heath, there was no

attribution of the theorem to Pythagoras for five centuries

after Pythagoras lived. However, when authors such as

Plutarch and Cicero attributed the theorem to Pythagoras,

they did so in a way which suggests that the attribution

was widely known and undoubted.[1]

Around 400 BC, according to Proklos, Plato gave

a method for finding Pythagorean triples that combined

algebra and geometry. Circa 300 BC, in Euclid's Elements ,

the oldest extant axiomatic proof of the theorem is

presented.

Written sometime between 500 BC and 200

AD, the Chinese text Chou Pei Suan Ching (周髀算经), (The

Arithmetical Classic of the Gnomon and the Circular Paths

of Heaven) gives a visual proof of the Pythagorean

theorem — in China it is called the "Gougu Theorem" (勾

股 定 理 ) — for the (3, 4, 5) triangle. During the Han

Dynasty, from 202 BC to 220 AD, Pythagorean triples

appear in The Nine Chapters on the Mathematical Art,

together with a mention of right triangles.[11]

The first recorded use is in China (where it is

alternately known as the "Shang Gao Theorem" (商高定理),

named after the Duke of Zhou's astrologer, and

described in the mathematical collection Zhou Bi Suan

Jing) and in India, where it is known as the Bhaskara

Theorem.

There is much debate on whether the

Pythagorean theorem was discovered once or many

times. Boyer (1991) thinks the elements found in the

Shulba Sutras may be of Mesopotamian derivation.[12]

7. Cultural references to the

Pythagorean Theorem

The Pythagorean Theorem has been

referenced in a variety of mass media throughout history.

A verse of the Major-General's Song in the Gilbert

and Sullivan musical The Pirates of Penzance, "About

binomial theorem I'm teeming with a lot o' news,

With many cheerful facts about the square of the

hypotenuse", with oblique reference to the theorem.

The Scarecrow of The Wizard of Oz makes a more

specific reference to the theorem when he receives

his diploma from the Wizard. He immediately exhibits

his "knowledge" by reciting a mangled and incorrect

version of the theorem: "The sum of the square roots

of any two sides of an isosceles triangle is equal to

the square root of the remaining side. Oh, joy, oh,

rapture. I've got a brain!" The "knowledge" exhibited

by the Scarecrow is incorrect. The accurate

statement would have been "The sum of the squares

of the legs of a right triangle is equal to the square of

the remaining side."[13]

In an episode of The Simpsons, after finding a pair of

Henry Kissinger's glasses in a toilet at the Springfield

Nuclear Power Plant, Homer puts them on and quotes

Oz Scarecrow's mangled version of the formula. A

man in a nearby toilet stall then yells out "That's a

right triangle, you idiot!" (The comment about square

roots remained uncorrected.)

Similarly, the Speech software on an Apple MacBook

references the Scarecrow's incorrect statement. It is

the sample speech when the voice setting 'Ralph' is

selected.

In Freemasonry, one symbol for a Past Master is the

diagram from the 47th Proposition of Euclid, used in

Euclid's proof of the Pythagorean Theorem.

In 2000, Uganda released a coin with the shape of a

right triangle. The coin's tail has an image of

Pythagoras and the Pythagorean Theorem,

accompanied with the mention "Pythagoras

Millennium".[14] Greece, Japan, San Marino, Sierra

Leone, and Suriname have issued postage stamps

depicting Pythagoras and the Pythagorean theorem.[15]

In Neal Stephenson's speculative fiction Anathem,

the Pythagorean Theorem is referred to as 'the

Adrakhonic theorem'. A geometric proof of the

theorem is displayed on the side of an alien ship to

demonstrate their understanding of mathematics.

Pythagorean TheoremThe sum of the squares of the lengths of the two legs of a right triangle is equal to the

square of the length of the hypotenuse.

There are 55 jobs that use Pythagorean Theorem.

