Pythagorean theorem

31
Name :- karan balchandani Std :- 10 th Subject :- maths Topic :- euclid’s theorem

description

 

Transcript of Pythagorean theorem

Page 1: Pythagorean theorem

Name :- karan balchandaniStd :- 10th Subject :- mathsTopic :- euclid’s theorem

Page 2: Pythagorean theorem

1 Euclid's proof 2 Euler's proof 3 Erdős' proof 4 Furstenberg's proof 5 Some recent proofs

5.1 Pinasco5.2 Whang

6 Proof using the irrationality of π

Page 3: Pythagorean theorem

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.

Page 4: Pythagorean theorem

Euclid offered the following proof published in his work Elements (Book IX, Proposition 20),[1] which is paraphrased here.

Consider any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then q is either prime or not:

If q is prime, then there is at least one more prime than is in the list.

If q is not prime, then some prime factor p divides q. If this factor p were on our list, then it would divide P (since P is the product of every number on the list); but p divides P + 1 = q. If p divides P and q, then p would have to divide the difference[2] of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, this would be a contradiction and so p cannot be on the list. This means that at least one more prime number exists beyond those in the list.

Page 5: Pythagorean theorem

This proves that for every finite list of prime numbers there is a prime number not on the list, and therefore there must be infinitely many prime numbers.Euclid is often erroneously reported to have proved this result by contradiction, beginning with the assumption that the set initially considered contains all prime numbers, or that it contains precisely the n smallest primes, rather than any arbitrary finite set of primes.[3] Although the proof as a whole is not by contradiction (it does not assume that only finitely many primes exist), a proof by contradiction is within it, which is that none of the initially considered primes can divide the number q above.

Page 6: Pythagorean theorem

Another proof, by the Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: that every integer has a unique prime factorization. If P is the set of all prime numbers, Euler wrote that:

The first equality is given by the formula for a geometric series in each term of the product. To show the second equality, distribute the product over the sum:

in the result, every product of primes appears exactly once and so by the fundamental theorem of arithmetic the sum is equal to the sum over all integers.

The sum on the right is the harmonic series, which diverges. Thus the product on the left must also diverge. Since each term of the product is finite, the number of terms must be infinite; therefore, there is an infinite number of primes.

Page 7: Pythagorean theorem

Paul Erdős gave a third proof that relies on the fundamental theorem of arithmetic. First note that every integer n can be uniquely written as

where r is square-free, or not divisible by any square numbers (let s2 be the largest square number that divides n and then let r=n/s2). Now suppose that there are only finitely many prime numbers and call the number of prime numbers k.

Fix a positive integer N and try to count the number of integers between 1 and N. Each of these numbers can be written as rs2 where r is square-free and r and s2 are both less than N. By the fundamental theorem of arithmetic, there are only 2k square-free numbers r (see Combination#Number of k-combinations for all k) as each of the prime numbers factorizes r at most once, and we must have s<√N. So the total number of integers less than N is at most 2k√N; i.e.:

Page 8: Pythagorean theorem

In the 1950s, Hillel Furstenberg introduced a proof using point-set topology. See Furstenberg's proof of the infinitude of primes.

Page 9: Pythagorean theorem

Juan Pablo Pinasco has written the following proof.[4]

Let p1, ..., pN be the smallest N primes. Then by the inclusion–exclusion principle, the number of positive integers less than or equal to x that are divisible by one of those primes is

Pinasco

Page 10: Pythagorean theorem
Page 11: Pythagorean theorem

EuclidGreek mathematician –

“Father of Geometry”Developed mathematical

proof techniques that we know today

Influenced by Plato’s enthusiasm for mathematics

On Plato’s Academy entryway: “Let no man ignorant of geometry enter here.”

