Euclidpeople.math.umass.edu/~tevelev/475_2018/Euclid_NT.pdf · 2018-02-26 · It is believed that Euclid, a Greek mathematician, was born around 365 B.C. ... Division of Figures,

Euclid

The Man Himself

● It is believed that Euclid, a Greek mathematician, was born around 365 B.C. in Alexandria, Egypt and lived until about 300 B.C.

● A Greek philosopher, Proclus, says that Euclid has taught at a school in Alexandria.

● Throughout his lifetime he has written many books, but most have not been found.

○ These books included material on conic sections, logical fallacies, and “porisms.”○ The works that we know about are The Elements, Data, Division of Figures, Phenomena, and

Optics.

● We only know about his unfound work because they are referred to in other ancient writers work, like Proclus’s.

● Studied mathematics for a sense of order, structure and the ideal form of reason

○ Actually had disdain for those who were seeking gain from the study of mathematics (whether it be practical application or financial reasons)

What do we know about him?

Historical Context

Cultural Context● Greeks emphasized mathematical rigor● Influenced by Egypt, moved even further after introduction of Babylonian

methods

Square Root

● The Greeks were in the midst of calculating the square root of 2● Many unsuccessful attempts in arithmetic led them to geometry

The Elements

Essentially a compilation of previous propositions from earlier mathematicians. Still, the book is widely popular for its organization of ideas.

- Includes the work of Pythagoras, Hippocrates, Theudius, Theaetetus, Eudoxus, and other Athenian mathematics

Euclid’s Approach:● Begin with intuitive approach and well crafted definitions● Do some problems that show how certain algorithms work

Structure of The Elements● Books I-VI -- Plane geometry● Books VII-IX -- Theory of Numbers● Book X -- Incommensurables● Book XI-XIII -- Solid Geometry

Overview of Number Theory

What is Number Theory?Number theory is the study of the set of natural numbers; more specifically the relationship between different types numbers.

Today our set of natural numbers starts with 1. However, in Euclid’s time, they did not consider “1” as a number. So, at that time the set of natural numbers began with the number 2.

These numbers can be assorted into many different types. Such as even, odd, prime, composite, square, etc..

Definitions for number theory of the time● A unit is that by virtue of which each of the things that exists is called one.● A number is a multitude made up of units.● A greater number is a multiple of the lesser when it is measured by the

lesser number.● A prime number is that which is measured by a unit alone.● Numbers prime to one another are those which are measured by a unit

alone as common measure. ● A composite number is that which is measured by some number.● A perfect number is that which is equal to its own parts.

Euclidean Algorithm

Euclid’s Elements Book X: Definition 1.11.1 Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

Two numbers are said to be commensurable if and only if they share a common divisor.

Two numbers are incommensurable if they do not share a common divisor.

Euclid’s Elements Book X: Proposition 33. To find the greatest common measure of two given commensurable magnitudes.

Corollary: If a magnitude measures two magnitudes, then it also measures their greatest common measure.

If two numbers share a common divisor that is not their greatest common divisor, then that divisor is also a divisor of the greatest common divisor.

Greatest Common DivisorHow do you find the greatest common divisor of two given integers?

Greatest common divisor of a and b is denoted by gcd(a,b)

List all the divisors a and b

Look for the greatest one they have in common

This requires a and b to be factorized

Can’t always be done efficiently

Euclid’s AlgorithmIf d divides a and d divides b, then d divides their sum

d must also divide their difference, a - b, where a is the larger of the two

Therefore, gcd(a,b) = gcd(a,a−b)

gcd(a,b) could be found with division and remainder

ExampleCompute gcd(27,33)

gcd(a,b) = gcd(a,a-b): 33 - 27 = 6 gcd(33,27)=gcd(27,6)

Divide the bigger one by the smaller one: 33=1×27+6 gcd(33,27)=gcd(27,6)

Subtraction with remainder: 27 - 6 = 21

21 - 6 = 15

15 - 6 = 9

9 - 6 = 3

Division with remainder: 27 = 4 x 6 + 3

gcd(27,6)=gcd(6,3)

Subtraction with Remainder: 6 - 3 = 3

3 - 3 = 0

Division with Remainder: 6 = 2 x 3 + 0

gcd(6,3) = 3

gcd(33,27) = gcd(27,6) = gcd(6,3) = 3

Since 6 is a perfect multiple of 3 gcd(33,27)=3

Extended Euclidean AlgorithmUsing the gcd of two numbers we can find integers m,n such that

3=33m+27n

Rearrange all the equations by isolating the remainder

6=33−1×27 3=27−4×6

Substitute the first equation into the last equation

3=27−4×(33−1×27)=(−4)×33+5×27)

m=−4,n=5

Practice ProblemsFind the gcd and linear combination

a. 270, 192

b. 210, 45

c. 180, 84

Prime Numbers

Euclid’s Elements Book VII: Definition 11

11. A prime number is that which is measured by a unit alone.

The Greeks did not think of 1 as a number. They took it as a unit that measures the other numbers. So, the natural numbers started with 2, rather than 1 like how we view it today.