Management

Management occupations

Computer and information systems managers

Construction managers

Engineering and natural sciences managers

Farmers, ranchers, and agricultural managers

Funeral directors

Industrial production managers

Medical and health services managers

Property, real estate, and community association

managers

Purchasing managers, buyers, and purchasing agents

Business and financial operations occupations

Insurance underwriters

Professional

Computer and mathematical occupations

Actuaries

Computer software engineers

Mathematicians

Statisticians

Architects, surveyors, and cartographers

Architects, except landscape and naval

Landscape architects

Surveyors, cartographers, photogrammetrists, and

surveying technicians

Engineers

Aerospace engineers

Chemical engineers

Civil engineers

Computer hardware engineers

Electrical engineers

Environmental engineers

Industrial engineers

Materials engineers

Mechanical engineers

Nuclear engineers

Drafters and engineering technicians

Drafters

Life scientists

Biological scientists

Conservation scientists and foresters

Physical scientists

Atmospheric scientists

Chemists and materials scientists

Environmental scientists and hydrologists

Physicists and astronomers

Social scientists and related occupations

Economists

Legal occupations

Lawyers

Education, training, library, and museum

occupations

Archivists, curators, and museum technicians

Teachers-preschool, kindergarten, elementary,

middle, and secondary

Media and communications-related

occupations

Writers and editors

Health diagnosing and treating occupations

Optometrists

Physicians and surgeons

Registered nurses

Veterinarians

Health technologists and technicians

Opticians, dispensing

Veterinary technologists and technicians

Farming

Farming

Agricultural workers

Construction

Construction

Carpenters

Construction and building inspectors

Electricians

Glaziers

Installation

Electrical and electronic equipment mechanics,

installers, and repairers

Electrical and electronics installers and repairers

Electronic home entertainment equipment installers

and repairers

Other installation, maintenance, and repair

occupations

Millwrights

Production

Metal workers and plastic workers

Machinists

Welding, soldering, and brazing workers

The Pythagorean Theorem

The Pythagorean Theorem was developed by the Greek

Philosopher Pythagoras and is used in every day situations. These tips

can ease the normal anxieties felt by students.

The words "Pythagorean Theorem" can sound incredibly

intimidating to typical middle school math students. What students may

not initially realize is that the Pythagorean Theorem is executed in

various daily situations, some that students may have already

experienced without realizing it. Introducing this concept using real life

examples can help set the stage for easier problem solving and a deeper

understanding and can hence help students appreciate the reasons why

it’s essential to learn this concept at all.

The Pythagorean Theorem consists of the following: The sum of

the squares of two legs of a right triangle is equal to the hypotenuse

squared.

For example, if one of the legs is 6 inches and the other leg is 7

inches, we can calculate how long the hypotenuse, or the third leg is. Let

a = 6, b=7, and c= the length of the hypotenuse. (6)^2 + (7)^2 = c^2.

6*6=36, and 7*7=49. Thus, 36 + 49 = 85. So, the square root of 85 is

approximately 9.2 inches. Therefore, we just calculated the length of the

hypotenuse using this theorem.

The following consists of real life applications to introduce to students

which can greatly ease their anxieties and further promote their

learning.

Baseball Diamond

If the teacher asks students how many of them play baseball or

enjoy baseball, the majority of boys in the classroom will more than

likely raise their hands. The teacher can utilize this concept by using an

overhead transparency, chalkboard, or other advanced technological

device. In a baseball diamond, the distance between each of the three

bases and home plate are 90 feet and all form right angles. If a teacher

draws a line from home plate to first base, then from first base to

second base and back to the home plate, the students can see a right

triangle has been formed. Using the Pythagorean Theorem, the teacher

can then pose the question, "How far does the second baseman have to

throw the ball in order to get the runner out before he slides into the

home plate?" (90)^2 + (90)^2 = c^2, or the distance from home plate

to second base. 8100 + 8100 = 16,200. The square root of 16,200 is

approximately 127, so the second baseman would have to throw it

about 127 feet.

