Euclid, fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the “Father of Geometry”. His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.
“Euclid” is the anglicized version of the Greek name meaning “Good Glory”.
Euclid’s Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. Euclid’s Elements has been referred to as the most successful and influential textbook ever written. It is one of the very earliest mathematical works to be printed.
Contents of the books
Books 1 through 4 deals with plane geometry:
* Book 1 contains Euclid’s 10 axioms (5 named postulates—including the parallel postulate—and 5 named axioms) and the basic propositions of geometry: the pons asinorum (proposition 5) , the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are “equal” (have the same area).
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* Book 2 is commonly called the “book of geometrical algebra,” because most of the propositions can be seen as geometric interpretations of algebraic identities, such as a(b + c + …) = ab + ac + … or (2a + b)2 + b2 = 2(a2 + (a + b)2).
* Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point, Thales’ theorem.
* Book 4 constructs the incircle and circumcircle of a triangle, and constructs regular polygons with 4, 5, 6, and 15 sides.
Books 5 through 10 introduce ratios and proportions:
* Book 5 is a treatise on proportions of magnitudes. Proposition 25 has as a special case the inequality of arithmetic and geometric means.
* Book 6 applies proportions to geometry: Similar figures.
* Book 7 deals strictly with elementary number theory: divisibility, prime numbers, Euclid’s algorithm for finding the greatest common divisor, least common multiple. Propositions 30 and 32 together are essentially equivalent to the fundamental theorem of arithmetic stating that every positive integer can be written as a product of primes in an essentially unique way, though Euclid would have had trouble stating it in this modern form as he did not use the product of more than 3 numbers.
* Book 8 deals with proportions in number theory and geometric sequences.
* Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers (proposition 20), the sum of a geometric series (proposition 35), and the construction of even perfect numbers (proposition 36).
* Book 10 attempts to classify incommensurable (in modern language, irrational) magnitudes by using the method of exhaustion, a precursor to integration.
Books 11 through to 13 deals with spatial geometry:
* Book 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of parallelepipeds.
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* Book 12 studies volumes of cones, pyramids, and cylinders in detail, and shows for example that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing the volume of a sphere is proportional to the cube of its radius by approximating it by a union of many pyramids.
* Book 13 constructs the five regular Platonic solids inscribed in a sphere, calculates the ratio of their edges to the radius of the sphere, and proves that there are no further regular solids.
FUNDAMENTAL THEOREM OF ARITHMETIC:
LHS: Every natural number is either prime or can be uniquely factored as a product of primes in a unique way.
In number theory, the fundamental theorem of arithmetic (or the unique-prime-factorization theorem) states that any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers. For example,
Are two examples of numbers satisfying the hypothesis of the theorem that can be written as the product of prime numbers. Proof of existence of a prime factorization is straightforward: proof of uniqueness is more challenging. Some proofs use the fact that if a prime number p divides the product of two natural numbers a and b, then p divides either a or b, a statement known as Euclid’s lemma. Since multiplication on the integers is both commutative and associative, it does not matter in what way we write a number greater than 1 as the product of primes; it is generally common to write the (prime) factors in the order of smallest to largest.
APPLICATION OF THEOREM:
The fundamental theorem of arithmetic establishes the importance of prime numbers. Prime numbers are the basic building blocks of any positive integer, in the sense that each positive integer can be constructed from the product of primes with one unique construction. Finding the prime factorization of an integer allows derivation of all its divisors, both prime and non-prime.
Once the prime factorizations of two numbers are known, their greatest common divisor and least common multiple can be found quickly. For instance, from the above it is shown that the greatest common divisor of 6936 and 1200 is 23 × 3 = 24. However, if the prime factorizations are not known, the use of the Euclidean algorithm generally requires much less calculation than factoring the two numbers.
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The fundamental theorem ensures that additive and multiplicative arithmetic functions are completely determined by their values on the powers of prime numbers.
EUCLID’S LEMMA:
In mathematics, Euclid’s lemma (Greek) is an important lemma regarding divisibility and prime numbers. In its simplest form, the lemma states that a prime number that divides a product of two integers must divide one of the two integers. This key fact requires a surprisingly sophisticated proof (using Bézout’s identity), and is a necessary step in the standard proof of the fundamental theorem of arithmetic.
Proof of uniqueness
The key step in proving uniqueness is Euclid’s proposition 30 of book 7 (known as Euclid’s lemma), which states that, for any prime number p and any natural numbers a, b: if p divides ab then p divides a or p divides b.
This may be proved as follows:
* Suppose that a prime p divides ab (where a, b are natural numbers) but does not divide a. We must prove that p divides b.
* Since p does not divide a, and p is prime, the greatest common divisor of p and a is 1.
* By Bézout’s identity, it follows that for some integers x, y (possibly negative),
* Multiplying both sides by b,
* Since p divides , p divides
* Since p divides both summands on the left, p divides b.
In mathematics, the Euclidean algorithm[a] (also called Euclid’s algorithm) is an efficient method for computing the greatest common divisor(GCD), also known as the greatest common factor (GCF) or highest common factor (HCF).
It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.
The GCD of two numbers is the largest number that divides both of them without leaving a remainder. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number. For example, 21 is the GCD of 252 and 105 (252 = 21 × 12; 105 = 21 × 5); since 252 − 105 = 147, the GCD of 147 and 105 is also 21. Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers until one of them is zero. When that occurs, the GCD is the remaining nonzero number. By reversing the steps in the Euclidean algorithm, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e.g., 21 = 5 × 105 + (−2) × 252. This important property is known as Bézout’s identity.
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Bézout’s identity
Bézout’s identity states that the greatest common divisor g of two integers a and b can be represented as a linear sum of the original two numbers a and b.[59] In other words, it is always possible to find integers s and t such that g = sa + tb
Principal ideals and related problems
Bézout’s identity provides yet another definition of the greatest common divisor g of two numbers a and b.
Extended Euclidean algorithm
The integers s and t of Bézout’s identity can be computed efficiently using the extended Euclidean algorithm. This extension adds two recursive equations to Euclid’s algorithm[64]
sk = sk−2 − qk−1sk−1
tk = tk−2 − qk−1tk−1
Matrix method
The integers s and t can also be found using an equivalent matrix method.[65] The sequence of equations of Euclid’s algorithm
a = q0 b + r0
b = q1 r0 + r1
Euclid’s lemma and unique factorization
Bézout’s identity is essential to many applications of Euclid’s algorithm, such as demonstrating the unique factorization of numbers into prime factors.
Linear Diophantine equations
Plot of a linear Diophantine equation, 9x + 12y = 483. The solutions are shown as blue circles.
Diophantine equations are equations in which the solutions are restricted to integers; they are named after the third-century Alexandrian mathematician Diophantus.[70] A typical linear Diophantine equation seeks integers x and y such that
ax + by = c
Where a, b and c are given integers
Chinese remainder theorem
Euclid’s algorithm can also be used to solve multiple linear Diophantine equations.[79] Such equations arise in the Chinese remainder theorem, which describes a novel method to represent an integer x.
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Factorization algorithms
Calculating a greatest common divisor is an essential step in several integer factorization algorithms, The Euclidean algorithm may be used to find this GCD efficiently. Continued fraction factorization uses continued fractions, which are determined using Euclid’s algorithm.[87].