Euclid believed that that unit, “1,” is the only proper divisor of a prime number.

For example, the number 5 is prime because it can be measured using the unit, 1, five times. It is not a composite number because it cannot be measured by some number.

Euclid’s Elements Book VII: Definition 1212. Numbers relatively prime as those which are measured by a unit alone as a common measure.

When Euclid says “which are measured by a unit alone as a common measure,” he specifically means that 1 (the unit) is the only proper divisor for all the numbers.

This does not mean the numbers that are being compared themselves are prime, but that in relation to each other they are.

For example, the numbers 6 and 35 are relatively prime, but neither of them are prime themselves. The numbers 3 and 17 are also relatively prime while alone they are both also prime.

Euclid’s Elements Book IX: Proposition 20

“Prime numbers are more than any assigned multitude of prime numbers.”

This is the theory for infinite many primes numbers.

In short, Euclid is saying that you could say there is a finite number of primes but that you will be able to find another prime number that is not one of the original.

For example, say there are n = 4 primes, 3, 5, 7, and 11. Euclid says that those are not the only primes. You would be able to find another, 13 (n+1). And this trend would continue infinitely.

Suppose you have n primes, p1, p2, …, pn. Let q be a new integer such that q = p1 * p2 * … * pn+1.

There are two cases you can consider:

Proof of Theory of Infinitely Many Prime Numbers

1. If q is a prime number, then you are done and you have indeed found a new prime.

2. If q is not a prime, then q must be divisible by a prime number r. But r cannot be any of the prime numbers p1, p2, …, pn because then there would leave a remainder of 1.

Thus, there are at least n + 1 primes.

A misconception of the product of primesSome think that if you have n primes, p1, p2, …, pn, then that their product + 1 would always yield a prime. However, this is not true. Even if you chose your primes to be the smallest primes it would still not hold true. As you can see:

2 * 3 + 1 = 7 prime

2 * 3 * 5 + 1 = 31 prime

2 * 3 * 5 * 7 + 1 = 211 prime

2 * 3 * 5 * 7 * 11 + 1 = 2311 prime

2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031 = 59 * 509 not prime

Fundamental Theorem of Arithmetic

Theorem 3.5.1 If n > 1 is an integer then it can be factored as a product of primes in exactly one way. In other words, in any two factorizations of into primes, every prime occurs the same number of times in each factorization.I.e. every number > 1 can be written:

p1n1p2

n2...pknk

Though he does not explicitly state the theorem in the Elements, the information needed to reach this conclusion is all stated

Example4312

= 2 · 2156 = 2 · 2 · 1078

= 2 · 2 · 2 · 539 = 2 · 2 · 2 · 7 · 77

= 2 · 2 · 2 · 7 · 7 · 11.

4312 = 23 ·72 ·11

Euclid’s Elements Book VII: Proposition 30“If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers.”

i.e. if a prime number, d, is a factor of a product of two numbers a * b, then d is also a factor of either a or b.

Example:

3 | 60 where 60 = 12 * 5

3∤5 but 3 | 12

Also known as Euclid’s Lemma (proof)

When proving, think about:

GCD, Linear Combination

Euclid’s Elements Book VII: Proposition 31“Any composite number is measured by some prime number.”

Euclid’s Elements Book VII: Proposition 32“Any number is either prime or is measured by some prime number.”

Essentially stating that numbers, prime or composite, can be broken into primes or is prime itself.

Euclid’s Elements Book IX: Proposition 14“If a number is the least that is measured by prime numbers, then it is not measured by any other prime number except those originally measuring it.”

Since prime numbers, by definition of a prime number, have no other factors that measure it.

Proof of FTA

Proof by contradiction using well-ordering principle

Works Cited● https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html● http://www.f.waseda.jp/sidoli/MI314_05_Euclid_02.pdf● https://www.ancient.eu/Euclid/● https://www.cs.nmsu.edu/historical-projects/Projects/Euclid2010.pdf● http://www.math.wichita.edu/history/Men/euclid.html● https://www.ancient.eu/article/606/greek-mathematics/● https://www.britannica.com/biography/Euclid-Greek-mathematician● https://www.math.uh.edu/~minru/spring11/fundamental-theorem.pdf● https://www.math.brown.edu/~jhs/frintch1ch6.pdf