Height of a Building

Firemen, construction workers, and other workers often rely on the

use of ladders in their line of work. They make use of the Pythagorean

Theorem in various situations. For example, the height to a second story

window may be 25 feet, and a window cleaner may need to put the

ladder ten feet away from the house in order to avoid the bushes or

flowers. How long of a ladder does the window cleaner need in order to

achieve this task? (25)^2 + (10)^2 = c^2, or the length of ladder

needed. 625 + 100 = 725. The square root of 725 is approximately 27,

so the window cleaner would need a ladder 27 feet long.

Two friends meeting at a

specific destination

Let’s say Bob and Larry are meeting at Blockbuster on the corner of

Park and Pleasant Street. Presently, Bob is on Park Street to and is 8

miles away. Meanwhile, Larry is on Pleasant Street 7 miles away. How

far away are they from each other? (8)^2 + (7)^2 = distance between

Bob and Larry. 64 + 49 = 113. The square root of 113 is approximately

10.6. Thus, this is how far apart Bob and Larry are from each other.

Ramp of a moving truck

The height of a moving truck is 4 feet. The distance from the bottom

edge of a ramp on the ground to the truck is 6 feet. How long is the

ramp? (4)^2 + (6)^2 = length of ramp. 16 + 36 = 52. The square root

of 52 is approximately 7.2, which is the length of the ramp.

Measurement of TV

Television sets are generally measured diagonally, thus classifying

them as 13 inches, 27 inches, 36 inches, and so forth. Suppose we want

to purchase an entertainment center, but it only holds enough room in

it’s cubicle for a 27 inch TV set. We initially know that the length of our

TV is 15 inches, and the height of our TV is 12 inches. Will our TV be able

to fit into the cubicle? (15)^2 + (12)^2 = 369. The square root of 369 is

approximately 19.2 inches. Therefore, our TV will fit with plenty of room

to spare. Introducing these real-life situations to students will ease their

mind on learning this powerful concept. The Pythagorean Theorem is

involved in many other everyday uses as well, which will help students

develop a thorough understanding of the fascinating complexities

behind the right triangle.

π Pi has been known for almost 4000 years—

but even if we calculated the number of seconds

in those 4000 years and calculated pi to that

number of places, we would still only be

approximating its actual value. Here’s a brief

history of finding Pi:

The ancient Babylonians calculated the area of a

circle by taking 3 times the square of its radius, which

gave a value of pi = 3. One Babylonian tablet (ca.

1900–1680 BC) indicates a value of 3.125 for pi, which

is a closer approximation.

In the Egyptian Rhind Papyrus (ca.1650 BC),

there is evidence that the Egyptians calculated the

area of a circle by a formula that gave the

approximate value of 3.1605 for pi.

The ancient cultures mentioned above found their

approximations by measurement. The first calculation

of pi was done by Archimedes of Syracuse (287–212

BC), one of the greatest mathematicians of the ancient

world. Archimedes approximated the area of a circle

by using the Pythagorean Theorem to find the areas of

two regular polygons: the polygon inscribed within the

circle and the polygon within which the circle was

circumscribed. Since the actual area of the circle lies

between the areas of the inscribed and circumscribed

polygons, the areas of the polygons gave upper and

lower bounds for the area of the circle. Archimedes

knew that he had not found the value of pi but only an

approximation within those limits. In this way,

Archimedes showed that pi is between 3 1/7 and 3

10/71.

A similar approach was used by Zu Chongzhi

(429–501), a brilliant Chinese mathematician and

astronomer. Zu Chongzhi would not have been familiar

with Archimedes’ method—but because his book has

been lost, little is known of his work. He calculated the

value of the ratio of the circumference of a circle to its

diameter to be 355/113. To compute this accuracy for

pi, he must have started with an inscribed regular

24,576-gon and performed lengthy calculations

involving hundreds of square roots carried out to 9

decimal places.