Almost all Greek mathematicians following Euclid had some connection with his school in Alexandria

Page 12: Pythagorean theorem

Euclid’s ElementsWritten in Alexandria around 300 BCE13 books on mathematics and geometryAxiomatic: began with 23 definitions, 5

postulates, and 5 common notionsBuilt these into 465 propositionsOnly the Bible has been more scrutinized

over timeNearly all propositions have stood the test of

time

Page 13: Pythagorean theorem

Preliminaries: DefinitionsBasic foundations of Euclidean geometryEuclid defines points, lines, straight lines,

circles, perpendicularity, and parallelismLanguage is often not acceptable for modern

definitionsAvoided using algebra; used only geometryEuclid never uses degree measure for angles

Page 14: Pythagorean theorem

Preliminaries: PostulatesSelf-evident truths of

Euclid’s systemEuclid only needed fiveThings that can be done

with a straightedge and compass

Postulate 5 caused some controversy

Page 15: Pythagorean theorem

Preliminaries: Common NotionsNot specific to geometrySelf-evident truthsCommon Notion 4: “Things which coincide

with one another are equal to one another”To accept Euclid’s Propositions, you must be

satisfied with the preliminaries

Page 16: Pythagorean theorem

Early PropositionsAngles produced by

trianglesProposition I.20: any

two sides of a triangle are together greater than the remaining one

This shows there were some omissions in his work

However, none of his propositions are false

Construction of triangles (e.g. I.1)

Page 17: Pythagorean theorem

Early Propositions: CongruenceSASASAAASSSSThese hold without reference to the angles of

a triangle summing to two right angles (180˚)Do not use the parallel postulate

Page 18: Pythagorean theorem

Parallelism and related topicsParallel lines produce

equal alternate angles (I.29)

Angles of a triangle sum to two right angles (I.32)

Area of a triangle is half the area of a parallelogram with same base and height (I.41)

How to construct a square on a line segment (I.46)

Page 19: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofConsider a right triangleWant to show a2 + b2 = c2

Page 20: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofEuclid’s idea was to use areas of squares in

the proof. First he constructed squares with the sides of the triangle as bases.

Page 21: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofEuclid wanted to show that the areas of the

smaller squares equaled the area of the larger square.

Page 22: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofBy I.41, a triangle with the same base and

height as one of the smaller squares will have half the area of the square. We want to show that the two triangles together are half the area of the large square.

Page 23: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofWhen we shear the triangle like this, the area

does not change because it has the same base and height.

Euclid also made certain to prove that the line along which the triangle is sheared was straight; this was the only time Euclid actually made use of the fact that the triangle is right.

Page 24: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofNow we can rotate the triangle without

changing it. These two triangles are congruent by I.4 (SAS).

Page 25: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofWe can draw a perpendicular (from A to L on

handout) by I.31

Now the side of the large square is the base of the triangle, and the distance between the base and the red line is the height (because the two are parallel).

Page 26: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofJust like before, we can do another shear

without changing the area of the triangle.

This area is half the area of the rectangle formed by the side of the square and the red line (AL on handout)

Page 27: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofRepeat these steps for the triangle that is half

the area of the other small square.

Then the areas of the two triangles together are half the area of the large square, so the areas of the two smaller squares add up to the area of the large square.

Therefore a2 + b2 = c2 !!!!

Page 28: Pythagorean theorem

Pythagorean Theorem: Euclid’s proofEuclid also proved the converse of the

Pythagorean Theorem; that is if two of the sides squared equaled the remaining side squared, the triangle was right.

Interestingly, he used the theorem itself to prove its converse!

Page 29: Pythagorean theorem

Other proofs of the TheoremMathematician Proof

Chou-pei Suan-ching (China), 3rd c. BCE

Bhaskara (India), 12th c. BCE

James Garfield (U.S. president), 1881

Page 30: Pythagorean theorem

Further issuesControversy over parallel postulateNobody could successfully prove itNon-Euclidean geometry: Bolyai, Gauss, and

LobachevskiGeometry where the sum of angles of a

triangle is less than 180 degreesGives you the AAA congruence

Page 31: Pythagorean theorem