Mathematicians began using the Greek letter π in the

1700s. Introduced by William Jones in 1706, use of the

symbol was popularized by Euler, who adopted it in

1737.

An 18th century French mathematician named

Georges Buffon devised a way to calculate pi based on

probability. You can try it yourself at the Exploratorium

exhibit Throwing Pi.

A history of Pi

A little known verse of the Bible reads

And he made a molten sea, ten cubits from the one

brim to the other: it was round all about, and his height

was five cubits: and a line of thirty cubits did compass it

about. (I Kings 7, 23)

The same verse can be found in II Chronicles 4, 2. It

occurs in a list of specifications for the great temple of

Solomon, built around 950 BC and its interest here is that

it gives π = 3. Not a very accurate value of course and

not even very accurate in its day, for the Egyptian and

Mesopotamian values of 25/8 = 3.125 and √10 = 3.162

have been traced to much earlier dates: though in

defence of Solomon's craftsmen it should be noted that

the item being described seems to have been a very

large brass casting, where a high degree of geometrical

precision is neither possible nor necessary. There are

some interpretations of this which lead to a much better

value.

The fact that the ratio of the circumference to the

diameter of a circle is constant has been known for so

long that it is quite untraceable. The earliest values of π

including the 'Biblical' value of 3, were almost certainly

found by measurement. In the Egyptian Rhind Papyrus,

which is dated about 1650 BC, there is good evidence for

4 (8/9)2 = 3.16 as a value for π.

The first theoretical calculation seems to have been

carried out by Archimedes of Syracuse (287-212 BC). He

obtained the approximation

223/71 < π < 22/7.

Before giving an indication of his proof, notice that very

considerable sophistication involved in the use of

inequalities here. Archimedes knew, what so many

people to this day do not, that π does not equal 22/7, and

made no claim to have discovered the exact value. If we

take his best estimate as the average of his two bounds

we obtain 3.1418, an error of about 0.0002.

Here is Archimedes' argument.

Consider a circle of radius 1, in which we inscribe a

regular polygon of 3 2n-1 sides, with semiperimeter bn,

and superscribe a regular polygon of 3 2n-1 sides, with

semiperimeter an.

The diagram for the case n = 2 is on the right.

The effect of this procedure is to define an increasing

sequence

b1 , b2 , b3 , ...

and a decreasing sequence

a1 , a2 , a3 , ...

such that both sequences have limit π.

Using trigonometrical notation, we see that the two semi

perimeters are given by

an = K tan(π/K), bn = K sin(π/K),

where K = 3 2n-1. Equally, we have

an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K),

and it is not a difficult exercise in trigonometry to show

that

(1/an + 1/bn) = 2/an+1   . . . (1)

an+1bn = (bn+1)2       . . . (2)

Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 =

3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using

(2), then a3 using (1), then b3 using (2), and so on until he

had calculated a6 and b6. His conclusion was that

b6 < π < a6 .

It is important to realize that the use of trigonometry here

is unhistorical: Archimedes did not have the advantage of

an algebraic and trigonometrical notation and had to

derive (1) and (2) by purely geometrical means. Moreover

he did not even have the advantage of our decimal

notation for numbers, so that the calculation of a6 and b6

from (1) and (2) was by no means a trivial task. So it was

a pretty stupendous feat both of imagination and of

calculation and the wonder is not that he stopped with

polygons of 96 sides, but that he went so far.

For of course there is no reason in principle why one

should not go on. Various people did, including:

Ptolemy

(c. 150

AD)

3.141

6

Zu

Chongzhi

(430-501

AD) 355/113

al-

Khwarizm

i

(c. 800 ) 3.141

6

al-Kashi (c. 1430) 14

places

Viète (1540-

1603)

9

places

Roomen (1561- 17

1615) places

Van

Ceulen (c. 1600)

35

places

Except for Zu Chongzhi, about whom next to nothing is

known and who is very unlikely to have known about

Archimedes' work, there was no theoretical progress

involved in these improvements, only greater stamina in

calculation. Notice how the lead, in this as in all scientific

matters, passed from Europe to the East for the

millennium 400 to 1400 AD.

Al-Khwarizmi lived in Baghdad, and incidentally gave his

name to 'algorithm', while the words al jabr in the title of

one of his books gave us the word 'algebra'. Al-Kashi lived

still further east, in Samarkand, while Zu Chongzhi, one

need hardly add, lived in China.

The European Renaissance brought about in due course a

whole new mathematical world. Among the first effects of

this reawakening was the emergence of mathematical

formulae for π. One of the earliest was that of Wallis

(1616-1703)

2/π = (1.3.3.5.5.7. ...)/(2.2.4.4.6.6. ...)

And one of the best-known is

π/4 = 1 - 1/3 + 1/5 - 1/7 + ....

This formula is sometimes attributed to Leibniz (1646-

1716) but is seems to have been first discovered by

James Gregory (1638- 1675).

These are both dramatic and astonishing formulae, for

the expressions on the right are completely arithmetical

in character, while π arises in the first instance from

geometry. They show the surprising results that infinite

processes can achieve and point the way to the

wonderful richness of modern mathematics.

From the point of view of the calculation of π, however,

neither is of any use at all. In Gregory's series, for

example, to get 4 decimal places correct we require the

error to be less than 0.00005 = 1/20000, and so we need

about 10000 terms of the series. However, Gregory also

showed the more general result

tan-1 x = x - x3/3 + x5/5 - ... (-1 ≤ x ≤ 1)   . . . (3)

from which the first series results if we put x = 1. So

using the fact that

tan-1(1/√3) = π/6 we get

π/6 = (1/√3)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...

Which converges much more quickly. The 10th term is

1/(19 39√3), which is less than 0.00005, and so we have

at least 4 places correct after just 9 terms.

An even better idea is to take the formula

π/4 = tan-1(1/2) + tan-1(1/3)   . . . (4)

and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).

Clearly we shall get very rapid convergence indeed if we

can find a formula something like

π/4 = tan-1(1/a) + tan-1(1/b)

With a and b large. In 1706 Machin found such a formula:

π/4 = 4 tan-1(1/5) - tan-1(1/239)   . . . (5)

Actually this is not at all hard to prove, if you know how to

prove (4) then there is no real extra difficulty about (5),

except that the arithmetic is worse. Thinking it up in the

first place is, of course, quite another matter.

With a formula like this available the only difficulty in

computing π is the sheer boredom of continuing the

calculation. Needless to say, a few people were silly

enough to devote vast amounts of time and effort to this

tedious and wholly useless pursuit. One of them, an

Englishman named Shanks, used Machin's formula to

calculate π to 707 places, publishing the results of many

years of labour in 1873. Shanks has achieved immortality

for a very curious reason which we shall explain in a

moment.

Here is a summary of how the improvement went:

169

9: Sharp used Gregory's result to get 71 correct digits

170

1:

Machin used an improvement to get 100 digits and

the following used his methods:

171

9: de Lagny found 112 correct digits

178

9: Vega got 126 places and in 1794 got 136

184

1: Rutherford calculated 152 digits and in 1853 got 440

187 Shanks calculated 707 places of which 527 were

3: correct

A more detailed Chronology is available.

Shanks knew that π was irrational since this had

been proved in 1761 by Lambert. Shortly after Shanks'

calculation it was shown by Lindemann that π is

transcendental, that is, π is not the solution of any

polynomial equation with integer coefficients. In fact this

result of Lindemann showed that 'squaring the circle' is

impossible. The transcendentality of π implies that there

is no ruler and compass construction to construct a

square equal in area to a given circle.

Very soon after Shanks' calculation a curious

statistical freak was noticed by De Morgan, who found

that in the last of 707 digits there was a suspicious

shortage of 7's. He mentions this in his Budget of

Paradoxes of 1872 and a curiosity it remained until 1945

when Ferguson discovered that Shanks had made an

error in the 528th place, after which all his digits were

wrong. In 1949 a computer was used to calculate π to

2000 places. In this and all subsequent computer

expansions the number of 7's does not differ significantly

from its expectation, and indeed the sequence of digits

has so far passed all statistical tests for randomness.

You can see 2000 places of π.

We should say a little of how the notation π arose.

Oughtred in 1647 used the symbol d/π for the ratio of the

diameter of a circle to its circumference. David Gregory

(1697) used π/r for the ratio of the circumference of a

circle to its radius. The first to use π with its present

meaning was an Welsh mathematician William Jones in

1706 when he states "3.14159 andc. = π". Euler adopted

the symbol in 1737 and it quickly became a standard

notation.

We conclude with one further statistical curiosity about

the calculation of π, namely Buffon's needle experiment.

If we have a uniform grid of parallel lines, unit distance

apart and if we drop a needle of length k < 1 on the grid,

the probability that the needle falls across a line is 2k/π.

Various people have tried to calculate π by throwing

needles. The most remarkable result was that of Lazzerini

(1901), who made 34080 tosses and got

π = 355/113 = 3.1415929

Which, incidentally, is the value found by Zu Chongzhi.

This outcome is suspiciously good, and the game is given

away by the strange number 34080 of tosses. Kendall

and Moran comment that a good value can be obtained

by stopping the experiment at an optimal moment. If you

set in advance how many throws there are to be then this

is a very inaccurate way of computing π. Kendall and

Moran comment that you would do better to cut out a

large circle of wood and use a tape measure to find its

circumference and diameter.

Still on the theme of phoney experiments, Gridgeman, in

a paper which pours scorn on Lazzerini and others,

created some amusement by using a needle of carefully

chosen length k = 0.7857, throwing it twice, and hitting a

line once. His estimate for π was thus given by

2 0.7857 / π = 1/2

from which he got the highly creditable value of π =

3.1428. He was not being serious!

It is almost unbelievable that a definition of π was used,

at least as an excuse, for a racial attack on the eminent

mathematician Edmund Landau in 1934. Landau had

defined π in this textbook published in Göttingen in that

year by the, now fairly usual, method of saying that π/2 is

the value of x between 1 and 2 for which cos x vanishes.

This unleashed an academic dispute which was to end in

Landau's dismissal from his chair at Göttingen.

Bieberbach, an eminent number theorist who disgraced

himself by his racist views, explains the reasons for

Landau's dismissal:-

Thus the valiant rejection by the Göttingen student body

which a great mathematician, Edmund Landau, has

experienced is due in the final analysis to the fact that

the un-German style of this man in his research and

teaching is unbearable to German feelings. A people who

have perceived how members of another race are

working to impose ideas foreign to its own must refuse

teachers of an alien culture.

G H Hardy replied immediately to Bieberbach in a

published note about the consequences of this un-

German definition of π

There are many of us, many Englishmen and many

Germans, who said things during the War which we

scarcely meant and are sorry to remember now. Anxiety

for one's own position, dread of falling behind the rising

torrent of folly, determination at all cost not to be

outdone, may be natural if not particularly heroic

excuses. Professor Bieberbach's reputation excludes such

explanations of his utterances, and I find myself driven to

the more uncharitable conclusion that he really believes

them true.

Not only in Germany did π present problems. In the USA

the value of π gave rise to heated political debate. In the

State of Indiana in 1897 the House of Representatives

unanimously passed a Bill introducing a new

mathematical truth.

Be it enacted by the General Assembly of the State of

Indiana: It has been found that a circular area is to the

square on a line equal to the quadrant of the

circumference, as the area of an equilateral rectangle is

to the square of one side.

(Section I, House Bill No. 246, 1897)

The Senate of Indiana showed a little more sense and

postponed indefinitely the adoption of the Act!

****END***